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CHAPTER 1 INTRODUCTION AND

MATHEMATICAL CONCEPTS ANSWERS TO FOCUS ON CONCEPTS QUESTIONS 1.

(d) The resultant vector R is drawn from the tail of the first vector to the head of the last vector.

2.

(c) Note from the drawing that the magnitude R of the resultant vector R is equal to the shortest distance between the tail of A and the head of B. Thus, R is less than the magnitude (length) of A plus the magnitude of B.

3.

(a) The triangle in the drawing is a right triangle. The lengths A and B of the two sides are known, so the Pythagorean theorem can be used to determine the length R of the hypotenuse. 4.0 km (b) The angle is found by using the inverse tangent function, θ = tan −1 = 53° . 3.0 km (b) In this drawing the vector –C is reversed relative to C, while vectors A and B are not reversed.

4. 5. 6.

(c) In this drawing the vectors –B and –C are reversed relative to B and C, while vector A is not reversed.

7.

(e) These vectors form a closed four-sided polygon, with the head of the fourth vector exactly meeting the tail of the first vector. Thus, the resultant vector is zero.

8.

(c) When the two vector components Ax and Ay are added by the tail-to-head method, the sum equals the vector A. Therefore, these vector components are the correct ones.

9.

(b) The three vectors form a right triangle, so the magnitude of A is given by the 2

Pythagorean theorem as A = A doubles:

(2 A ) + (2 A ) 2

x

2

Ax + Ay . If Ax and Ay double in size, then the magnitude of y

2

=

2

2

2

2

4 Ax + 4 Ay = 2 Ax + Ay = 2 A. −1

Ay . If Ax and A x

10. (a) The angle θ is determined by the inverse tangent function, θ = tan

Ay both become twice as large, the ratio does not change, and θ remains the same.

11. (b) The displacement vector A points in the –y direction. Therefore, it has no scalar component along the x axis (Ax = 0 m) and its scalar component along the y axis is negative.

2

INTRODUCTION AND MATHEMATICAL CONCEPTS

12. (e) The scalar components are given by Ax′ = −(450 m) sin 35.0° = −258 m and Ay′ = −(450 m) cos 35.0° = −369 m.

13. (d) The distance (magnitude) traveled by each runner is the same, but the directions are different. Therefore, the two displacement vectors are not equal. 14. (c) Ax and Bx point in opposite directions, and Ay and By point in the same direction. 15. (d) 16. Ay = 3.4 m, By = 3.4 m 17. Rx = 0 m, Ry = 6.8 m 18. R = 7.9 m, θ = 21 degrees

Chapter 1 Problems

3

CHAPTER 1 INTRODUCTION AND

MATHEMATICAL CONCEPTS PROBLEMS ____________________________________________________________________________________________

1.

REASONING We use the fact that 1 m = 3.28 ft to form the following conversion factor: (1 m)/(3.28 ft) = 1. 2

2

SOLUTION To convert ft into m , we apply the conversion factor twice:

(

)

1 m 1 m 2 Area = 1330 ft 2 3.28 ft = 124 m 3.28 ft _____________________________________________________________________________ 2.

REASONING The word “per” indicates a ratio, so “0.35 mm per day” means 0.35 mm/d, which is to be expressed as a rate in ft/century. These units differ from the given units in both length and time dimensions, so both must be converted. For length, 1 m = 103 mm, and 1 ft = 0.3048 m. For time, 1 year = 365.24 days, and 1 century = 100 years. Multiplying the resulting growth rate by one century gives an estimate of the total length of hair a long-lived adult could grow over his lifetime. SOLUTION Multiply the given growth rate by the length and time conversion factors, making sure units cancel properly:

mm 1 m 1 ft 365.24 d 100 y Growth rate = 0.35 = 42 ft/century 3 century d 10 mm 0.3048 m 1 y ______________________________________________________________________________ 3.

SSM REASONING We use the facts that 1 mi = 5280 ft, 1 m = 3.281 ft, and 1 yd = 3 ft. With these facts we construct three conversion factors: (5280 ft)/(1 mi) = 1, (1 m)/(3.281 ft) = 1, and (3 ft)/(1 yd) = 1. SOLUTION By multiplying by the given distance d of the fall by the appropriate conversion factors we find that 5280 ft d = 6 mi 1 mi

(

)

1 m 3.281 ft

3 ft + 551 yd 1 yd

(

)

1 m = 10 159 m 3.281 ft

4

4.

INTRODUCTION AND MATHEMATICAL CONCEPTS

REASONING a. To convert the speed from miles per hour (mi/h) to kilometers per hour (km/h), we need to convert miles to kilometers. This conversion is achieved by using the relation 1.609 km = 1 mi (see the page facing the inside of the front cover of the text). b. To convert the speed from miles per hour (mi/h) to meters per second (m/s), we must convert miles to meters and hours to seconds. This is accomplished by using the conversions 1 mi = 1609 m and 1 h = 3600 s. SOLUTION a. Multiplying the speed of 34.0 mi/h by a factor of unity, (1.609 km)/(1 mi) = 1, we find the speed of the bicyclists is mi mi 1.609 km km Speed = 34.0 (1) = 34.0 = 54.7 h 1 mi h h b. Multiplying the speed of 34.0 mi/h by two factors of unity, (1609 m)/(1 mi) = 1 and (1 h)/(3600 s) = 1, the speed of the bicyclists is

mi mi 1609 m 1 h m Speed = 34.0 (1)(1) = 34.0 = 15.2 h s h 1 mi 3600s ______________________________________________________________________________ 5.

REASONING In order to calculate d, the units of a and b must be, respectively, cubed and squared along with their numerical values, then combined algebraically with each other and the units of c. Ignoring the values and working first with the units alone, we have

( m ) = m 3 2 = m2 a3 → cb2 ( m/s )( s )2 m / s ⋅ s 2 1 s 3

d=

(

)

Therefore, the units of d are m2/s.

SOLUTION With the units known, the numerical value may be calculated: d=

6.

( 9.7 )3 ( 69 )( 4.2 )2

m2 /s = 0.75 m2 /s

REASONING AND SOLUTION x has the dimensions of [L], v has the dimensions of [L]/[T], and a has the dimensions of [L]/[T]2. The equation under consideration is vn = 2ax.

Chapter 1 Problems

L

The dimensions of the right hand side are

T

2

L =

L

2

T

2

5

, while the dimensions of the left

[ L] [ L] . The right side will equal the left side only when n = 2 . hand side are = [T ] [T ]n n

n

____________________________________________________________________________________________

7.

SSM REASONING This problem involves using unit conversions to determine the number of magnums in one jeroboam. The necessary relationships are 1.0 magnum = 1.5 liters 1.0 jeroboam = 0.792 U. S. gallons 1.00 U. S. gallon = 3.785 × 10 –3 m 3 = 3.785 liters

These relationships may be used to construct the appropriate conversion factors.

SOLUTION By multiplying one jeroboam by the appropriate conversion factors we can determine the number of magnums in a jeroboam as shown below:

(1.0

jeroboam

)

0.792 gallons 1.0 jeroboam

3.785 liters 1.0 gallon

1.0 magnum = 2.0 magnums 1.5 liters

____________________________________________________________________________________________

8.

REASONING In the expression for the volume flow rate, the dimensions on the left side of the equals sign are [L]3/[T]. If the expression is to be valid, the dimensions on the right side of the equals sign must also be [L]3/[T]. Thus, the dimensions for the various symbols on the right must combine algebraically to yield [L]3/[T]. We will substitute the dimensions for each symbol in the expression and treat the dimensions of [M], [L], and [T] as algebraic variables, solving the resulting equation for the value of the exponent n. SOLUTION We begin by noting that the symbol π and the number 8 have no dimensions. It follows, then, that

Q=

π R n ( P2 − P1 ) 8η L

[ L]3 = [ L]n [T ] [ L] [T ]

or

[ L] [T]

or

=

n L] [ [ L] = [ L] 3

[M] [ L][ T ]2 = [ L]n [ T ] = [ L]n 2 [ L][T ] [M] L ][ T ] [ [ L] [ L] [T]

[ L]n

3

or

[ L]3 [ L] = [ L]4 = [ L]n

6

INTRODUCTION AND MATHEMATICAL CONCEPTS

Thus, we find that n = 4 .

9. REASONING AND SOLUTION a. F = [M][L]/[T]2; ma = [M][L]/[T]2 = [M][L]/[T]2 so F = ma is dimensionally correct . b. x = [L]; at3 = ([L]/[T]2)[T]3 = [L][T] so x = (1/2)at3 is not dimensionally correct . c. E = [M][L]2/[T]2; mv = [M][L]/[T] so E = (1/2)mv is not dimensionally correct . d. E = [M][L]2/[T]2; max = [M]([L]/[T]2)[L] = [M][L]2/[T]2 so E = max is dimensionally correct . e. v = [L]/[T]; (Fx/m)1/2 = {([M][L]/[T]2)([L]/[M])}1/2 = {[L]2/[T]2}1/2 = [L]/[T] so v = (Fx/m)1/2 is dimensionally correct . ____________________________________________________________________________________________

10. REASONING To convert from gallons to cubic meters, use the equivalence 1 U.S. gal = 3.785×10−3 m3. To find the thickness of the painted layer, we use the fact that the paint’s volume is the same, whether in the can or painted on the wall. The layer of paint on the wall can be thought of as a very thin “box” with a volume given by the product of the surface area (the “box top”) and the thickness of the layer. Therefore, its thickness is the ratio of the volume to the painted surface area: Thickness = Volume/Area. That is, the larger the area it’s spread over, the thinner the layer of paint. SOLUTION a. The conversion is

( 0.67

3.785 × 10−3 m3 U.S. gallons = 2.5 × 10−3 m3 U.S. gallons

)

b. The thickness is the volume found in (a) divided by the area,

Thickness =

Volume 2.5 × 10−3 m3 = = 1.9 × 10−4 m 2 Area 13 m

7

Chapter 1 Problems

11.

SSM REASONING The dimension of the spring constant k can be determined by first solving the equation T = 2π m / k for k in terms of the time T and the mass m. Then, the dimensions of T and m can be substituted into this expression to yield the dimension of k. SOLUTION Algebraically solving the expression above for k gives k = 4 π 2 m / T 2 . The term 4π 2 is a numerical factor that does not have a dimension, so it can be ignored in this analysis. Since the dimension for mass is [M] and that for time is [T], the dimension of k is Dimension of k =

[M] [ T ]2

____________________________________________________________________________________________

12. REASONING The shortest distance between the tree and the termite mound is equal to the magnitude of the chimpanzee's displacement r.

SOLUTION a. From the Pythagorean theorem, we have r=

51 m

θ

(51 m) 2 + (39 m) 2 = 64 m

b. The angle θ is given by

θ = tan −1

39 m I F G H51 m JK=

39 m

r

37 ° south of east

_____________________________________________________________________________________________

13. SSM WWW REASONING The shortest distance between the two towns is along the line that joins them. This distance, h, is the hypotenuse of a right triangle whose other sides are ho = 35.0 km and ha = 72.0 km, as shown in the figure below. SOLUTION The angle θ is given by tan θ = ho / ha so that −1 35.0 km

θ = tan = 25.9° S of W 72.0 km

W

θ h

h

θ

We can then use the Pythagorean theorem to find h.

h a S

h = ho2 + ha2 = (35.0 km) 2 + ( 72 .0 km) 2 = 80.1 km

o

8

INTRODUCTION AND MATHEMATICAL CONCEPTS

14. REASONING When the monkey has climbed as far up the pole as it can, its leash is taut, making a straight line from the stake to the monkey, that is, L = 3.40 m long. The leash is the hypotenuse of a right triangle, and the other sides are a line drawn from the stake to the base of the pole (d = 3.00 m), and a line from the base of the pole to the monkey (height = h).

Stake L h d

SOLUTION These three lengths are related by the Pythagorean theorem (Equation 1.7):

15.

h 2 + d 2 = L2

or

h 2 = L2 − d 2

h = L2 − d 2 =

( 3.40 m )2 − ( 3.00 m )2

= 1.6 m

REASONING Using the Pythagorean theorem (Equation 1.7), we find that the relation between the length D of the diagonal of the square (which is also the diameter of the circle) and the length L of one side of the square is D = L2 + L2 = 2 L . SOLUTION Using the above relation, we have

D 0.35 m = = 0.25 m 2 2 ______________________________________________________________________________ D = 2L

or

L=

16. REASONING In both parts of the drawing the line of sight, the horizontal dashed line, and the vertical form a right triangle. The angles θa = 35.0° and θb = 38.0° at which the person’s line of sight rises above the horizontal are known, as is the horizontal distance d = 85.0 m from the building. The unknown vertical sides of the right triangles correspond, respectively, to the heights Ha and Hb of the bottom and top of the antenna relative to the person’s eyes. The antenna’s height H is the difference between Hb and Ha: H = H b − H a . The horizontal side d of the triangle is adjacent to the angles θa and θb, while the vertical

sides Ha and Hb are opposite these angles. Thus, in either triangle, the angle θ is related to

h the horizontal and vertical sides by Equation 1.3 tan θ = o : ha H tan θa = a d tan θ b =

Hb d

(1)

(2)

Chapter 1 Problems

9

H Hb

Ha

θa

θb d

d

(a)

(b)

SOLUTION Solving Equations (1) and (2) for the heights of the bottom and top of the antenna relative to the person’s eyes, we find that H a = d tan θa

H b = d tan θ b

and

The height of the antenna is the difference between these two values:

H = H b − H a = d tan θb − d tan θa = d ( tan θb − tan θa )

(

)

H = (85.0 m ) tan 38.0 − tan 35.0 = 6.9 m

17. REASONING AND SOLUTION Consider the following views of the cube.

Bottom View

Side View

Na

L

c a

Na

a

C

a

Na

C

L

Na

The length, L, of the diagonal of the bottom face of the cube can be found using the Pythagorean theorem to be 2

2

2

2

2

L = a + a = 2(0.281 nm) = 0.158 nm

or

L = 0.397 nm

10

INTRODUCTION AND MATHEMATICAL CONCEPTS

The required distance c is also found using the Pythagorean theorem. 2

2

2

2

2

2

c = L + a = (0.397 nm) + (0.281 nm) = 0.237 nm Then, c = 0.487 nm

18. REASONING The drawing shows the heights of the two balloonists and the horizontal distance x between them. Also shown in dashed lines is a right triangle, one angle of which is 13.3°. Note that the side adjacent to the 13.3° angle is the horizontal distance x, while the side opposite the angle is the distance between the two heights, 61.0 m − 48.2 m. Since we know the angle and the length of one side of the right triangle, we can use trigonometry to find the length of the other side.

13.3°

x

61.0 m

48.2 m

SOLUTION The definition of the tangent function, Equation 1.3, can be used to find the horizontal distance x, since the angle and the length of the opposite side are known: tan13.3° =

length of opposite side length of adjacent side (= x )

Solving for x gives

x=

length of opposite side 61.0 m − 48.2 m = = 54.1 m tan13.3° tan13.3°

____________________________________________________________________________________________

19. REASONING Note from the drawing that the shaded right triangle contains the angle θ , the side opposite the angle (length = 0.281 nm), and the side adjacent to the angle (length = L). If the length L can be determined, we can use trigonometry to find θ. The bottom face of the cube is a square whose diagonal has a length L. This length can be found from the Pythagorean theorem, since the lengths of the two sides of the square are known.

0.281 nm

θ 0.281 nm

0.281 nm L

SOLUTION The angle can be obtained from the inverse tangent function, Equation 1.6, as θ = tan −1 ( 0.281 nm ) / L . Since L is the length of the hypotenuse of a right triangle whose

sides have lengths of 0.281 nm, its value can be determined from the Pythagorean theorem: L=

( 0.281 nm )2 + ( 0.281 nm )2

= 0.397 nm

Chapter 1 Problems

11

Thus, the angle is

0.281 nm −1 0.281 nm = tan = 35.3° L 0.397 nm ______________________________________________________________________________

θ = tan −1

20. REASONING There are two right triangles in the drawing. Each H2 contains the common side that is 21° 52° shown as a dashed line and is D labeled D, which is the distance between the buildings. The H1 hypotenuse of each triangle is one of the lines of sight to the top and base of the taller building. The remaining (vertical) sides of the triangles are labeled H1 and H2. Since the height of the taller building is H1 + H2 and the height of the shorter building is H1, the ratio that we seek is (H1 + H2)/H1. We will use the tangent function to express H1 in terms of the 52° angle and to express H2 in terms of the 21° angle. The unknown distance D will be eliminated algebraically when the ratio (H1 + H2)/H1 is calculated. SOLUTION The ratio of the building heights is

H + H2 Height of taller building = 1 H1 Height of shorter building Using the tangent function, we have that tan 52° =

H1 D

tan 21° =

H2 D

or

H1 = D tan 52°

or

H 2 = D tan 21°

Substituting these results into the expression for the ratio of the heights gives H + H 2 D tan 52° + D tan 21° Height of taller building = 1 = H1 D tan 52° Height of shorter building = 1+

tan 21° = 1.30 tan 52°

12

INTRODUCTION AND MATHEMATICAL CONCEPTS

If the taller building were half again as tall as the shorter building, this ratio would have been 1.50. Therefore, your friend is wrong .

21. SSM REASONING The drawing at the right shows the location of each deer A, B, and C. From the problem statement it follows that

C

c B

b = 62 m

β

α

a

γ

b

77°

51°

c = 95 m

A

γ = 180° − 51° − 77° = 52° Applying the law of cosines (given in Appendix E) to the geometry in the figure, we have 2

2

2

a − 2ab cos γ + (b − c ) = 0

which is an expression that is quadratic in a. It can be simplified to Aa 2 + Ba + C = 0 , with A=1 B = –2b cos γ = – 2(62 m) cos 52° = –76 m 2

2

2

2

2

C = (b − c ) = (62 m) − (95 m) = –5181 m

This quadratic equation can be solved for the desired quantity a.

SOLUTION Suppressing units, we obtain from the quadratic formula a=

− ( −76 ) ± ( −76 ) 2 − 4 (1)(–5181) = 1.2 × 10 2 m and – 43 m 2 (1)

Discarding the negative root, which has no physical significance, we conclude that the 2

distance between deer A and C is 1.2 × 10 m .

Chapter 1 Problems

22. REASONING The trapeze cord is L = 8.0 m long, so the trapeze is initially h1 = L cos 41° meters below the support. At the instant he releases the trapeze, it is h2 = L cos θ meters below the support. The difference in heights is d = h2 – h1 =

13

θ 41° h1

L

h2 L

41°

θ

0.75 m. Given that the trapeze is released at a lower elevation than the platform, we expect to find θ < 41°.

0.75 m

SOLUTION Putting the above relationships together, we have

d = h2 − h1 = L cos θ − L cos 41 cos θ =

or

d + L cos 41 = L cos θ

d + cos 41 L d 0.75 m + cos 41 = cos−1 + cos 41 = 32 L 8.0 m

θ = cos−1

23. SSM REASONING AND SOLUTION A single rope must supply the resultant of the two forces. Since the forces are perpendicular, the magnitude of the resultant can be found from the Pythagorean theorem. a. Applying the Pythagorean theorem, 475 N

F = ( 475 N ) + (315 N) = 5.70 × 10 N 2

2

2

b. The angle θ that the resultant makes with the westward direction is

θ = tan −1

θ

W

315 N I F G H475 N JK= 33.6°

F

315 N S

Thus, the rope must make an angle of 33.6° south of west . _____________________________________________________________________________________________

14

INTRODUCTION AND MATHEMATICAL CONCEPTS

24. REASONING The Pythagorean theorem (Equation 1.7) can be used to find the magnitude of the resultant vector, and trigonometry can be employed to determine its direction. a. Arranging the vectors in tail-to-head fashion, we can see that the vector A gives the resultant a westerly direction and vector B gives the resultant a southerly direction. Therefore, the resultant A + B points south of west. b. Arranging the vectors in tail-to-head fashion, we can see that the vector A gives the resultant a westerly direction and vector –B gives the resultant a northerly direction. Therefore, the resultant A + (–B) points north of west. SOLUTION Using the Pythagorean theorem and trigonometry, we obtain the following results:

a.

Magnitude of A + B =

( 63 units )2 + ( 63 units )2

= 89 units

63 units = 45° south of west 63 units

θ = tan −1

b.

Magnitude of A − B =

( 63 units )2 + ( 63 units )2

= 89 units

63 units = 45° north of west 63 units ______________________________________________________________________________

θ = tan −1

25. SSM WWW REASONING a. Since the two force vectors A and B have directions due west and due north, they are perpendicular. Therefore, the resultant vector F = A + B has a magnitude given by the 2 2 2 Pythagorean theorem: F = A + B . Knowing the magnitudes of A and B, we can calculate the magnitude of F. The direction of the resultant can be obtained using trigonometry. b. For the vector F′ = A – B we note that the subtraction can be regarded as an addition in the following sense: F′ = A + (–B). The vector –B points due south, opposite the vector B, so the two vectors are once again perpendicular and the magnitude of F′ again is given by the Pythagorean theorem. The direction again can be obtained using trigonometry.

Chapter 1 Problems

SOLUTION a. The drawing shows the two vectors and the resultant vector. According to the Pythagorean theorem, we have F 2 = A2 + B 2

F=

A2 + B 2

F=

b445 N g+ b325 N g

North

North F B

θ

θ

F′′

A 2

15

–B A

2

(b)

(a)

= 551 N Using trigonometry, we can see that the direction of the resultant is tan θ =

B A

or

θ = tan −1

325 N I F G H445 N JK=

36.1° north of west

b. Referring to the drawing and following the same procedure as in part a, we find

bg

F ′2 = A2 + − B

2

tan θ =

F′ =

or B A

or

b g = b445 N g+ b−325 N g = 325 N I F G H445 N JK= 36.1° south of west

A2 + − B

θ = tan −1

2

2

2

551 N

_____________________________________________________________________________________________

26.

REASONING The triple jump consists of a double jump in one direction, followed by a perpendicular single jump, which we can represent with displacement vectors J and K (see the drawing). K These two perpendicular vectors form a right triangle with their resultant D = J + K, which is the displacement of the colored D checker. In order to find the magnitude D of the displacement, we first need to find the magnitudes J and K of the double jump and the single jump. As the three sides of a right triangle, J, K, and D (the hypotenuse) are related to one another by the Pythagorean d s J theorem (Equation 1.7) The double jump moves the colored s checker a straight-line distance equal to the length of four square’s diagonals d, and the single jump moves a length equal to two square’s diagonals. Therefore, J = 4d and K = 2d (1)

Let the length of a square’s side be s. Any two adjacent sides of a square form a right triangle with the square’s diagonal (see the drawing). The Pythagorean theorem gives the diagonal length d in terms of the side length s:

16

INTRODUCTION AND MATHEMATICAL CONCEPTS

d = s2 + s2 = 2s2 = s 2

(2)

SOLUTION First, we apply the Pythagorean theorem to the right triangle formed by the three displacement vectors, using Equations (1) for J and K: D=

( 4d )2 + ( 2d )2

J 2 + K2 =

= 16d 2 + 4d 2 = 20d 2 = d 20

(3)

Substituting Equation (2) into Equation (3) gives

(

D = d 20 = s 2

)

20 = s 40 = ( 4.0 cm ) 40 = 25 cm

27. REASONING For convenience, we can assign due east to be the positive direction and due west to be the negative direction. Since all the vectors point along the same east-west line, the vectors can be added just like the usual algebraic addition of positive and negative scalars. We will carry out the addition for all of the possible choices for the two vectors and identify the resultants with the smallest and largest magnitudes. SOLUTION There are six possible choices for the two vectors, leading to the following resultant vectors:

F1 + F2 = 50.0 newtons + 10.0 newtons = +60.0 newtons = 60.0 newtons, due east F1 + F3 = 50.0 newtons − 40.0 newtons = +10.0 newtons = 10.0 newtons, due east F1 + F4 = 50.0 newtons − 30.0 newtons = +20.0 newtons = 20.0 newtons, due east F2 + F3 = 10.0 newtons − 40.0 newtons = −30.0 newtons = 30.0 newtons, due west F2 + F4 = 10.0 newtons − 30.0 newtons = −20.0 newtons = 20.0 newtons, due west F3 + F4 = −40.0 newtons − 30.0 newtons = −70.0 newtons = 70.0 newtons, due west The resultant vector with the smallest magnitude is F1 + F3 = 10.0 newtons, due east .

The resultant vector with the largest magnitude is F3 + F4 = 70.0 newtons, due west .

Chapter 1 Problems

28. REASONING Both P and Q and the vector sums K and M can be drawn with correct magnitudes and directions by counting grid squares. To add vectors, place them tail-to-head and draw the resultant vector from the tail of the first vector to the head of the last. The vector 2P is equivalent to P + P, and −Q is a vector that has the same magnitude as Q, except it is directed in the opposite direction.

17

P Q M

−Q

8.00 cm

K

P The vector M runs 11 squares P horizontally and 3 squares vertically, and the vector K runs 4 squares horizontally and 9 squares vertically. These distances can be converted from grid squares to centimeters with the grid scale: 1 square = 4.00 cm. Once the distances are calculated in centimeters, the Pythagorean theorem (Equation 1.7) will give the magnitudes of the vectors. SOLUTION a. The vector M = P + Q runs 11 squares horizontally and 3 squares vertically, and these cm distances are equivalent to, respectively, 4.00 (11 squares ) = 44.0 cm and square cm 4.00 ( 3 squares ) = 12.0 cm . Thus, the magnitude of M is square M =

( 44.0 cm )2 + (12.0 cm )2

= 45.6 cm

b. Similarly, the lengths of the horizontal and vertical distances of K = 2P − Q are 4 horizontal squares and 9 vertical squares, or 16.0 cm and 36.0 cm, respectively. The magnitude of K is then K=

(16.0 cm )2 + ( 36.0 cm )2

= 39.4 cm

18

INTRODUCTION AND MATHEMATICAL CONCEPTS

29. REASONING At the turning point, the distance to the campground is labeled d in the drawing. Note that d is the length of the hypotenuse of a right triangle. Since we know the lengths of the other two sides of the triangle, the Pythagorean theorem can be used to find d. The direction that cyclist #2 must head during the last part of the trip is given by the angle θ. It can be determined by using the inverse tangent function.

Turning point

N

θ θ

d Campground

1950 m 1430 m Start

SOLUTION E W 1080 m a. The two sides of the triangle have lengths of 1080 m S and 520 m (1950 m − 1430 m = 520 m). The length d of the hypotenuse can be determined from the Pythagorean theorem, Equation (1.7), as d=

(1080 m )2 + ( 520 m )2

= 1200 m

b. Since the lengths of the sides opposite and adjacent to the angle θ are known, the inverse tangent function (Equation 1.6) can be used to find θ. 520 m = 26° south of east 1080 m ______________________________________________________________________________

θ = tan −1

30. REASONING a. Since the two displacement vectors A and B have directions due south and due east, they are perpendicular. Therefore, the resultant vector R = A + B has a 2

2

2

magnitude given by the Pythagorean theorem: R = A + B . Knowing the magnitudes of R and A, we can calculate the magnitude of B. The direction of the resultant can be obtained using trigonometry. b. For the vector R′ = A – B we note that the subtraction can be regarded as an addition in the following sense: R′ = A + (–B). The vector –B points due west, opposite the vector B, so the two vectors are once again perpendicular and the magnitude of R′ again is given by the Pythagorean theorem. The direction again can be obtained using trigonometry.

SOLUTION a. The drawing shows the two vectors and the resultant vector. According to the Pythagorean theorem, we have R 2 = A2 + B 2 B=

( 3.75 km )

or 2

B = R 2 − A2

− ( 2.50 km ) = 2.8 km 2

North

North

A

θ

R

R′′

B

–B

(a)

(b)

θ

A

Chapter 1 Problems

19

Using trigonometry, we can see that the direction of the resultant is tan θ =

B A

or

θ = tan −1

2.8 km I F G H2.50 km JK=

48 ° east of south

b. Referring to the drawing and following the same procedure as in part a, we find

bg

R′ 2 = A2 + − B

2

tan θ =

B=

or B A

or

b3.75 km g− b2.50 kmg = 2.8 km I F G H2.50 km JK= 48° west of south 2

R′ 2 − A2 =

θ = tan −1

2

2 .8 km

_____________________________________________________________________________________________

31.

SSM WWW REASONING AND SOLUTION The single force needed to produce the same effect is equal to the resultant of the forces provided by the two ropes. The figure below shows the force vectors drawn to scale and arranged tail to head. The magnitude and direction of the resultant can be found by direct measurement using the scale factor shown in the figure. 2900 N

2900 N 1000 N Scale: Resultant

a. From the figure, the magnitude of the resultant is 5600 N . b. The single rope should be directed along the dashed line in the text drawing. ____________________________________________________________________________________________

32. REASONING a. and b. The drawing shows the two vectors A and B, as well as the resultant vector A + B. The three vectors form a right triangle, of which two of the sides are known. We can employ the Pythagorean theorem, Equation 1.7, to find the length of the third side. The angle θ in the drawing can be determined by using the inverse cosine function, Equation 1.5, since the side adjacent to θ and the length of the hypotenuse are known. N B 15.0 units A+B W

A

θ

E

12.3 units S

20

INTRODUCTION AND MATHEMATICAL CONCEPTS

c. and d. The drawing illustrates the two vectors A and −B, as well as the resultant vector A − B. The three vectors form a right triangle, which is identical to the one above, except for the orientation. Therefore, the lengths of the hypotenuses and the angles are equal. N 12.3 units

W A

A−B

E

θ 15.0 units

−B

S

SOLUTION a. Let R = A + B. The Pythagorean theorem (Equation 1.7) states that the square of the 2 2 2 hypotenuse is equal to the sum of the squares of the sides, so that R = A + B . Solving for B yields 2

(15.0 units )2 − (12.3 units )2

2

B= R −A =

= 8.6 units

b. The angle θ can be found from the inverse cosine function, Equation 1.5: 12.3 units = 34.9° north of west 15.0 units

θ = cos −1

c. Except for orientation, the triangles in the two drawings are the same. Thus, the value for B is the same as that determined in part (a) above: B = 8.6 units d. The angle θ is the same as that found in part (a), except the resultant vector points south of west, rather than north of west: θ = 34.9° south of west ______________________________________________________________________________

33. REASONING AND SOLUTION The figures below are scale diagrams of the forces drawn tail-to-head. The scale factor is shown in the figure. a. From the figure on the left, we see that FA − FB = 142 N, θ = 67° south of east . b. Similarly, from the figure on the right, FB − FA = 142 N, θ = 67° north of west .

Chapter 1 Problems

–FA = 90.0 N

FA = 90.0 N

θ

21

75°

–FB = 135 N FA – FB

FB = 135 N FB – FA

Scale Factor:

20.0 N

θ

75°

_____________________________________________________________________________________________

34.

REASONING The magnitude of the x-component of the force vector is the product of the magnitude of the force times the cosine of the angle between the vector and the x axis. Since the x-component points in the +x direction, it is positive. Likewise, the magnitude of the y-component of the force vector is the product of the magnitude of the force times the sine of the angle between the vector and the x axis. Since the vector points 36.0º below the positive x axis, the y component of the vector points in the −y direction; thus, a minus sign must be assigned to the y-component to indicate this direction. SOLUTION The x and y scalar components are a.

Fx = (575 newtons) cos 36.0° = 465 newtons

b.

Fy = –(575 newtons) sin 36.0° = –338 newtons

22

INTRODUCTION AND MATHEMATICAL CONCEPTS

35. SSM REASONING The ostrich's velocity vector v and the desired components are shown in the figure at the right. The components of the velocity in the directions due west and due north are v W and v N , respectively. The sine and cosine functions can be used to find the components.

v vN 68.0° vW

SOLUTION a. According to the definition of the sine function, we have for the vectors in the figure sin θ =

vN v

or

vN = v sin θ = (17.0 m/s) sin 68° = 15.8 m/s

or

vW = v cos θ = (17.0 m/s) cos 68.0° = 6.37 m/s

b. Similarly, cos θ =

vW v

_____________________________________________________________________________________________

36. REASONING The triangle in the drawing is a right triangle. We know one of its angles is 30.0°, and the length of the hypotenuse is 8.6 m. Therefore, the sine and cosine functions can be used to find the magnitudes of Ax and Ay. The directions of these vectors can be found by examining the diagram. SOLUTION a. The magnitude Ax of the displacement vector Ax is related to the length of the hypotenuse and the 30.0° angle by the sine function (Equation 1.1). The drawing shows that the direction of Ax is due east. Ax = A sin 30.0° = ( 8.6 m ) sin 30.0° = 4.3 m, due east

N E S 30.0°

8.6 m

Ay

b. In a similar manner, the magnitude Ay of Ay can be found by using the cosine function (Equation 1.2). Its direction is due south.

Ay = A cos 30.0° = ( 8.6 m ) cos 30.0° = 7.4 m, due south

W

#1

A 90° #2

Net

Ax

______________________________________________________________________________ 37. REASONING AND SOLUTION The first three rows of the following table give the components of each of the three individual displacements. The fourth row gives the components of the resultant displacement. The directions due east and due north have been taken as the positive directions.

Chapter 1 Problems

Displacement

East/West Component

North/South Component

A

–52 paces

0

B C

–(42 paces) cos 30.0° = –36 paces 0

(42 paces) sin 30.0° = 21 paces 25 paces

R=A+B+C

–88 paces

46 paces

23

a. Therefore, the magnitude of the displacement in the direction due north is 46 paces . b. Similarly, the magnitude of the displacement in the direction due west is 88 paces . _____________________________________________________________________________________________

38. REASONING AND SOLUTION a. From the Pythagorean theorem, we have F=

(150 N) 2 + (130 N) 2 = 2.0 × 10 2 N

130 N I F G H150 N JK=

130 N

θ

b. The angle θ is given by

θ = tan −1

F

41°

150 N

_____________________________________________________________________________________________

39. SSM REASONING The x and y components of r are mutually perpendicular; therefore, the magnitude of r can be found using the Pythagorean theorem. The direction of r can be found using the definition of the tangent function. SOLUTION According to the Pythagorean theorem, we have r=

x

x 2 + y 2 = ( −125 m ) 2 + ( −184 m) 2 = 222 m

θ y

The angle θ is

r

184 m = 55.8° 125 m

θ = tan −1

____________________________________________________________________________________________

24

INTRODUCTION AND MATHEMATICAL CONCEPTS

40. REASONING Using trigonometry, we can determine the angle θ from the relation tan θ = Ay /Ax: SOLUTION Ay −1 12 m = tan = 45 12 m A x A 12 m θ = tan −1 y = tan −1 = 35 17 m Ax

θ = tan −1

a. b.

Ay −1 17 m = tan = 55 12 m Ax ______________________________________________________________________________

θ = tan −1

c.

41. REASONING Two vectors that are equal must have the same magnitude and direction. Equivalently, they must have identical x components and identical y components. We will begin by examining the given information with respect to these criteria, in order to see if there are obvious reasons why some of the vectors could not be equal. Then we will compare our choices for the equal vectors by calculating the magnitude and direction and the scalar components, as needed. SOLUTION Vectors A and B cannot possibly be equal, because they have different x scalar components of Ax = 80.0 m and Bx = 60.0 m. Furthermore, vectors B and C cannot possibly be equal, because they have different magnitudes of B = 75.0 m and C = 100.0 m. Therefore, we conclude that vectors A and C are the equal vectors. To verify that this is indeed the case we have two choices. We can either calculate the magnitude and direction of A (and compare it to the given magnitude and direction of C) or determine the scalar components of C (and compare them to the given components of A). Either choice will do, although both are shown below. +y The magnitude and direction of A are A

A=

Ax2

+

Ay2

=

( 80.0 m )

2

+ ( 60.0 m ) = 100.0 m 2

θ

Ay

+x

Ax

Ay −1 60.0 m = tan = 36.9° 80.0 m Ax

θ = tan −1

+y These results are identical to those given for C. C

The scalar components of C are 36.9°

Cx

Cy

+x

Chapter 1 Problems

25

C x = C cos 36.9° = (100.0 m ) cos 36.9° = 80.0 m C y = C sin 36.9° = (100.0 m ) sin 36.9° = 60.0 m

These results are identical to the components given for A.

42. REASONING Because both boats travel at Bwest W 101 km per hour, each one ends up 25.0° (0.500 h)(101 km/h) = 50.5 km from the dock Bsouth 53.0° B after a half-hour. They travel along straight paths, so this is the magnitude of both Gsouth G displacement vectors: B = G = 50.5 km. Since the displacement vector G makes an angle of 37° south of due west, its direction can also be expressed as 90° − 37° = 53° west of south. Gwest S With these angles and the magnitudes of both vectors in hand, we can consider the westward and southward components of each vector. SOLUTION a. First we calculate the magnitudes of the westward component of the displacement of each boat and then subtract them to find the difference:

Magnitude of Bwest = B cos 25.0 = ( 50.5 km ) cos 25.0 = 45.8 km Magnitude of Gwest = G sin 53.0 = ( 50.5 km ) sin 53.0 = 40.3 km The blue boat travels farther by the following amount:

45.8 km − 40.3 km = 5.5 km b. Similarly, we find for the magnitudes of the southward components that

Magnitude of Bsouth = B sin 25.0 = ( 50.5 km ) sin 25.0 = 21.3 km Magnitude of Gsouth = G cos53.0 = ( 50.5 km ) cos53.0 = 30.4 km The green boat travels farther by the following amount:

30.4 km − 21.3 km = 9.1 km

26

INTRODUCTION AND MATHEMATICAL CONCEPTS

43.

SSM WWW REASONING AND SOLUTION We take due north to be the direction of the +y axis. Vectors A and B are the components of the resultant, C. The angle that C makes with the x axis is then θ = tan −1 (B / A) . The symbol u denotes the units of the vectors. a. Solving for B gives B = A tan θ = (6.00 u) tan 60.0° = 10.4 u b. The magnitude of C is

C = A 2 + B 2 = ( 6. 00 u) 2 + ( 10 .4 u) 2 = 12 .0 u ______________________________________________________________________________

44. REASONING AND SOLUTION The force F can be first resolved into two components; the z component Fz and the projection onto the x-y plane, Fp as shown in the figure below on the left. According to that figure, Fp = F sin 54.0° = (475 N) sin 54.0°= 384 N The projection onto the x-y plane, Fp, can then be resolved into x and y components. z y F F

54.0° 54.0°

F

p

F

y

z

33.0° F

x

Fx

p

a. From the figure on the right, Fx = Fp cos 33.0° = (384 N) cos 33.0°=

322 N

b. Also from the figure on the right, Fy = Fp sin 33.0° = (384 N) sin 33.0°=

209 N

c. From the figure on the left, Fz = F cos 54.0° = (475 N) cos 54.0°=

279 N

Chapter 1 Problems

27

______________________________________________________________________________ 45. SSM REASONING The individual displacements of the golf ball, A, B, and C are shown in the figure. Their resultant, R, is the displacement that would have been needed to "hole the ball" on the very first putt. We will use the component method to find R. N R

C B

θ

20.0

o

E

A

SOLUTION The components of each displacement vector are given in the table below.

Vector

x Components

y Components

A B C

(5.0 m) cos 0° = 5.0 m (2.1 m) cos 20.0° = 2.0 m (0.50 m) cos 90.0° = 0

(5.0 m) sin 0° = 0 (2.1 m) sin 20.0° = 0.72 m (0.50 m) sin 90.0° = 0.50 m

R = A+B+C

7.0 m

1.22 m

The resultant vector R has magnitude R = ( 7 .0 m) 2 + ( 1.22 m) 2 = 7 .1 m

and the angle θ is

1.22 m = 9.9° 7.0 m

θ = tan −1

Thus, the required direction is 9.9° north of east . ______________________________________________________________________________ 46. REASONING Using the component method for vector addition, we will find the x component of the resultant force vector by adding the x components of the individual vectors. Then we will find the y component of the resultant vector by adding the y components of the individual vectors. Once the x and y components of the resultant are known, we will use the Pythagorean theorem to find the magnitude of the resultant and trigonometry to find its direction. We will take east as the +x direction and north as the +y direction. SOLUTION The x component of the resultant force F is

28

INTRODUCTION AND MATHEMATICAL CONCEPTS

b

g

b

g

b

g

Fx = 2240 N cos 34 .0 ° + 3160 N cos 90.0 ° = 2240 N cos 34 .0 ° FAx FBx The y component of the resultant force F is

b

g

b

Fy = − 2240 N sin 34 .0 ° + −3160 N FAy

g

FBy

Using the Pythagorean theorem, we find that the magnitude of the resultant force is F=

Fx2 + Fy2 =

b2240 N gcos 34.0°

2

b

g

+ − 2240 N sin 34 .0 °−3160 N

2

= 4790 N

Using trigonometry, we find that the direction of the resultant force is

θ = tan −1

L b2240 N gsin 34.0°+3160 N O = 67 .2 ° south of east M N b2240 N gcos 34.0° P Q

______________________________________________________________________________ 47. REASONING AND SOLUTION The horizontal component of the resultant vector is Rh = Ah + Bh +Ch = (0.00 m) + (15.0 m) + (18.0 m) cos 35.0° = 29.7 m

Similarly, the vertical component is Rv = Av + Bv + Cv = (5.00 m) + (0.00 m) + (– 18.0 m) sin 35.0° = – 5.32 m

The magnitude of the resultant vector is

R=

R h2 + Rv2 =

b29.7 mg+ b–5.32 mg = 2

2

30.2 m

The angle θ is obtained from

θ = tan–1[(5.32 m)/(29.7 m)] = 10.2° ____________________________________________________________________________________________

48. REASONING The resultant force in part a is FA, because that is the only force applied. The resultant force R in part b, where the two additional forces with identical magnitudes (FB = FC = F) are applied, is the sum of all three vectors: R = FA + FB + FC. Because the magnitude R of the resultant force R is k times larger than the magnitude FA of FA, we have R = kFA

(1)

Chapter 1 Problems

29

To solve the problem, we need to find an expression for R in terms of FA and F. Let the +x direction be in the direction of FA and the +y direction be upward (see the drawing), and consider the resultant of the two additional force vectors in part b. Because FB and FC are directed symmetrically about the x axis, and have the same magnitude F, their y components are equal and opposite. Therefore, they cancel out of the vector sum R, leaving only the x components of FB and FC. Now, since FA also has only an x component, the resultant R can be written as the sum of FA and the vector x components of FB and FC: R = FA + FBx + FCx. Because these vectors all point in the same direction, we can write down a first expression for the magnitude of the resultant: R = FA + FBx + FCx

(2)

+y

FB FA

FC

20.0°

20.0°

FBx

FCx

+x

R

The x components of the vectors FB and FC are adjacent to the 20.0°-angles, and so are

h related to the common magnitude F of both vectors by Equation 1.2 cos θ = a , with h ha = FBx and h = F: FBx = F cos θ

and

FCx = F cos θ

(3)

SOLUTION First, we use Equations (3) to replace FBx and FCx in Equation (2): R = FA + F cos θ + F cos θ = FA + 2 F cos θ

(4)

Then, we combine Equation (4) with Equation (1) (R = kFA) to eliminate R, and solve for the desired ratio F/FA: FA + 2 F cos θ = kFA

or

2 F cos θ = kFA − FA

or

2.00 − 1 F k −1 = = = 0.532 FA 2 cos θ 2 cos 20.0

2 F cos θ = ( k − 1) FA

30

INTRODUCTION AND MATHEMATICAL CONCEPTS

49. REASONING Using the component method, we find the components of the resultant R that are due east and due north. The magnitude and direction of the resultant R can be determined from its components, the Pythagorean theorem, and the tangent function. SOLUTION The first four rows of the table below give the components of the vectors A, B, C, and D. Note that east and north have been taken as the positive directions; hence vectors pointing due west and due south will appear with a negative sign.

Vector

East/West Component

North/South Component

A B C D

+ 2.00 km 0 – 2.50 km 0

0 + 3.75 km 0 –3.00 km

R=A+B+C+D

– 0.50 km

+ 0.75 km

The fifth row in the table gives the components of R. The magnitude of R is given by the Pythagorean theorem as R = (– 0.50 km) 2 + ( + 0.75 km) 2 = 0 . 90 km

The angle θ that R makes with the direction due west is R

0.75 km = 56° north of west 0.50 km

θ = tan −1

RN

θ

RE _____________________________________________________________________________________________

50.

REASONING According to the component method for vector addition, the x component of the resultant vector is the sum of the x component of A and the x component of B. Similarly, the y component of the resultant vector is the sum of the y component of A and the y component of B. The magnitude R of the resultant can be obtained from the x and y components of the resultant by using the Pythagorean theorem. The directional angle θ can be obtained using trigonometry.

North B R A

θ 30.0°

Chapter 1 Problems

31

SOLUTION We find the following results: Rx = ( 244 km ) cos 30.0 +

( −175 km ) = 36 km B

Ax

x

R y = ( 244 km ) sin 30.0 +

B

Ay R = Rx2 + R y2 =

( 0 km ) = 122 km y

( 36 km )2 + (122 km )2

= 127 km

Ry 122 km = tan −1 = 74 R 36 km x ______________________________________________________________________________

θ = tan −1

51. REASONING If we let the directions due east and due north be the positive directions, then the desired displacement A has components

A E = (4.8 km ) cos 42° = 3.57 km A N = ( 4.8 km ) sin 42° = 3.21 km while the actual displacement B has components B E = (2.4 km ) cos 22° = 2.23 km B N = ( 2.4 km ) sin 22° = 0.90 km Therefore, to reach the research station, the research team must go 3.57 km – 2.23 km = 1.34 km, eastward

and 3.21 km – 0.90 km = 2.31 km, northward

SOLUTION a. From the Pythagorean theorem, we find that the magnitude of the displacement vector required to bring the team to the research station is R = (1.34 km) 2 + (2.31 km) 2 = 2.7 km

b. The angle θ is given by

N

R θ 1.34 km

2.31 km

32

INTRODUCTION AND MATHEMATICAL CONCEPTS

θ = tan –1

2.31 km = 6.0 × 10 1 degrees, north of east 1.34 km

_____________________________________________________________________________________________

52. REASONING Let A be the vector from base camp to the first team, B the vector Second team from base camp to the second team, and N (y) C the vector from the first team’s position First team C to the second team’s position. C is the vector whose magnitude and direction are θ given by the first team’s GPS unit. Since B=A+C 35° A you can get from the base camp to the 19° second team’s position either by traveling W E (x) along vector B alone, or by traveling first Base camp along A and then along C, we know that B is the vector sum of the other two: B = A + C. The reading on the first team’s GPS unit is then C = B − A. The components of C are found from the components of A and B: Cx = Bx − Ax, Cy = By − Ay. Once we have these components, we can calculate the magnitude and direction of C, as shown on the GPS readout. Because the first team is northwest of camp, and the second team is northeast, we expect the vector C to be directed east and either north or south.

SOLUTION Let east serve as the positive x direction and north as the positive y direction. We then calculate the components of C, noting that B’s components are both positive, and A’s x-component is negative: C x = (29 km) sin 35 − ( −38 km) cos 19 = 53 km B A x

x

C y = (29 km) cos 35 − (38 km) sin 19 = 11 km By A y

The second team is therefore 53 km east of the first team (since Cx is positive), and 11 km north (since Cy is positive). The straight-line distance between the teams can be calculated with the Pythagorean theorem (Equation 1.7):

C = Cx2 + C y 2 =

( 53 km )2 + (11 km )2 =

54 km

The direction of the vector C is to be measured relative to due east, so we apply the inverse tangent function (Equation 1.6) to get the angle θ:

Chapter 1 Problems

33

Cy −1 11 km = tan = 12 53 km Cx

θ = tan −1

53.

SSM REASONING Since the finish line is coincident with the starting line, the net displacement of the sailboat is zero. Hence the sum of the components of the displacement vectors of the individual legs must be zero. In the drawing in the text, the directions to the right and upward are taken as positive. SOLUTION In the horizontal direction Rh = Ah + Bh + Ch + Dh = 0 Rh = (3.20 km) cos 40.0° – (5.10 km) cos 35.0° – (4.80 km) cos 23.0° + D cos θ = 0 D cos θ = 6.14 km

(1)

In the vertical direction Rv = Av + Bv + Cv + Dv = 0. Rv = (3.20 km) sin 40.0° + (5.10 km) sin 35.0° – (4.80 km) sin 23.0° – D sin θ = 0. D sin θ = 3.11 km

(2)

Dividing (2) by (1) gives tan θ = (3.11 km)/(6.14 km)

or

θ = 26.9°

Solving (1) gives D = (6.14 km)/cos 26.9° =

6.88 km

54. REASONING We know that the three displacement vectors have a resultant of zero, so that A + B + C = 0. This means that the sum of the x components of the vectors and the sum of the y components of the vectors are separately equal to zero. From these two equations we will be able to determine the magnitudes of vectors B and C. The directions east and north are, respectively, the +x and +y directions. SOLUTION Setting the sum of the x components of the vectors and the sum of the y components of the vectors separately equal to zero, we have

34

INTRODUCTION AND MATHEMATICAL CONCEPTS

b1550 m gcos 25.0° + B sin 41.0° + b− C cos 35.0° g= 0 Ax

Bx

Cx

b1550 m gsin 25.0° + b− B cos 41.0° g+ C sin 35.0° = 0 Ay

Cy

By

These two equations contain two unknown variables, B and C. They can be solved simultaneously to show that a. B = 5550 m and b. C = 6160 m ______________________________________________________________________________ 55. REASONING The drawing shows the vectors A, B, and C. Since these vectors add to give a resultant that is zero, we can write that A + B + C = 0. This addition will be carried out by the component method. This means that the x-component of this equation must be zero (Ax + Bx + Cx = 0) and the y-component must be zero (Ay + By + Cy = 0). These two equations will allow us to find the magnitudes of B and C.

+y (north) A −x

145 units 35.0°

B 65.0°

+x (east)

15.0°

C −y

SOLUTION The x- and y-components of A, B, and C are given in the table below. The plus and minus signs indicate whether the components point along the positive or negative axes. Vector

x Component

y Component

A

–(145 units) cos 35.0° = –119 units

+(145 units) sin 35.0° = +83.2 units

B C

+B sin 65.0° = +B (0.906) –C sin 15.0° = –C (0.259)

+B cos 65.0° = +B (0.423) –C cos 15.0° = –C (0.966)

A+B+C

–119 units + B (0.906) – C (0.259)

+83.2 units + B (0.423) – C (0.966)

Setting the separate x- and y- components of A + B + C equal to zero gives x-component

(–119 units) + B (0.906) – C (0.259) = 0

y-component

(+83.2 units) + B (0.423) – C (0.966) = 0

Solving these two equations simultaneously, we find that

a. B = 178 units

b. C = 164 units

Chapter 1 Problems

35

56. REASONING The following table shows the components of the individual displacements and the components of the resultant. The directions due east and due north are taken as the positive directions. East/West Component

Displacement (1) (2) (3) (4)

North/South Component

0 –27.0 cm –(23.0 cm) cos 35.0° = –18.84 cm –(23.0 cm) sin 35.0° = –13.19 cm (28.0 cm) cos 55.0° = 16.06 cm –(28.0 cm) sin 55.0° = –22.94 cm (35.0 cm) sin 63.0° = 31.19 cm (35.0 cm) cos 63.0° = 15.89 cm –13.89 cm

Resultant

–4.94 cm

SOLUTION a. From the Pythagorean theorem, we find that the magnitude of the resultant displacement vector is 13.89 cm

R = (13.89 cm) 2 + (4.94 cm) 2 = 14.7 cm

R

b. The angle θ is given by

θ = tan −1

4.94 cm I F G H13.89 cm JK=

θ

4.94 cm

19 .6° , south of west

57. REASONING AND SOLUTION The following figure (not drawn to scale) shows the geometry of the situation, when the observer is a distance r from the base of the arch. The angle θ is related to r and h by tan θ = h/ r . Solving for r, we find h = 192 m

θ

h 192 m r= = = 5.5 × 10 3 m = 5.5 km r tanθ tan 2.0° ______________________________________________________________________________ 58. REASONING AND SOLUTION In the diagram below, θ = 14.6° and h = 2830 m. We know that sin θ = H/h and, therefore,

h

H = h sin θ = (2830 m) sin 14.6° = 713 m

H θ

36

INTRODUCTION AND MATHEMATICAL CONCEPTS

59.

SSM REASONING AND SOLUTION In order to determine which vector has the largest x and y components, we calculate the magnitude of the x and y components explicitly and compare them. In the calculations, the symbol u denotes the units of the vectors. Ax = (100.0 u) cos 90.0° = 0.00 u 2 Bx = (200.0 u) cos 60.0° = 1.00 × 10 u Cx = (150.0 u) cos 0.00° = 150.0 u

2

Ay = (100.0 u) sin 90.0° = 1.00 × 10 u By = (200.0 u) sin 60.0° = 173 u Cy = (150.0 u) sin 0.00° = 0.00 u

a. C has the largest x component.

b. B has the largest y component. ______________________________________________________________________________ 60. REASONING Multiplying an equation by a factor of 1 does not alter the equation; this is the basis of our solution. We will use factors of 1 in the following forms: 1 gal = 1 , since 1 gal = 128 oz 128 oz

3.785 ×10−3 m3 = 1 , since 3.785 × 10−3 m3 = 1 gal 1 gal 1 mL = 1 , since 1 mL = 10−6 m3 −6 3 10 m

SOLUTION The starting point for our solution is the fact that

Volume = 1 oz Multiplying this equation on the right by factors of 1 does not alter the equation, so it follows that 1 gal 3.785 ×10−3 m3 1 mL Volume = (1 oz )(1)(1)(1) = 1 oz = 29.6 mL 128 oz 10−6 m3 1 gal

(

)

Note that all the units on the right, except one, are eliminated algebraically, leaving only the desired units of milliliters (mL).

37

Chapter 1 Problems

61. REASONING AND SOLUTION The east and north components are, respectively a.

Ae = A cos θ = (155 km) cos 18.0° =

b.

An = A sin θ = (155 km) sin 18.0° =

147 km 47.9 km

_____________________________________________________________________________________________

62. REASONING We will use the scalar x and y components of the resultant vector to obtain its magnitude and direction. To obtain the x component of the resultant we will add together the x components of each of the vectors. To obtain the y component of the resultant we will add together the y components of each of the vectors. +y SOLUTION The x and y components of the resultant vector R are Rx and Ry, respectively. In terms of these components, the magnitude R and the B directional angle θ (with respect to the x axis) A for the resultant are

R=

Rx2

+ R y2

and

Ry θ = tan −1 R x

20.0° 35.0°

(1)

C

The following table summarizes the components of the individual vectors shown in the drawing: Vector

+x

50.0°

x component

D

y component

A

Ax = − (16.0 m ) cos 20.0° = −15.0 m

Ay = (16.0 m ) sin 20.0° = 5.47 m

B

Bx = 0 m

B y = 11.0 m

C

C x = − (12.0 m ) cos 35.0° = −9.83 m

C y = − (12.0 m ) sin 35.0° = −6.88 m

D

Dx = ( 26.0 m ) cos 50.0° = 16.7 m

Dy = − ( 26.0 m ) sin 50.0° = −19.9 m

R

Rx = −15.0 m + 0 m − 9.83 m + 16.7 m

R y = 5.47 m + 11.0 m − 6.88 m − 19.9 m

= −8.1 m

= −10.3 m

38

INTRODUCTION AND MATHEMATICAL CONCEPTS

Note that the component Bx is zero, because B points along the y axis. Note also that the

components Cx and Cy are both negative, since C points between the −x and −y axes. Finally, note that the component Dy is negative since D points below the +x axis. Using equation (1), we find that +y Rx 2 2 +x R = Rx2 + Ry2 = ( −8.1 m ) + ( −10.3 m ) = 13 m θ

Ry θ = tan Rx −1

Ry

−1 −10.3 m = tan = 52° −8.1 m

R

Since both Rx and Ry are negative, the resultant points between the −x and −y axes.

63. SSM REASONING The performer walks out on the wire a distance d, and the vertical distance to the net is h. Since these two distances are perpendicular, the magnitude of the displacement is given by the Pythagorean theorem as s = d 2 + h 2 . Values for s and h are given, so we can solve this expression for the distance d. The angle that the performer’s displacement makes below the horizontal can be found using trigonometry. SOLUTION a. Using the Pythagorean theorem, we find that

s = d 2 + h2

or

d = s2 − h2 =

b26.7 ft g− b25.0 ft g = 2

2

9 .4 ft

b. The angle θ that the performer’s displacement makes below the horizontal is given by tan θ =

h d

or

θ = tan −1

hI F F25.0 ft IJ= = tan G G J Hd K H9.4 ft K −1

69 °

64. REASONING The force vector F points at an angle of θ above the +x axis. Therefore, its x and y components are given by Fx = F cos θ and Fy = F sin θ. SOLUTION follows:

a. The magnitude of the vector can be obtained from the y component as Fy = F sin θ

or

F=

Fy sin θ

=

290 N = 370 N sin 52 °

Chapter 1 Problems

39

b. Now that the magnitude of the vector is known, the x component of the vector can be calculated as follows: Fx = F cosθ = 370 N cos 52 ° = +230 N

b

g

_____________________________________________________________________________________________

65. SSM REASONING The force F and its two components form a right triangle. The hypotenuse is 82.3 newtons, and the side parallel to the +x axis is Fx = 74.6 newtons. Therefore, we can use the trigonometric cosine and sine functions to determine the angle of F relative to the +x axis and the component Fy of F along the +y axis. SOLUTION a. The direction of F relative to the +x axis is specified by the angle θ as 74.6 newtons = 25.0° 82.3 newtons

θ = cos −1

(1.5)

b. The component of F along the +y axis is

Fy = F sin 25.0° = ( 82.3 newtons ) sin 25.0° = 34.8 newtons

(1.4)

______________________________________________________________________________ 66. REASONING We are given that the vector sum of the three forces is zero, so F1 + F2 + F3 = 0 N. Since F1 and F2 are known, F3 can be found from the relation F3 = −(F1 + F2). We will use the x- and y-components of this equation to find the magnitude and direction of F3.

SOLUTION The x- and y-components of the equation F3 = −( F1 + F2) are: x-component

F3 x = − ( F1x + F2 x )

y-component

F3 y = − F1 y + F2 y

(

(1)

)

(2)

The table below gives the x- and y-components of F1 and F2:

Vector

x component

y component

F1 F2

F1x = −(21.0 N) sin 30.0° = −10.5 N F2x = +15.0 N

F1y = −(21.0 N) cos 30.0° = +18.2 N F2y = 0 N

Substituting the values for F1x and F2x into Equation (1) gives

40

INTRODUCTION AND MATHEMATICAL CONCEPTS

F3 x = − ( F1x + F2 x ) = − ( −10.5 N + 15.2 N ) = −4.5 N Substituting F1y and F2y into Equation (2) gives

(

)

F3 y = − F1 y + F2 y = − ( +18.2 N + 0 N ) = −18.2 N The magnitude of F3 can now be obtained by employing the Pythagorean theorem:

F3 = F32x + F32y =

( −4.5 N )2 + ( −18.2 N )2

= 18.7 N

The angle θ that F3 makes with respect to the −x axis can be determined from the inverse tangent function (Equation 1.6),

F3 y −1 −18.2 N = tan = 76° −4.5 N F3 x ______________________________________________________________________________

θ = tan −1

67. REASONING AND SOLUTION The following figure is a scale diagram of the forces drawn tail-to-head. The scale factor is shown in the figure. The head of F3 touches the tail of F1, because the resultant of the three forces is zero. a. From the figure, F3 must have a

θ

magnitude of 78 N if the resultant force acting on the ball is zero. b. Measurement with a protractor indicates that the angle θ = 34° .

60.0° F1 = 50.0 N

F3

θ

60.0° F2 = 90.0 N

Scale Factor:

20.0 N

_____________________________________________________________________________________________

68. REASONING a. The drawing shows the person standing on the earth and looking at the horizon. Notice the right triangle, the sides of which are R, the radius of the earth, and d, the distance from the person’s eyes to the horizon. The length of the hypotenuse is R + h, where h is the height of the person’s eyes above the water. Since we know the lengths of two sides of the triangle, the Pythagorean

h

d R

90°

R Earth

Horizon

Chapter 1 Problems

41

theorem can be employed to find the length of the third side. b. To convert the distance from meters to miles, we use the relation 1609 m = 1 mi (see the page facing the inside of the front cover of the text). SOLUTION a. The Pythagorean theorem (Equation 1.7) states that the square of the hypotenuse is equal to the sum of the squares of the sides, or ( R + h ) = d + R . Solving this equation for d yields 2

d=

( R + h )2 − R 2

2

2

= R + 2 Rh + h − R

(

2

2

2

)

= 2 Rh + h 2 = 2 6.38 ×106 m (1.6 m ) + (1.6 m ) = 4500 m 2

b. Multiplying the distance of 4500 m by a factor of unity, (1 mi)/(1609 m) = 1, the distance (in miles) from the person's eyes to the horizon is

1 mi d = ( 4500 m )(1) = 4500 m = 2.8 mi 1609 m ______________________________________________________________________________

(

)

69. REASONING AND SOLUTION If D is the unknown vector, then A + B + C + D = 0 requires that DE = –(AE + BE + CE) or DE = (113 u) cos 60.0° – (222 u) cos 35.0° – (177 u) cos 23.0° =

–288 units

The minus sign indicates that DE has a direction of due west. Also, DN = –(AN + BN + CN) or DN = (113 u) sin 60.0° + (222 u) sin 35.0° – (177 u) sin 23.0° = 156 units ______________________________________________________________________________

CHAPTER 2 KINEMATICS IN

ONE DIMENSION ANSWERS TO FOCUS ON CONCEPTS QUESTIONS

1. (b) Displacement, being a vector, conveys information about magnitude and direction. Distance conveys no information about direction and, hence, is not a vector. 2. (c) Since each runner starts at the same place and ends at the same place, the three displacement vectors are equal. 3. (c) The average speed is the distance of 16.0 km divided by the elapsed time of 2.0 h. The average velocity is the displacement of 0 km divided by the elapsed time. The displacement is 0 km, because the jogger begins and ends at the same place. 4. (a) Since the bicycle covers the same number of meters per second everywhere on the track, its speed is constant. 5. (e) The average velocity is the displacement (2.0 km due north) divided by the elapsed time (0.50 h), and the direction of the velocity is the same as the direction of the displacement. 6. (c) The average acceleration is the change in velocity (final velocity minus initial velocity) divided by the elapsed time. The change in velocity has a magnitude of 15.0 km/h. Since the change in velocity points due east, the direction of the average acceleration is also due east. 7. (d) This is always the situation when an object at rest begins to move. 8. (b) If neither the magnitude nor the direction of the velocity changes, then the velocity is constant, and the change in velocity is zero. Since the average acceleration is the change in velocity divided by the elapsed time, the average acceleration is also zero. 9. (a) The runners are always moving after the race starts and, therefore, have a non-zero average speed. The average velocity is the displacement divided by the elapsed time, and the displacement is zero, since the race starts and finishes at the same place. The average acceleration is the change in the velocity divided by the elapsed time, and the velocity changes, since the contestants start at rest and finish while running. 10. (c) The equations of kinematics can be used only when the acceleration remains constant and cannot be used when it changes from moment to moment. 11. (a) Velocity, not speed, appears as one of the variables in the equations of kinematics. Velocity is a vector. The magnitude of the instantaneous velocity is the speed.

Chapter 2 Answers to Focus on Concepts Questions

(

43

)

12. (b) According to one of the equation of kinematics v = v0 + 2 ax, with v0 = 0 m/s , the 2

2

displacement is proportional to the square of the velocity.

(

)

13. (d) According to one of the equation of kinematics x = v0t + 12 at , with v0 = 0 m/s , the 2

displacement is proportional to the acceleration. 14. (b) For a single object each equation of kinematics contains four variables, one of which is the unknown variable. 15. (e) An equation of kinematics ( v = v0 + at ) gives the answer directly, since the initial velocity, the final velocity, and the time are known. 16. (c) An equation of kinematics x =

1 2

( v0 + v ) t gives the answer directly, since the initial

velocity, the final velocity, and the time are known.

(

)

17. (e) An equation of kinematics v = v0 + 2ax gives the answer directly, since the initial 2

2

velocity, the final velocity, and the acceleration are known. 18. (d) This statement is false. Near the earth’s surface the acceleration due to gravity has the approximate magnitude of 9.80 m/s2 and always points downward, toward the center of the earth. 19. (b) Free-fall is the motion that occurs while the acceleration is solely the acceleration due to gravity. While the rocket is picking up speed in the upward direction, the acceleration is not just due to gravity, but is due to the combined effect of gravity and the engines. In fact, the effect of the engines is greater than the effect of gravity. Only when the engines shut down does the free-fall motion begin.

(

)

20. (c) According to an equation of kinematics v = v0 + 2ax, with v = 0 m/s , the launch 2

2

speed v0 is proportional to the square root of the maximum height. 21. (a) An equation of kinematics ( v = v0 + at ) gives the answer directly. 22. (d) The acceleration due to gravity points downward, in the same direction as the initial velocity of the stone thrown from the top of the cliff. Therefore, this stone picks up speed as it approaches the nest. In contrast, the acceleration due to gravity points opposite to the initial velocity of the stone thrown from the ground, so that this stone loses speed as it approaches the nest. The result is that, on average, the stone thrown from the top of the cliff travels faster than the stone thrown from the ground and hits the nest first. 23. 1.13 s

44

KINEMATICS IN ONE DIMENSION

24. (a) The slope of the line in a position versus time graph gives the velocity of the motion. The slope for part A is positive. For part B the slope is negative. For part C the slope is positive. 25. (b) The slope of the line in a position versus time graph gives the velocity of the motion. Section A has the smallest slope and section B the largest slope. 26. (c) The slope of the line in a position versus time graph gives the velocity of the motion. Here the slope is positive at all times, but it decreases as time increases from left to right in the graph. This means that the positive velocity is decreasing as time increases, which is a condition of deceleration.

Chapter 2 Problems

45

CHAPTER 2 KINEMATICS IN

ONE DIMENSION PROBLEMS ______________________________________________________________________________ 1.

SSM REASONING AND SOLUTION a. The total distance traveled is found by adding the distances traveled during each segment of the trip. 6.9 km + 1.8 km + 3.7 km = 12.4 km

b. All three segments of the trip lie along the east-west line. Taking east as the positive direction, the individual displacements can then be added to yield the resultant displacement. 6.9 km + (–1.8 km) + 3.7 km = +8.8 km The displacement is positive, indicating that it points due east. Therefore, Displacement of the whale = 8.8 km, due east ______________________________________________________________________________ 2.

REASONING The displacement is a vector that points from an object’s initial position to its final position. If the final position is greater than the initial position, the displacement is positive. On the other hand, if the final position is less than the initial position, the displacement is negative. (a) The final position is greater than the initial position, so the displacement will be positive. (b) The final position is less than the initial position, so the displacement will be negative. (c) The final position is greater than the initial position, so the displacement will be positive. SOLUTION The displacement is defined as Displacement = x – x0, where x is the final position and x0 is the initial position. The displacements for the three cases are: (a) Displacement = 6.0 m − 2.0 m = +4.0 m (b) Displacement = 2.0 m − 6.0 m = −4.0 m (c) Displacement = 7.0 m − (−3.0 m) = +10.0 m

3.

SSM REASONING The average speed is the distance traveled divided by the elapsed time (Equation 2.1). Since the average speed and distance are known, we can use this relation to find the time.

46

KINEMATICS IN ONE DIMENSION

SOLUTION The time it takes for the continents to drift apart by 1500 m is Distance 1500 m = = 5 × 104 yr Average speed cm 1 m 3 yr 100 cm ______________________________________________________________________________ Elapsed time =

4.

REASONING Average speed is the ratio of distance to elapsed time (Equation 2.1), so the elapsed time is distance divided by average speed. Both the average speed and the distance are given in SI base units, so the elapsed time will come out in seconds, which can then be converted to minutes (1 min = 60 s). SOLUTION First, calculate the elapsed time ∆t in seconds:

∆t =

Distance 1.5 m = = 140 s Average speed 1.1×10−2 m/s

(2.1)

Converting the elapsed time from seconds to minutes, we find that 1 min ∆t = 140 s = 2.3 min 60 s

(

5.

)

x − x0 REASONING According to Equation 2.2 v = , the average velocity ( v ) is equal t − t0 to the displacement ( x − x0 ) divided by the elapsed time ( t − t0 ) , and the direction of the average velocity is the same as that of the displacement. The displacement is equal to the difference between the final and initial positions. SOLUTION Equation 2.2 gives the average velocity as

v=

x − x0 t − t0

Therefore, the average velocities for the three cases are: (a) Average velocity = (6.0 m − 2.0 m)/(0.50 s) = +8.0 m/s (b) Average velocity = (2.0 m − 6.0 m)/(0.50 s) = −8.0 m/s (c) Average velocity = [7.0 m − (−3.0 m)]/(0.50 s) = +20.0 m/s

Chapter 2 Problems

47

The algebraic sign of the answer conveys the direction in each case. ______________________________________________________________________________ 6.

REASONING The distance traveled by the Space Shuttle is equal to its speed multiplied by the time. The number of football fields is equal to this distance divided by the length L of one football field. SOLUTION The number of football fields is

(

)(

−3

)

7.6 ×10 m / s 110 ×10 s x vt = = 9.1 Number = = L L 91.4 m ______________________________________________________________________________ 7.

3

REASONING In a race against el-Guerrouj, Bannister would run a distance given by his average speed times the time duration of the race (see Equation 2.1). The time duration of the race would be el-Guerrouj’s winning time of 3:43.13 (223.13 s). The difference between Bannister’s distance and the length of the race is el-Guerrouj’s winning margin. SOLUTION From the table of conversion factors on the page facing the front cover, we find that one mile corresponds to 1609 m. According to Equation 2.1, Bannister’s average speed is Distance 1609 m Average speed = = Elapsed time 239.4 s Had he run against el-Guerrouj at this average speed for the 223.13-s duration of the race, he would have traveled a distance of 1609 m Distance = Average speed × Time = ( 223.13 s ) 239.4 s

while el-Guerrouj traveled 1609 m. Thus, el-Guerrouj would have won by a distance of 1609 m 1609 m − ( 223.13 s ) = 109 m 239.4 s

8.

REASONING The younger (and faster) runner should start the race after the older runner, the delay being the difference between the time required for the older runner to complete the race and that for the younger runner. The time for each runner to complete the race is equal to the distance of the race divided by the average speed of that runner (see Equation 2.1). SOLUTION The difference between the times for the two runners to complete the race is t50 − t18 , where

48

KINEMATICS IN ONE DIMENSION

t50 =

Distance ( Average Speed )50-yr-old

and t18 =

Distance ( Average Speed )18-yr-old

(2.1)

The difference between these two times (which is how much later the younger runner should start) is Distance Distance t50 − t18 = − ( Average Speed )50-yr-old ( Average Speed )18-yr-old

10.0 × 103 m 10.0 ×103 m = − = 64 s 4.27 m/s 4.39 m/s ______________________________________________________________________________ 9.

REASONING In order for the bear to catch the tourist over the distance d, the bear must reach the car at the same time as the tourist. During the time t that it takes for the tourist to reach the car, the bear must travel a total distance of d + 26 m. From Equation 2.1, vtourist =

d t

(1)

and

vbear =

d + 26 m t

(2)

Equations (1) and (2) can be solved simultaneously to find d. SOLUTION Solving Equation (1) for t and substituting into Equation (2), we find vbear =

d + 26 m (d + 26 m)vtourist = d / vtourist d

26 m vbear = 1 + v d tourist

Solving for d yields:

26 m 26 m = = 52 m 6.0 m/s vbear −1 −1 4.0 m/s vtourist ______________________________________________________________________________ d=

10. REASONING AND SOLUTION Let west be the positive direction. The average velocity of the backpacker is

x +x e v= w t +t w e

where

t

x = w w v w

and

x t = e e v e

Chapter 2 Problems

49

Combining these equations and solving for xe (suppressing the units) gives xe =

– (1 – v / vw ) xw

(1 – v / ve )

=

– 1 – (1.34 m/s ) / ( 2.68 m/s ) ( 6.44 km ) 1 – (1.34 m/s ) / ( 0.447 m/s )

= –0.81 km

The distance traveled is the magnitude of xe, or 0.81 km . ______________________________________________________________________________ 11.

SSM WWW REASONING Since the woman runs for a known distance at a known constant speed, we can find the time it takes for her to reach the water from Equation 2.1. We can then use Equation 2.1 to determine the total distance traveled by the dog in this time.

SOLUTION The time required for the woman to reach the water is Elapsed time =

d woman vwoman

4.0 km 1000 m = = 1600 s 2.5 m/s 1.0 km

In 1600 s, the dog travels a total distance of

ddog = vdog t = (4.5 m/s)(1600 s) = 7.2 × 103 m ______________________________________________________________________________ 12. REASONING The definition of average velocity is given by Equation 2.2 as Average velocity = Displacement/(Elapsed time). The displacement in this expression is the total displacement, which is the sum of the displacements for each part of the trip. Displacement is a vector quantity, and we must be careful to account for the fact that the displacement in the first part of the trip is north, while the displacement in the second part is south. SOLUTION According to Equation 2.2, the displacement for each part of the trip is the average velocity for that part times the corresponding elapsed time. Designating north as the positive direction, we find for the total displacement that Displacement =

( 27 m/s ) tNorth Northward

+

( −17 m/s ) tSouth Southward

where tNorth and tSouth denote, respectively, the times for each part of the trip. Note that the minus sign indicates a direction due south. Noting that the total elapsed time is tNorth + tSouth, we can use Equation 2.2 to find the average velocity for the entire trip as follows:

50

KINEMATICS IN ONE DIMENSION

Average velocity =

Displacement ( 27 m/s ) t North + ( −17 m/s ) tSouth = Elapsed time tNorth + tSouth

t North = ( 27 m/s ) t North + tSouth t North But t North + tSouth

3 tSouth = and 4 t North + tSouth

tSouth + ( −17 m/s ) t North + tSouth

1 = . Therefore, we have that 4

3 1 Average velocity = ( 27 m/s ) + ( −17 m/s ) = +16 m/s 4 4

The plus sign indicates that the average velocity for the entire trip points north.

13. REASONING AND SOLUTION The upper edge of the wall will disappear after the train has traveled the distance d in the figure below.

0.90 m

B

B A

A

12°

b

2.0 m

12°

d The distance d is equal to the length of the window plus the base of the 12° right triangle of height 0.90 m. The base of the triangle is given by b =

0.90 m = 4.2 m tan12°

0.90 m

12°

b

Thus, d = 2.0 m + 4.2 m = 6.2 m. The time required for the train to travel 6.2 m is, from the definition of average speed, x 6.2 m = = 2.1 s v 3.0 m/s ______________________________________________________________________________ t=

Chapter 2 Problems

51

v − v0 14. REASONING The average acceleration ( a ) is defined by Equation 2.4 a = as t − t 0 the change in velocity ( v − v0 ) divided by the elapsed time ( t − t0 ) . The change in velocity

is equal to the final velocity minus the initial velocity. Therefore, the change in velocity, and hence the acceleration, is positive if the final velocity is greater than the initial velocity. The acceleration is negative if the final velocity is less than the initial velocity. (a) The final velocity is greater than the initial velocity, so the acceleration will be positive. (b) The final velocity is less than the initial velocity, so the acceleration will be negative. (c) The final velocity is greater than the initial velocity (–3.0 m/s is greater than –6.0 m/s), so the acceleration will be positive. (d) The final velocity is less than the initial velocity, so the acceleration will be negative. SOLUTION Equation 2.4 gives the average acceleration as a=

v − v0 t − t0

Therefore, the average accelerations for the four cases are: (a) a = (5.0 m/s − 2.0 m/s)/(2.0 s) = +1.5 m/s 2 (b) a = (2.0 m/s − 5.0 m/s)/(2.0 s) = −1.5 m/s 2 (c) a = [−3.0 m/s − (−6.0 m/s)]/(2.0 s) = +1.5 m/s 2 (d) a = (−4.0 m/s − 4.0 m/s)/(2.0 s) = −4.0 m/s 2 ______________________________________________________________________________ v − v0 15. REASONING The average acceleration ( a ) is defined by Equation 2.4 a = as t − t0 the change in velocity ( v − v0 ) divided by the elapsed time ( t − t0 ) . The change in velocity

is equal to the final velocity minus the initial velocity. Therefore, the change in velocity, and hence the acceleration, is positive if the final velocity is greater than the initial velocity. The acceleration is negative if the final velocity is less than the initial velocity. The acceleration is zero if the final and initial velocities are the same. SOLUTION Equation 2.4 gives the average acceleration as a=

v − v0 t − t0

a. The initial and final velocities are both +82 m/s, since the velocity is constant. The average acceleration is

52

KINEMATICS IN ONE DIMENSION

a = (82 m/s − 82 m/s)/(t – t0) = 0 m/s 2 b. The initial velocity is +82 m/s, and the final velocity is –82 m/s. The average acceleration is

a = (−82 m/s − 82 m/s)/(12 s) = −14 m/s 2 ______________________________________________________________________________ 16. REASONING Although the planet follows a curved, two-dimensional path through space, this causes no difficulty here because the initial and final velocities for this period are in opposite directions. Thus, the problem is effectively a problem in one dimension only. ∆v Equation 2.4 a = relates the change ∆v in the planet’s velocity to its average ∆t acceleration and the elapsed time ∆t = 2.16 years. It will be convenient to convert the elapsed time to seconds before calculating the average acceleration. SOLUTION a. The net change in the planet’s velocity is the final minus the initial velocity: ∆v = v − v 0 = −18.5 km/s − 20.9 km/s = −39.4 km/s km 1000 m 4 ∆v = −39.4 = −3.94 ×10 m/s s 1 km

b. Although the planet’s velocity changes by a large amount, the change occurs over a long time interval, so the average acceleration is likely to be small. Expressed in seconds, the interval is 365 d 24 h 60 min 60 s = 6.81×107 s ∆t = 2.16 yr 1 yr 1d 1 h 1 min

(

)

Then the average acceleration is a=

17.

∆v −3.94 × 104 m/s = = −5.79 × 10−4 m/s 2 7 ∆t 6.81× 10 s

SSM REASONING The average acceleration is defined by Equation 2.4 as the change in velocity divided by the elapsed time. We can find the elapsed time from this relation because the acceleration and the change in velocity are given. Since the acceleration of the spacecraft is constant, it is equal to the average acceleration.

Chapter 2 Problems

53

SOLUTION a. The time ∆t that it takes for the spacecraft to change its velocity by an amount ∆v = +2700 m/s is ∆v +2700 m / s ∆t = = = 3.0 ×102 days m / s a +9.0 day b. Since 24 hr = 1 day and 3600 s = 1 hr, the acceleration of the spacecraft (in m/s2) is +9.0 m / s = +1.04 × 10−4 m / s 2 24 hr 3600 s (1 day ) 1 day 1 hr ______________________________________________________________________________ a=

18. REASONING

∆v = t

We can use the definition of average acceleration a = ( v − v 0 ) / ( t − t0 )

(Equation 2.4) to find the sprinter’s final velocity v at the end of the acceleration phase, because her initial velocity ( v 0 = 0 m/s , since she starts from rest), her average acceleration

a , and the time interval t − t0 are known. SOLUTION a. Since the sprinter has a constant acceleration, it is also equal to her average acceleration, so a = +2.3 m/s 2 Her velocity at the end of the 1.2-s period is

v = v0 + a ( t − t0 ) = ( 0 m/s ) + ( +2.3 m/s 2 ) (1.2 s ) = +2.8 m/s b. Since her acceleration is zero during the remainder of the race, her velocity remains constant at +2.8 m/s . ______________________________________________________________________________ 19. REASONING When the velocity and acceleration vectors are in the same direction, the speed of the object increases in time. When the velocity and acceleration vectors are in opposite directions, the speed of the object decreases in time. (a) The initial velocity and acceleration are in the same direction, so the speed is increasing. (b) The initial velocity and acceleration are in opposite directions, so the speed is decreasing. (c) The initial velocity and acceleration are in opposite directions, so the speed is decreasing. (d) The initial velocity and acceleration are in the same direction, so the speed is increasing.

SOLUTION The final velocity v is related to the initial velocity v0, the acceleration a, and the elapsed time t through Equation 2.4 (v = v0 + a t). The final velocities and speeds for the four moving objects are:

54

KINEMATICS IN ONE DIMENSION

a. v = 12 m/s + (3.0 m/s2)(2.0 s) = 18 m/s. The final speed is 18 m/s . b. v = 12 m/s + (−3.0 m/s2)(2.0 s) = 6.0 m/s. The final speed is 6.0 m/s . c. v = −12 m/s + (3.0 m/s2)(2.0 s) = −6.0 m/s. The final speed is 6.0 m/s . d. v = −12 m/s + (−3.0 m/s2)(2.0 s) = −18 m/s. The final speed is 18 m/s . ______________________________________________________________________________ 20. REASONING AND SOLUTION from Equation 2.4 (v = v0 + at) as v − v0

The magnitude of the car's acceleration can be found

26.8 m/s – 0 m/s = 8.18 m/s 2 t 3.275 s ______________________________________________________________________________

a=

21.

=

SSM REASONING AND SOLUTION The velocity of the automobile for each stage is given by Equation 2.4: v = v0 + at . Therefore, v1 = v0 + a1t = 0 m/s + a1t

and

v2 = v1 + a2t

Since the magnitude of the car's velocity at the end of stage 2 is 2.5 times greater than it is at the end of stage 1, v2 = 2.5v1 . Thus, rearranging the result for v2, we find v2 – v1 2.5v1 – v1 1.5v1 1.5( a1t ) = = = = 1.5a1 = 1.5(3.0 m/s 2 ) = 4.5 m/s 2 t t t t ______________________________________________________________________________ a2 =

22. REASONING According to Equation 2.4, the average acceleration of the car for the first twelve seconds after the engine cuts out is a1 =

v1f − v10

(1)

∆t1

and the average acceleration of the car during the next six seconds is a2 =

v2f − v20 ∆t2

=

v2f − v1f ∆t2

(2)

The velocity v1f of the car at the end of the initial twelve-second interval can be found by solving Equations (1) and (2) simultaneously.

Chapter 2 Problems

55

SOLUTION Dividing Equation (1) by Equation (2), we have a1 a2

=

(v1f − v10 ) / ∆t1

(v2f − v1f ) / ∆t2

=

(v1f − v10 )∆t2

(v2f − v1f )∆t1

Solving for v1f , we obtain v1f =

v1f =

a1∆t1v2f + a2 ∆t2v10 a1∆t1 + a2 ∆t2

=

(a1 / a2 )∆t1v2f + ∆t2v10 (a1 / a2 )∆t1 + ∆t2

1.50(12.0 s)(+28.0 m/s) + (6.0 s)( + 36.0 m/s) = +30.0 m/s 1.50(12.0 s) + 6.0 s

23. REASONING AND SOLUTION Both motorcycles have the same velocity v at the end of the four second interval. Now v = v0A + aAt for motorcycle A and v = v0B + aBt for motorcycle B. Subtraction of these equations and rearrangement gives v0A – v0B = (4.0 m/s2 – 2.0 m/s2)(4 s) =

+8.0 m/s

The positive result indicates that motorcycle A was initially traveling faster. ______________________________________________________________________________ 24. REASONING AND SOLUTION a = v / t , so

The average acceleration of the basketball player is

2 6.0 m/s x = 12 at 2 = 12 (1.5 s ) = 4.5 m 1.5 s ______________________________________________________________________________

25.

SSM REASONING AND SOLUTION

(

found by solving Equation 2.9 v

2

= v02

The average acceleration of the plane can be

)

+ 2ax for a. Taking the direction of motion as

positive, we have a=

v 2 − v02 2x

=

(+6.1 m/s) 2 − (+69 m/s)2 = – 3.1 m/s 2 2(+750 m)

56

KINEMATICS IN ONE DIMENSION

The minus sign indicates that the direction of the acceleration is opposite to the direction of motion, and the plane is slowing down. ______________________________________________________________________________ 26. REASONING The average acceleration is defined by Equation 2.4 as the change in velocity divided by the elapsed time. We can find the elapsed time from this relation because the acceleration and the change in velocity are given. SOLUTION a. The time ∆t that it takes for the VW Beetle to change its velocity by an amount ∆v = v – v0 is (and noting that 0.4470 m/s = 1 mi/h)

∆t =

v − v0

=

a

0.4470 m / s −0 m/s 1 mi / h

( 60.0 mi / h )

2.35 m / s 2

= 11.4 s

b. From Equation 2.4, the acceleration (in m/s2) of the dragster is 0.4470 m / s −0 m/s v − v0 1 mi / h a= = = 44.7 m / s 2 0.600 s − 0 s t − t0 ______________________________________________________________________________

( 60.0 mi / h )

27. REASONING

We know the initial and final velocities of the blood, as well as its

(

)

displacement. Therefore, Equation 2.9 v 2 = v02 + 2ax can be used to find the acceleration of the blood. The time it takes for the blood to reach it final velocity can be found by using x Equation 2.7 t = 1 . v + v ) 2( 0 SOLUTION a. The acceleration of the blood is

a=

v 2 − v02 2x

2 2 26 cm / s ) − ( 0 cm / s ) ( = 2 ( 2.0 cm )

= 1.7 × 102 cm / s 2

b. The time it takes for the blood, starting from 0 cm/s, to reach a final velocity of +26 cm/s is 2.0 cm x t= 1 =1 = 0.15 s 0 cm / s + 26 cm / s v + v ( ) ( ) 2 2 0 ______________________________________________________________________________

Chapter 2 Problems

57

28. REASONING AND SOLUTION a. From Equation 2.4, the definition of average acceleration, the magnitude of the average acceleration of the skier is a=

v − v0 t − t0

=

8.0 m/s – 0 m/s = 1.6 m/s 2 5.0 s

b. With x representing the displacement traveled along the slope, Equation 2.7 gives: x = 12 (v0 + v )t = 12 (8.0 m/s + 0 m/s)(5.0 s) = 2.0 × 101 m

______________________________________________________________________________ 29. SSM WWW REASONING AND SOLUTION a. The magnitude of the acceleration can be found from Equation 2.4 (v = v0 + at) as a=

v − v0 t

=

3.0 m/s – 0 m/s = 1.5 m/s 2 2.0 s

b. Similarly the magnitude of the acceleration of the car is a=

v − v0 t

=

41.0 m/s – 38.0 m/s = 1.5 m/s 2 2.0 s

c. Assuming that the acceleration is constant, the displacement covered by the car can be found from Equation 2.9 (v2 = v02 + 2ax):

x=

v 2 − v02 2a

=

(41.0 m/s)2 − (38.0 m/s)2 = 79 m 2(1.5 m/s 2 )

Similarly, the displacement traveled by the jogger is

x=

v 2 − v02 2a

=

(3.0 m/s) 2 − (0 m/s) 2 = 3.0 m 2(1.5 m/s 2 )

Therefore, the car travels 79 m – 3.0 m = 76 m further than the jogger. ______________________________________________________________________________

58

KINEMATICS IN ONE DIMENSION

30. REASONING The cheetah and its prey run the same distance. The prey runs at a constant velocity, so that its distance is the magnitude of its displacement, which is given by Equation 2.2 as the product of velocity and time. The distance for the cheetah can be expressed using Equation 2.8, since the cheetah’s initial velocity (zero, since it starts from rest) and the time are given, and we wish to determine the acceleration. The two expressions for the distance can be equated and solved for the acceleration. SOLUTION We begin by using Equation 2.2 and assuming that the initial position of the prey is x0 = 0 m. The distance run by the prey is ∆x = x − x0 = x = vPreyt

The distance run by the cheetah is given by Equation 2.8 as x = v0, Cheetah t + 12 aCheetah t 2

Equating the two expressions for x and using the fact that v0, Cheetah = 0 m/s, we find that

vPreyt = 12 aCheetah t 2 Solving for the acceleration gives aCheetah =

31.

2vPrey t

=

2 ( +9.0 m/s ) 3.0 s

= +6.0 m/s 2

SSM WWW REASONING Since the belt is moving with constant velocity, the displacement (x0 = 0 m) covered by the belt in a time tbelt is giving by Equation 2.2 (with x0 assumed to be zero) as x = vbelt tbelt

(1 )

Since Clifford moves with constant acceleration, the displacement covered by Clifford in a time tCliff is, from Equation 2.8, 2 2 x = v0tCliff + 12 atCliff = 12 atCliff

(2 )

The speed vbelt with which the belt of the ramp is moving can be found by eliminating x between Equations (1) and (2).

Chapter 2 Problems

59

SOLUTION Equating the right hand sides of Equations (1) and (2), and noting that tCliff = 14 tbelt , we have

vbelt tbelt = 12 a vbelt =

1 at 32 belt

( 14 tbelt )

2

1 (0.37 m/s 2 )(64 s) = 0.74 m/s = 32

______________________________________________________________________________ 32. REASONING At time t both rockets return to their starting points and have a displacement of zero. This occurs, because each rocket is decelerating during the first half of its journey. However, rocket A has a smaller initial velocity than rocket B. Therefore, in order for rocket B to decelerate and return to its point of origin in the same time as rocket A, rocket B must have a deceleration with a greater magnitude than that for rocket A. Since we know that the displacement of each rocket is zero at time t, since both initial velocities are given, and since we seek information about the acceleration, we begin our solution with Equation 2.8, for it contains just these variables. SOLUTION Applying Equation 2.8 to each rocket gives

xA = v0A t + 12 aA t 2

xB = v0Bt + 12 aBt 2

0 = v0A t + 12 aA t 2

0 = v0Bt + 12 aBt 2

0 = v0A + 12 aA t

0 = v0B + 12 aBt

t=

−2v0A

t=

aA

−2v0B aB

The time for each rocket is the same, so that we can equate the two expressions for t, with the result that −2v0A −2v0B v0A v0B or = = aA aB aA aB Solving for aB gives

aB =

aA v0A

v0B =

−15 m/s2 ( 8600 m/s ) = −22 m/s2 5800 m/s

As expected, the magnitude of the acceleration for rocket B is greater than that for rocket A.

60

KINEMATICS IN ONE DIMENSION

33. REASONING The definition of average velocity is given by Equation 2.2 as the displacement divided by the elapsed time. When the velocity is constant, as it is for car A, the average velocity is the same as the constant velocity. We note that, since both displacement and time are the same for each car, this equation gives the same value for car B’s average velocity and car A’s constant velocity. Since the acceleration of car B is constant, we know that its average velocity is given by Equation 2.6 as vB = 12 ( vB + vB0 ) , where vB is the final velocity and vB0 = 0 m/s is the initial velocity (car B starts from rest). Thus, we can use Equation 2.6 to find the final velocity. Car B’s constant acceleration can be calculated from Equation 2.4 (vB = vB0 + aBt), which is one of the equations of kinematics and gives the acceleration as [aB = (vB − vB0)/t]. Since car B starts from rest, we know that vB0 = 0 m/s. Furthermore, t is given. Therefore, calculation of the acceleration aB requires that we use the value calculated for the final velocity vB. SOLUTION a. According to Equation 2.2, the velocity of car A is the displacement L divided by the time t. Thus, we obtain L 460 m vA = = = 2.2 m/s t 210 s b. The average velocity of car B is given by Equation 2.6 as vB =

1 2

( vB + vB0 ) , where vB is

the final velocity and vB0 is the initial velocity. Solving for the final velocity and using the fact that car B starts from rest (vB0 = 0 m/s) gives vB = 2vB − vB0 = 2vB

(1)

As discussed in the REASONING, the average velocity of car B is equal to the constant velocity of car A. Substituting this result into Equation (1), we find that vB = 2vB = 2vA = 2 ( 2.2 m/s ) = 4.4 m/s c. Solving Equation 2.4 (vB = vB0 + aBt) for the acceleration shows that

aB =

vB − vB0 t

=

4.4 m/s − 0 m/s = 0.021 m/s 2 210 s

Chapter 2 Problems

61

34. REASONING The entering car maintains a constant acceleration of a1 = 6.0 m/s2 from the time it starts from rest in the pit area until it catches the other car, but it is convenient to separate its motion into two intervals. During the first interval, lasting t1 = 4.0 s, it accelerates from rest to the velocity v01 with which it enters the main speedway. This velocity is found from Equation 2.4 ( v = v0 + at ) , with v0 = 0 m/s, a = a1, t = t1, and v = v01: v10 = a1t1

(1)

The second interval begins when the entering car enters the main speedway with velocity v01, and ends when it catches up with the other car, which travels with a constant velocity v02 = 70.0 m/s. Since both cars begin and end the interval side-by-side, they both undergo the same displacement x during this interval. The displacement of each car is given by

(

)

Equation 2.8 x = v0t + 12 at 2 . For the accelerating car, v0 = v10, and a = a1, so x = v10t + 12 a1t 2

(2)

For the other car, v0 = v02 and a = 0 m/s2, and so Equation 2.8 yields x = v20t

(3)

SOLUTION The displacement during the second interval is not required, so equating the right hand sides of Equations (2) and (3) eliminates x, leaving an equation that may be solved for the elapsed time t, which is now the only unknown quantity: v10t + 12 a1t 2 = v20t v10 t + 12 a1t 2 = v20 t 1at 2 1

t=

(4)

= v20 − v10

2 ( v20 − v10 ) a1

Substituting Equation (1) for v10 into Equation (4), we find that

t=

2 ( v20 − a1t1 ) a1

(

2 70.0 m/s − 6.0 m/s 2 = 6.0 m/s 2

(

)

) ( 4.0 s ) = 15 s

62

KINEMATICS IN ONE DIMENSION

35. REASONING The drawing shows the two knights, initially separated by the displacement d, traveling toward each other. At any moment, Sir George’s displacement is xG and that of Sir Alfred is xA. When they meet, their displacements are the same, so xG = xA.

−

Starting point for Sir George

xG

+

Starting point for Sir Alfred

xA d

According to Equation 2.8, Sir George's displacement as a function of time is xG = v0,G t + 12 aG t 2 = ( 0 m/s ) t + 12 aG t 2 = 12 aG t 2

(1)

where we have used the fact that Sir George starts from rest ( v0,G = 0 m/s ) . Since Sir Alfred starts from rest at x = d at t = 0 s, we can write his displacement as (again, employing Equation 2.8) xA = d + v0,A t + 12 aA t 2 = d + ( 0 m/s ) t + 12 aA t 2 = d + 12 aA t 2 2

Solving Equation 1 for t

( t 2 = 2 xG / aG )

(2)

and substituting this expression into Equation 2

yields

2x xA = d + 12 aA G aG

xG = d + aA aG

Noting that xA = xG when the two riders collide, we see that Equation 3 becomes x xG = d + aA G aG Solving this equation for xG gives xG =

d . aA 1− aG

(3)

Chapter 2 Problems

(

63

)

SOLUTION Sir George’s acceleration is positive aG = +0.300 m/s2 since he starts from rest and moves to the right (the positive direction). Sir Alfred’s acceleration is negative

( aA = −0.200 m/s2 ) since he starts from rest and moves to the left (the negative direction).

The displacement of Sir George is, then, xG =

88.0 m d = = 52.8 m 2) aA ( − 0.200 m/s 1− 1− aG ( +0.300 m/s2 )

____________________________________________________________________________________________

36. REASONING At a constant velocity the time required for the first car to travel to the next exit is given by Equation 2.2 as the magnitude of the displacement (2.5 × 103 m) divided by the magnitude of the velocity. This is also the travel time for the second car to reach the next exit. The acceleration for the second car can be determined from Equation 2.8, since the initial velocity, the displacement, and the time are known. This equation applies, because the acceleration is constant. SOLUTION According to Equation 2.2, with the assumption that the initial time is t0 = 0 s, the time for the first car to reach the next exit at a constant velocity is ∆t = t − t 0 = t =

∆x 2.5 × 103 m = = 76 s v 33 m/s

Remembering that the initial velocity v0 of the second car is zero, we can solve Equation 2.8

( x = v0t + 12 at 2 = 12 at 2 ) for the acceleration to show that a=

2x t

2

=

2(2.5 × 103 m) (76 s)

Since the second car’s speed in the same direction as the velocity .

is

2

= 0.87 m/s 2

increasing,

this

acceleration

must

be

37. REASONING AND SOLUTION The speed of the car at the end of the first (402 m) phase can be obtained as follows: v12 = v02 + 2a1x1 v1 = 2 (17.0 m/s 2 ) ( 402 m )

64

KINEMATICS IN ONE DIMENSION

The speed after the second phase (3.50 × 102 m) can be obtained in a similar fashion. v22 = v022 + 2a2x2

v2 = v12 + 2 ( −6.10 m/s2 )( 3.50 ×102 m ) v2 = 96.9 m/s ______________________________________________________________________________ 38. REASONING AND SOLUTION a. The velocity at the end of the first (7.00 s) period is v1 = v0 + a1 = (2.01 m/s2)(7.00 s) At the end of the second period the velocity is v2 = v1 + a2t2 = v1 + (0.518 m/s2)(6.00 s) And the velocity at the end of the third (8.00 s) period is v3 = v2 + a3t3 = v2 + (–1.49 m/s2)(8.00 s) = +5.26 m/s b. The displacement for the first time period is found from x1 = v0t1 + x1 = (0 m/s)(7.00 s) +

1 (2.01 2

1a t 2 2 11

m/s2)(7.00 s)2 = +49.2 m

Similarly, x2 = +93.7 m and x3 = +89.7 m, so the total displacement of the boat is x = x1 + x2 + x3 =

+233 m

______________________________________________________________________________ 39. REASONING Because the car is traveling in the +x direction and decelerating, its acceleration is negative: a = −2.70 m/s2. The final velocity for the interval is given (v = +4.50 m/s), as well as the elapsed time (t = 3.00 s). Both the car’s displacement x and its initial velocity v0 at the instant braking begins are unknown.

Chapter 2 Problems

65

Compare the list of known kinematic quantities (v, a, t) to the equations of kinematics for constant acceleration: v = v0 + at (Equation 2.4), x = 12 ( v0 + v ) t (Equation 2.7), x = v0t + 12 at 2 (Equation 2.8), and v 2 = v02 + 2 ax (Equation 2.9). None of these four

equations contains all three known quantities and the desired displacement x, and each of them contains the initial velocity v0. Since the initial velocity is neither known nor requested, we can combine two kinematic equations to eliminate it, leaving an equation in which x is the only unknown quantity. SOLUTION For the first step, solve Equation 2.4 ( v = v0 + at ) for v0: v0 = v − at

(1)

(

)

Substituting the expression for v0 in Equation (1) into Equation 2.8 x = v0t + 12 at 2 yields an expression for the car’s displacement solely in terms of the known quantities v, a, and t: x = ( v − at ) t + 12 at 2 = vt − at 2 + 12 at 2 x = vt − 12 at 2

(2)

Substitute the known values of v, a, and t into Equation (2):

(

)

x = ( +4.50 m/s )( 3.00 s ) − 12 −2.70 m/s2 ( 3.00 s ) = +25.7 m 2

Note: Equation (2) can also be obtained by combining Equation (1) with Equation 2.7 x = 1 ( v0 + v ) t , or, with more effort, by combining Equation (1) with Equation 2.9 2

( v2 = v02 + 2ax ) .

40. REASONING AND SOLUTION As the plane decelerates through the intersection, it covers a total distance equal to the length of the plane plus the width of the intersection, so x = 59.7 m + 25.0 m = 84.7 m The speed of the plane as it enters the intersection can be found from Equation 2.9. Solving Equation 2.9 for v0 gives v0 = v 2 − 2ax = (45.0 m) 2 − 2(−5.70 m/s 2 )(84.7 m) = 54.7 m/s

66

KINEMATICS IN ONE DIMENSION

The time required to traverse the intersection can then be found from Equation 2.4. Solving Equation 2.4 for t gives v − v0

45.0 m/s − 54.7 m/s = 1.7 s a −5.70 m/s 2 ______________________________________________________________________________ t=

41.

SSM REASONING

=

As the train passes through the crossing, its motion is described by

1 Equations 2.4 (v = v0 + at) and 2.7 x = (v + v0 )t , which can be rearranged to give 2

v − v0 = at

v + v0 =

and

2x t

These can be solved simultaneously to obtain the speed v when the train reaches the end of the crossing. Once v is known, Equation 2.4 can be used to find the time required for the train to reach a speed of 32 m/s. SOLUTION

Adding the above equations and solving for v, we obtain

2x 2(20.0 m) v = 12 at + = 12 (1.6 m/s 2 )(2.4 s) + = 1.0 × 101 m/s 2.4 s t

The motion from the end of the crossing until the locomotive reaches a speed of 32 m/s requires a time v − v0 32 m/s − 1.0 ×101 m/s = = 14 s t= a 1.6 m/s 2 ______________________________________________________________________________ 42. REASONING Since 1 mile = 1609 m, a quarter-mile race is L = 402 m long. If a car crosses the finish line before reaching its maximum speed, then there is only one interval of constant acceleration to consider. We will first determine whether this is true by calculating the car’s displacement x1 while accelerating from rest to top speed from Equation 2.9

(v

2

)

= v02 + 2ax , with v0 = 0 m/s and v = vmax: 2 vmax

= ( 0 m/s ) + 2ax1 2

or

x1 =

2 vmax

2a

(1)

Chapter 2 Problems

67

If x1 > L, then the car crosses the finish line before reaching top speed, and the total time for

(

its race is found from Equation 2.8 x = v0t + 12 at

L = ( 0 m/s ) t + 12 at 2 = 12 at 2

2

) , with x = L and v

0

t=

or

2L a

= 0 m/s: (2)

On the other hand, if a car reaches its maximum speed before crossing the finish line, the race divides into two intervals, each with a different constant acceleration. The displacement x1 is found as given in Equation (1), but the time t1 to reach the maximum speed is most easily found from Equation 2.4 ( v = v0 + at ) , with v0 = 0 m/s and v = vmax: vmax = 0 m/s + at1

t1 =

or

vmax

(3)

a

The time t2 that elapses during the rest of the race is found by solving Equation 2.8

(x = v t + 0

1 at 2 2

).

Let x2 = L − x1 represent the displacement for this part of the race. With

the aid of Equation (1), this becomes x2 = L −

2 vmax

. Then, since the car is at its maximum 2a speed, the acceleration is a = 0 m/s2, and the displacement is 2

(

)

x2 = vmax t2 + 12 0 m/s 2 t22 = vmax t2

t2 =

or

x2 vmax

v L − max 2a = L − vmax (4) = vmax vmax 2a

Using this expression for t2 and Equation (3) for t1 gives the total time for a two-part race: t = t1 + t2 =

L v + − max v a 2a max

vmax

v L + max = 2a vmax

(5)

SOLUTION First, we use Equation (1) to determine whether either car finishes the race while accelerating:

vmax (106 m/s ) = 511 m = 2 2a 2 11.0 m/s 2

2

Car A

Car B

x1 = x1 =

(

2 vmax

2a

=

)

( 92.4 m/s )2

(

2 11.6 m/s

2

)

= 368 m

Therefore, car A finishes the race before reaching its maximum speed, but car B has 402 m − 368 m = 34 m to travel at its maximum speed. Equation (2) gives the time for car A to reach the finish line as

68

KINEMATICS IN ONE DIMENSION

t=

Car A

2 ( 402 m ) 2L = = 8.55 s 2 a 11.0 m/s

Equation (5) gives the time for car B to reach the finish line as

t=

Car B

L vmax

+

vmax 2a

=

402 m 92.4 m/s + = 8.33 s 92.4 m/s 2 11.6 m/s 2

(

)

Car B wins the race by 8.55 s − 8.33 s = 0.22 s .

43.

SSM REASONING AND SOLUTION a. Once the pebble has left the slingshot, it is subject only to the acceleration due to gravity.

Since the downward direction is negative, the acceleration of the pebble is

–9.80 m/s 2 .

The pebble is not decelerating. Since its velocity and acceleration both point downward, the magnitude of the pebble’s velocity is increasing, not decreasing. b. The displacement y traveled by the pebble as a function of the time t can be found from Equation 2.8. Using Equation 2.8, we have

y = v0t + 12 a y t 2 = (–9.0 m/s)(0.50 s) + 12 (–9.80 m/s 2 )(0.50 s)2 = –5.7 m Thus, after 0.50 s, the pebble is 5.7 m beneath the cliff-top. ______________________________________________________________________________ 44. REASONING Because there is no effect due to air resistance, the rock is in free fall from its launch until it hits the ground, so that the acceleration of the rock is always −9.8 m/s2, assuming upward to be the positive direction. In (a), we will consider the interval beginning at launch and ending 2.0 s later. In (b), we will consider the interval beginning at launch and ending 5.0 s later. Since the displacement isn’t required, Equation 2.4 ( v = v0 + at ) suffices to solve both parts of the problem. The stone slows down as it rises, so we expect the speed in (a) to be larger than 15 m/s. The speed in (b) could be smaller than 15 m/s (the rock does not reach its maximum height) or larger than 15 m/s (the rock reaches its maximum height and falls back down below its height at the 2.0-s point). SOLUTION a. For the interval from launch to t = 2.0 s, the final velocity is v = 15 m/s, the acceleration is a = −9.8 m/s2, and the initial velocity is to be found. Solving Equation 2.4 ( v = v0 + at )

for v0 gives

Chapter 2 Problems

69

v0 = v − at = 15 m/s − ( −9.8 m/s 2 )(2.0 s) = 35 m/s

Therefore, at launch, Speed = 35 m/s

b. Now we consider the interval from launch to t = 5.0 s. The initial velocity is that found in part (a), v0 = 35 m/s. The final velocity is v = v0 + at = 35 m/s + ( −9.8 m/s 2 )(5.0 s) = − 14 m/s

(2.4)

Instantaneous speed is the magnitude of the instantaneous velocity, so we drop the minus sign and find that Speed = 14 m/s

45. REASONING AND SOLUTION The figure at the right shows the paths taken by the pellets fired from gun A and gun B. The two paths differ by the extra distance covered by the pellet from gun A as it rises and falls back to the edge of the cliff. When it falls back to the edge of the cliff, the pellet from gun A will have the same speed as the pellet fired from gun B, as Conceptual Example 15 discusses. Therefore, the flight time of pellet A will be greater than that of B by the amount of time that it takes for pellet A to cover the extra distance.

A ____ B

The time required for pellet A to return to the cliff edge after being fired can be found from Equation 2.4: v = v0 + a t. If "up" is taken as the positive direction then v0 = +30.0 m/s and v = –30.0 m/s. Solving Equation 2.4 for t gives t =

v − v0 a

=

(−30.0 m/s) − (+30.0 m/s) = −9.80 m/s2

6.12 s

Notice that this result is independent of the height of the cliff. ______________________________________________________________________________ 46. REASONING Assuming that air resistance can be neglected, the acceleration is the same for both the upward and downward parts, namely −9.80 m/s2 (upward is the positive direction). Moreover, the displacement is y = 0 m, since the final and initial positions of the ball are the

70

KINEMATICS IN ONE DIMENSION

same.

The time is given as t = 8.0 s.

( y = v0 t +

1 a t2 2

Therefore, we may use Equation 2.8

) to find the initial velocity v0 of the ball. (

)

SOLUTION Solving Equation 2.8 y = v0 t + 12 a t 2 for the initial velocity v0 gives

v0 =

y − 12 a t 2

(

)

0 m − 12 −9.80 m/s2 ( 8.0 s )

2

= = +39 m/s t 8.0 s ______________________________________________________________________________ 47. REASONING AND SOLUTION In a time t the card will undergo a vertical displacement y given by y = 12 at2 where a = –9.80 m/s2. When t = 60.0 ms = 6.0 × 10–2 s, the displacement of the card is 0.018 m, and the distance is the magnitude of this value or d1 = 0.018 m . Similarly, when t = 120 ms, d 2 = 0.071 m , and when t = 180 ms, d3 = 0.16 m . ______________________________________________________________________________ 48. REASONING The minimum time that a player must wait before touching the basketball is the time required for the ball to reach its maximum height. The initial and final velocities are known, as well as the acceleration due to gravity, so Equation 2.4 ( v = v0 + a t ) can be used to find the time. SOLUTION Solving Equation 2.4 for the time yields

v − v0

0 m / s − 4.6 m / s = 0.47 s a −9.8 m / s 2 ______________________________________________________________________________

t=

49.

=

SSM REASONING AND SOLUTION Equation 2.8 can be used to determine the displacement that the ball covers as it falls halfway to the ground. Since the ball falls from rest, its initial velocity is zero. Taking down to be the negative direction, we have

y = v0t + 12 at 2 = 12 at 2 = 12 (–9.80 m/s 2 )(1.2 s) 2 = –7.1 m

Chapter 2 Problems

71

In falling all the way to the ground, the ball has a displacement of y = –14.2 m . Solving Equation 2.8 with this displacement then yields the time 2y 2(–14.2 m) = = 1.7 s a –9.80 m/s 2 ______________________________________________________________________________ t=

50. REASONING The initial velocity of the compass is +2.50 m/s. The initial position of the compass is 3.00 m and its final position is 0 m when it strikes the ground. The displacement of the compass is the final position minus the initial position, or y = –3.00 m. As the compass falls to the ground, its acceleration is the acceleration due to gravity, a = –9.80 m/s2. Equation 2.8 y = v0t + 1 a t 2 can be used to find how much time elapses

(

2

)

before the compass hits the ground. SOLUTION Starting with Equation 2.8, we use the quadratic equation to find the elapsed time.

t=

( 12 a ) ( − y ) = − ( 2.50 m/s ) ± ( 2.50 m/s )2 − 4 ( −4.90 m/s2 ) − ( −3.00 m ) 2 ( 12 a ) 2 ( −4.90 m/s 2 )

−v0 ± v02 − 4

There are two solutions to this quadratic equation, t1 = 1.08 s and t2 = –0.568 s. The second solution, being a negative time, is discarded. ______________________________________________________________________________ 51. REASONING AND SOLUTION a.

v 2 = v02 + 2ay

(

)

v = ± (1.8 m/s ) + 2 –9.80 m/s 2 ( –3.0 m ) = ±7.9 m/s 2

The minus is chosen, since the diver is now moving down. Hence, v = −7.9 m/s . b. The diver's velocity is zero at his highest point. The position of the diver relative to the board is y=–

v02 2a

=–

(1.8 m/s )2

(

2 –9.80 m/s 2

)

= 0.17 m

The position above the water is 3.0 m + 0.17 m = 3.2 m . ______________________________________________________________________________

72

KINEMATICS IN ONE DIMENSION

(

)

Equation 2.9 v 2 = v02 + 2ay can be used to find out how far above the

52. REASONING

cliff's edge the pellet would have gone if the gun had been fired straight upward, provided that we can determine the initial speed imparted to the pellet by the gun. This initial speed can be found by applying Equation 2.9 to the downward motion of the pellet described in the problem statement. SOLUTION If we assume that upward is the positive direction, the initial speed of the pellet is, from Equation 2.9, v0 = v 2 − 2ay = ( – 27 m/s) 2 − 2(–9.80 m/s 2 )(– 15 m) = 20.9 m/s

Equation 2.9 can again be used to find the maximum height of the pellet if it were fired straight up. At its maximum height, v = 0 m/s, and Equation 2.9 gives

−v02

−(20.9 m/s) 2 = 22 m 2a 2(–9.80 m/s2 ) ______________________________________________________________________________ y=

=

53. SSM REASONING AND SOLUTION Since the balloon is released from rest, its initial velocity is zero. The time required to fall through a vertical displacement y can be found from Equation 2.8

( y = v0t + 12 at 2 ) with v0 = 0 m/s.

Assuming upward to be the positive

direction, we find 2y 2(–6.0 m) = = 1.1 s a –9.80 m/s 2 ______________________________________________________________________________

t=

54. REASONING

(v

2

= v02

)

The initial speed of the ball can be determined from Equation 2.9

+ 2ay . Once the initial speed of the ball is known, Equation 2.9 can be used a

second time to determine the height above the launch point when the speed of the ball has decreased to one half of its initial value. SOLUTION When the ball has reached its maximum height, its velocity is zero. If we take upward as the positive direction, we have from Equation 2.9

v0 = v 2 − 2ay =

( 0 m/s )2 − 2(–9.80 m/s 2 )(12.0 m) = +15.3 m/s

When the speed of the ball has decreased to one half of its initial value, v = 12 v0 , and Equation 2.9 gives

Chapter 2 Problems

y=

v 2 − v02 2a

=

( 12 v0 ) 2 − v02 2a

73

v02 1 (+15.3 m/s)2 1 = − 1 = − 1 = 8.96 m 2a 4 2(–9.80 m/s 2 ) 4

55. REASONING The displacement y of the diver is equal to her average velocity v multiplied by the time t, or y = v t . Since the diver has a constant acceleration (the acceleration due to gravity), her average velocity is equal to v =

1 2

( v0 + v ) ,

where v0 and v

are, respectively, the initial and final velocities. Thus, according to Equation 2.7, the displacement of the diver is (2.7) y = 12 ( v0 + v ) t The final velocity and the time in this expression are known, but the initial velocity is not. To determine her velocity at the beginning of the 1.20-s period (her initial velocity), we turn to her acceleration. The acceleration is defined by Equation 2.4 as the change in her velocity, v − v0, divided by the elapsed time t: a = ( v − v0 ) / t . Solving this equation for the initial velocity v0 yields v0 = v − at

Substituting this relation for v0 into Equation 2.7, we obtain y=

1 2

( v0 + v ) t = 12 ( v − at + v ) t = vt − 12 at 2 2

SOLUTION The diver’s acceleration is that due to gravity, or a = −9.80 m/s . The acceleration is negative because it points downward, and this direction is the negative direction. The displacement of the diver during the last 1.20 s of the dive is

y = vt − 12 at 2 = ( −10.1 m/s )(1.20 s ) − 12 ( −9.80 m/s2 ) (1.20 s ) = −5.06 m 2

The displacement of the diver is negative because she is moving downward. ______________________________________________________________________________ 56. REASONING The ball is initially in free fall, then collides with the pavement and rebounds, which puts it into free fall again, until caught by the boy. We don’t have enough information to analyze its collision with the pavement, but we’re only asked to calculate the time it spends in the air, undergoing free-fall motion. The motion can be conveniently divided into three intervals: from release (h1 = 9.50 m) to impact, from impact to the second highest point (h2 = 5.70 m), and from the second highest point to h3 = 1.20 m above the pavement. For each of the intervals, the acceleration is that due to gravity. For the first and last interval, the ball’s initial velocity is zero, so the time to fall a given distance can be

(

found from Equation 2.8 y = v0t + 12 at

2

).

74

KINEMATICS IN ONE DIMENSION

The second interval begins at the pavement and ends at h2, so the initial velocity isn’t zero. However, the symmetry of free-fall motion is such that it takes the ball as much time to rise from the ground to a maximum height h2 as it would take for a ball dropped from h2 to fall to the pavement, so we can again use Equation 2.8 to find the duration of the second interval. SOLUTION Taking upward as the positive direction, we have a = −9.80 m/s2 for the acceleration in each of the three intervals. Furthermore, the initial velocity for each of the intervals is v0 = 0 m/s. Remember, we are using symmetry to treat the second interval as if the ball were dropped from rest at a height of 5.70 m and fell to the pavement. Using

(

Equation 2.8 y = v0t + 12 at

2

) , with v = 0 m/s, we can solve for the time to find that 0

2y a

t=

Applying this result to each interval gives the total time as

ttotal =

2 ( −9.50 m ) −9.80 m/s

2

+

st

1 interval

2 ( −5.70 m ) −9.80 m/s 2

nd

2

+

2 − ( 5.70 m − 1.20 m ) −9.80 m/s

2

= 3.43 s

rd

interval

3 interval

Note that the displacement y for each interval is negative, because upward has been designated as the positive direction.

57. REASONING To calculate the speed of the raft, it is necessary to determine the distance it travels and the time interval over which the motion occurs. The speed is the distance divided by the time, according to Equation 2.1. The distance is 7.00 m – 4.00 m = 3.00 m. The time is the time it takes for the stone to fall, which can be obtained from Equation 2.8 y = v0t + 12 a t 2 , since the displacement y, the initial velocity

(

)

v0, and the acceleration a are known. SOLUTION During the time t that it takes the stone to fall, the raft travels a distance of 7.00 m – 4.00 m = 3.00 m, and according to Equation 2.1, its speed is speed =

3.00 m t

Chapter 2 Problems

75

The stone falls downward for a distance of 75.0 m, so its displacement is y = –75.0 m, where the downward direction is taken to be the negative direction. Equation 2.8 can be used to find the time of fall. Setting v0 = 0 m/s, and solving Equation 2.8 for the time t, we have

t=

2 ( −75.0 m ) 2y = = 3.91 s a −9.80 m/s 2

Therefore, the speed of the raft is speed =

3.00 m = 0.767 m/s 3.91 s

58. REASONING AND SOLUTION a. We can use Equation 2.9 to obtain the speed acquired as she falls through the distance H. Taking downward as the positive direction, we find

v 2 = v02 + 2ay = ( 0 m/s ) + 2aH 2

or

v = 2aH

To acquire a speed of twice this value or 2 2aH , she must fall an additional distance H ′.

(

)

According to Equation 2.9 v 2 = v02 + 2ay , we have

(2

2aH

) =( 2

2aH

)

2

+ 2aH ′

or

4 ( 2aH ) = 2aH + 2aH ′

The acceleration due to gravity a can be eliminated algebraically from this result, giving

4H = H + H ′

or

H ′ = 3H

b. In the previous calculation the acceleration due to gravity was eliminated algebraically. Thus, a value other than 9.80 m/s2 would not have affected the answer to part (a) . ______________________________________________________________________________ 59.

SSM WWW REASONING AND SOLUTION The stone requires a time, t1 , to reach the bottom of the hole, a distance y below the ground. Assuming downward to be the positive direction, the variables are related by Equation 2.8 with v0 = 0 m/s:

y = 12 at12

(1)

76

KINEMATICS IN ONE DIMENSION

The sound travels the distance y from the bottom to the top of the hole in a time t2 . Since the sound does not experience any acceleration, the variables y and t2 are related by Equation 2.8 with a = 0 m/s2 and vsound denoting the speed of sound: y = vsound t2

(2)

Equating the right hand sides of Equations (1) and (2) and using the fact that the total elapsed time is t = t1 + t2 , we have 1 at 2 2 1

= vsound t2

or

1 at 2 2 1

= vsound (t − t1 )

Rearranging gives 1 at 2 2 1

+ vsound t1 – vsound t = 0

Substituting values and suppressing units for brevity, we obtain the following quadratic equation for t1 : 4.90t12 + 343t1 – 514 = 0 From the quadratic formula, we obtain

t1 =

–343 ± (343) 2 − 4(4.90)(–514) = 1.47 s or –71.5 s 2(4.90)

The negative time corresponds to a nonphysical result and is rejected. The depth of the hole is then found using Equation 2.8 with the value of t1 obtained above:

y = v0t1 + 12 at12 = ( 0 m/s )(1.47 s ) + 12 (9.80 m/s2 )(1.47 s)2 = 10.6 m ______________________________________________________________________________ 60. REASONING The stone that is thrown upward loses speed on the way up. The stone that is thrown downward gains speed on the way down. The stones cross paths below the point that corresponds to half the height of the cliff. To see why, consider where they would cross paths if they each maintained their initial speed as they moved. Then, they would cross paths exactly at the halfway point. However, the stone traveling upward begins immediately to lose speed, while the stone traveling downward immediately gains speed. Thus, the upward moving stone travels more slowly than the downward moving stone. Consequently, the stone thrown downward has traveled farther when it reaches the crossing point than the stone thrown upward.

Chapter 2 Problems

77

The initial velocity v0 is known for both stones, as is the acceleration a due to gravity. In addition, we know that at the crossing point the stones are at the same place at the same time t. Furthermore, the position of each stone is specified by its displacement y from its starting point. The equation of kinematics that relates the variables v0, a, t and y is

(

)

Equation 2.8 y = v0t + 12 at 2 , and we will use it in our solution. In using this equation, we will assume upward to be the positive direction. SOLUTION Applying Equation 2.8 to each stone, we have yup = v0up t + 12 at 2

ydown = v0down t + 12 at 2

and

Downward moving stone

Upward moving stone

In these expressions t is the time it takes for either stone to reach the crossing point, and a is the acceleration due to ydown gravity. Note that yup is the displacement of the upward H moving stone above the base of the cliff, ydown is the displacement of the downward moving stone below the yup top of the cliff, and H is the displacement of the cliff-top above the base of the cliff, as the drawing shows. The distances above and below the crossing point must add to equal the height of the cliff, so we have yup − ydown = H where the minus sign appears because the displacement ydown points in the negative direction. Substituting the two expressions for yup and ydown into this equation gives

(

)

v0up t + 12 at 2 − v0down t + 12 at 2 = H This equation can be solved for t to show that the travel time to the crossing point is t=

H v0up

− v0down

Substituting this result into the expression from Equation 2.8 for yup gives

78

KINEMATICS IN ONE DIMENSION

yup = v0up t

+

1 at 2 = v up 0 2

H up down v −v 0 0

H + 12 a up down v0 − v0

2

1 6.00 m 2 = ( 9.00 m/s ) + 2 −9.80 m/s 9.00 m/s − ( −9.00 m/s )

(

)

6.00 m 9.00 m/s − ( −9.00 m/s )

2

= 2.46 m

Thus, the crossing is located a distance of 2.46 m above the base of the cliff, which is below the halfway point of 3.00 m, as expected.

61.

SSM REASONING Once the man sees the block, the man must get out of the way in the time it takes for the block to fall through an additional 12.0 m. The velocity of the block at the instant that the man looks up can be determined from Equation 2.9. Once the velocity is known at that instant, Equation 2.8 can be used to find the time required for the block to fall through the additional distance.

SOLUTION When the man first notices the block, it is 14.0 m above the ground and its displacement from the starting point is y = 14.0 m – 53.0 m . Its velocity is given by

(

)

Equation 2.9 v 2 = v02 + 2ay . Since the block is moving down, its velocity has a negative

value, v = − v0 + 2ay = –

( 0 m/s )2 + 2(–9.80 m/s 2 )(14.0 m – 53.0 m) = –27.7 m/s

The block then falls the additional 12.0 m to the level of the man's head in a time t which satisfies Equation 2.8: y = v0t + 12 at 2 where y = –12.0 m and v0 = –27.7 m/s . Thus, t is the solution to the quadratic equation

4.90t 2 + 27.7t –12.0 = 0 where the units have been suppressed for brevity. From the quadratic formula, we obtain –27.7 ± (27.7) 2 − 4(4.90)(–12.0) t= = 0.40 s or –6.1 s 2(4.90) The negative solution can be rejected as nonphysical, and the time it takes for the block to reach the level of the man is 0.40 s . ______________________________________________________________________________

Chapter 2 Problems

79

62. REASONING Once its fuel is gone, the rocket is in free fall, so its motion consists of two intervals of constant but different acceleration. We will take upward as the positive direction. From launch to engine burn-out, the acceleration is a1 = +86.0 m/s2, and the rocket’s displacement is y1. Its velocity at the end of the burn, v1, is also the initial velocity for the second portion of its flight: engine burn-out to maximum altitude. During this second portion, the rocket slows down with the acceleration of gravity a2 = −9.80 m/s2 and undergoes an additional displacement of y2 in reaching its maximum height. Its maximum altitude is the sum of these two vertical displacements: h = y1 + y2. SOLUTION First we consider the time period t1 = 1.70 s from the ignition of the engine until the fuel is gone. The rocket accelerates from v0 = 0 m/s to v = v1, rising a displacement

(

y1, as given by Equation 2.8 y = v0t + 12 at

2

):

y1 = v0t1 + 12 a1t12 = ( 0 m/s ) t1 + 12 a1t12 = 12 a1t12

(1)

Equation 2.4 ( v = v0 + at ) gives its velocity v1 at the instant the fuel runs out: v1 = v0 + a1t1 = 0 m/s + a1t1 = a1t1

(2)

From that moment onward, the second part of the rocket’s motion is free fall (a2 = −9.80 m/s2). It takes a time t2 for the rocket’s velocity to decrease from v0 = v1 to

(

2

2

)

v2 = 0 m/s at its maximum altitude. We solve Equation 2.9 v = v0 + 2ay to find its upward displacement y2 during this time:

( 0 m/s ) = v12 + 2a2 y2 2

y2 =

or

−v12 2a2

Substituting for v1 from Equation (2), we find for y2 that

y2 =

− ( a1t1 ) 2a2

2

(3)

Using Equations (1) and (3), we find that the rocket’s maximum altitude, relative to the ground, is

h = y1 + y2 =

1 a t2 2 11

2 a1t1 ) a ( − = 1 a t2 1− 1

2a2

2 11

a2

80

KINEMATICS IN ONE DIMENSION

Using the values given, we find that

h=

63. REASONING

1 2

(

86.0 m/s

2

) (1.70 s )2 1 −

86.0 m/s 2 = 1210 m −9.80 m/s 2

To find the initial velocity v0,2 of the second stone, we will employ

Equation 2.8, y = v0,2t2 +

1 2

at2 2 . In this expression t2 is the time that the second stone is in

the air, and it is equal to the time t1 that the first stone is in the air minus the time t3.20 it takes for the first stone to fall 3.20 m: t2 = t1 − t3.20

We can find t1 and t3.20 by applying Equation 2.8 to the first stone. SOLUTION To find the initial velocity v0,2 of the second stone, we employ Equation 2.8, y = v0,2t2 + 12 at2 2 . Solving this equation for v0,2 yields

v0,2 =

y − 12 at2 2 t2

The time t1 for the first stone to strike the ground can be obtained from Equation 2.8, y = v0,1t1 +

1 2

at12 . Noting that v0,1 = 0 m/s since the stone is dropped from rest and solving

this equation for t1, we have t1 =

2y = a

2 ( −15.0 m ) −9.80 m/s 2

= 1.75 s

(1)

Note that the stone is falling down, so its displacement is negative (y = − 15.0 m). Also, its 2 acceleration a is that due to gravity, so a = −9.80 m/s . The time t3.20 for the first stone to fall 3.20 m can also be obtained from Equation 1: t3.20 =

2y = a

2 ( −3.20 m ) −9.80 m/s 2

The time t2 that the second stone is in the air is

= 0.808 s

Chapter 2 Problems

81

t2 = t1 − t3.20 = 1.75 s − 0.808 s = 0.94 s

The initial velocity of the second stone is

v0,2 =

y − 12 at2 2

( −15.0 m ) − 1 ( −9.80 m/s2 ) ( 0.94 s )2

2 = = −11 m/s t2 0.94 s ______________________________________________________________________________

64. REASONING AND SOLUTION We measure the positions of the balloon and the pellet relative to the ground and assume up to be positive. The balloon has no acceleration, since it travels at a constant velocity vB, so its displacement in time t is vBt. Its position above the ground, therefore, is yB = H 0 + vBt where H0 = 12 m. The pellet moves under the influence of gravity (a = –9.80 m/s2), so its position above the ground is given by Equation 2.8 as

yP = v0t + 12 at 2 But yP = yB at time t, so that v0t + 12 at 2 = H 0 + vBt

Rearranging this result and suppressing the units gives 1 at 2 2

+ ( v0 – vB ) t – H 0 =

1 2

( –9.80 ) t 2 + ( 30.0 – 7.0 ) t –12.0 = 0

4.90t 2 – 23.0t + 12.0 = 0 t=

23.0 ± 23.02 – 4 ( 4.90 )(12.0 ) 2 ( 4.90 )

= 4.09 s or 0.602 s

Substituting each of these values in the expression for yB gives

yB = 12.0 m + ( 7.0 m/s )( 4.09 s ) = 41 m yB = 12.0 m + ( 7.0 m/s )( 0.602 s ) = 16 m ______________________________________________________________________________

82

KINEMATICS IN ONE DIMENSION

65. REASONING AND SOLUTION The average acceleration for each segment is the slope of that segment. 40 m/s − 0 m/s = 1.9 m/s 2 aA = 21 s − 0 s aB =

40 m/s − 40 m/s = 0 m/s 2 48 s − 21 s

80 m/s − 40 m/s = 3.3 m/s 2 60 s − 48 s ______________________________________________________________________________ aC =

66. REASONING On a position-versus-time graph, the velocity is the slope. Since the object’s velocity is constant and it moves in the +x direction, the graph will be a straight line with a positive slope, beginning at x = −16 m when t = 0 s. At t = 18 s, its position should be x = −16 m + 48 m = +32 m. Once the graph is constructed, the object’s velocity is found by ∆x calculating the slope of the graph: v = . ∆t SOLUTION The position-versus-time graph for the motion is as follows: 40

Position x (m)

32 24 16 8.0 0.0 −8.0 −16 −24 0

3.0

6.0

9.0

12

15

18

Time t (s) The object’s displacement is +48 m, and the elapsed time is 18 s, so its velocity is v=

67.

∆x +48 m = = +2.7 m/s ∆t 18 s

SSM REASONING The slope of a straight-line segment in a position-versus-time graph is the average velocity. The algebraic sign of the average velocity, therefore, corresponds to the sign of the slope.

Chapter 2 Problems

83

SOLUTION a. The slope, and hence the average velocity, is positive for segments A and C, negative for segment B, and zero for segment D. b. In the given position-versus-time graph, we find the slopes of the four straight-line segments to be 1.25 km − 0 km vA = = +6.3 km/h 0.20 h − 0 h vB =

0.50 km − 1.25 km = −3.8 km/h 0.40 h − 0.20 h

vC =

0.75 km − 0.50 km = +0.63 km/h 0.80 h − 0.40 h

0.75 km − 0.75 km = 0 km/h 1.00 h − 0.80 h ______________________________________________________________________________ vD =

68. REASONING segment.

The average velocity for each segment is the slope of the line for that

SOLUTION Taking the direction of motion as positive, we have from the graph for segments A, B, and C, 10.0 km – 40.0 km vA = = –2.0 × 101 km/h 1.5 h – 0.0 h vB =

20.0 km – 10.0 km = 1.0 × 101 km/h 2.5 h – 1.5 h

40.0 km – 20.0 km = 40 km/h 3.0 h – 2.5 h ______________________________________________________________________________ vC =

69. REASONING The slope of the position-time graph is the velocity of the bus. Each of the three segments of the graph is a straight line, so the bus has a different constant velocity for each part of the trip: vA, vB, and vC . The slope of each segment may be calculated from ∆x Equation 2.2 v = , where x is the difference between the final and initial positions of ∆t the bus and t is the elapsed time during each segment. The average acceleration of the bus v − v0 is the change in its velocity divided by the elapsed time, as in Equation 2.4 a = . ∆t The trip lasts from t = 0 h (the initial instant on the graph) to t = 3.5 h (the final instant on the graph), so the total elapsed time is t = 3.5 h. The initial velocity of the bus is its

84

KINEMATICS IN ONE DIMENSION

velocity at t = 0, which is its constant velocity for segment A: v0 = vA. Similarly, the velocity of the bus at the last instant of segment C is its final velocity for the trip: v = vC. ∆x SOLUTION In using Equation 2.2 v = to calculate the slopes of segments A and C, ∆t any displacement x within a segment may be chosen, so long as the corresponding elapsed time t is used in the calculation. If the full displacements for each segment are chosen, then

∆x A 24 km − 0 km = = 24 km/h ∆t A 1.0 h − 0 h

vA = vC =

∆xC ∆tC

=

27 km − 33 km = −5 km/h 3.5 h − 2.2 h

Apply these results to Equation 2.4:

a=

70. REASONING

(v

2

)

v − v0 ∆t

=

( −5 km/h ) − ( 24 km/h ) = 3.5 h

−8.3 km/h 2

The car’s initial velocity v0 may be calculated from Equation 2.9

= v02 + 2ax , since we know the final velocity v = +4.0 m/s, the displacement

x = +120 m, and the acceleration a = −3.2 m/s2. The acceleration is negative for the following reason: the car is slowing down, so the acceleration points opposite to the velocity, which is positive. To construct the graph, we need to know the elapsed time t, which may be found from the initial and final velocities and Equation 2.4 ( v = v0 + at ) . Because the car’s acceleration is constant and negative, it should be a straight line with a negative slope. SOLUTION

(

2

2

)

a. Solving Equation 2.9 v = v0 + 2ax for the initial velocity gives v02 = v 2 − 2ax v0 = + v 2 − 2ax = +

( 4.0 m/s )2 − 2 ( −3.2 m/s )(120 m ) =

+28 m/s

The positive root is appropriate, since the car is traveling in the +x direction.

Chapter 2 Problems

85

b. We know that at t = 0 s, the car’s velocity is +28 m/s. Equation 2.4 ( v = v0 + at ) can be solved to give the time necessary to slow down to +4.0 m/s: t=

v − v0 a

=

4.0 m/s − 28 m/s −3.2 m/s 2

= 7.5 s

Therefore, the graph should run from t = 0 s to t = 7.5 s: 32

Velocity v (m/s)

28 24 20 16 12 8.0 4.0 0 0

1.5

3.0

4.5

6.0

7.5

Time t (s)

71.

SSM REASONING The two runners start one hundred meters apart and run toward each other. Each runs ten meters during the first second and, during each second thereafter, each runner runs ninety percent of the distance he ran in the previous second. While the velocity of each runner changes from second to second, it remains constant during any one second.

SOLUTION The following table shows the distance covered during each second for one of the runners, and the position at the end of each second (assuming that he begins at the origin) for the first eight seconds. Time t (s)

Distance covered (m)

Position x (m)

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

10.00 9.00 8.10 7.29 6.56 5.90 5.31 4.78

0.00 10.00 19.00 27.10 34.39 40.95 46.85 52.16 56.94

86

KINEMATICS IN ONE DIMENSION

The following graph is the position-time graph constructed from the data in the table above. 60

50

40

30

20

10

0 0

2

4

6

8

10

Time t (s)

a. Since the two runners are running toward each other in exactly the same way, they will meet halfway between their respective starting points. That is, they will meet at x = 50.0 m. According to the graph, therefore, this position corresponds to a time of 6.6 s . b. Since the runners collide during the seventh second, the speed at the instant of collision can be found by taking the slope of the position-time graph for the seventh second. The speed of either runner in the interval from t = 6.00 s to t = 7.00 s is v=

∆x 52.16 m – 46.85 m = = 5.3 m/s ∆t 7.00 s – 6.00 s

Therefore, at the moment of collision, the speed of either runner is 5.3 m/s . ______________________________________________________________________________ 72. REASONING AND SOLUTION Since v = v0 + at, the acceleration is given by a = ( v – v0 ) / t . Since the direction of travel is in the negative direction throughout the

problem, all velocities will be negative. a.

a=

( − 29.0 m/s) − ( − 27.0 m/s) = −0.40 m/s 2 5.0 s

Since the acceleration is negative, it is in the same direction as the velocity and the car is speeding up.

Chapter 2 Problems

b.

a=

87

( − 23.0 m/s) − ( − 27.0 m/s) = +0.80 m/s 2 5.0 s

Since the acceleration is positive, it is in the opposite direction to the velocity and the car is slowing down or decelerating. ______________________________________________________________________________ 73.

SSM REASONING AND SOLUTION When air resistance is neglected, free fall conditions are applicable. The final speed can be found from Equation 2.9;

v 2 = v02 + 2ay where v0 is zero since the stunt man falls from rest. If the origin is chosen at the top of the hotel and the upward direction is positive, then the displacement is y = –99.4 m. Solving for v, we have

v = − 2ay = – 2(–9.80 m/s 2 )(–99.4 m) = – 44.1 m/s The speed at impact is the magnitude of this result or 44.1 m/s . ______________________________________________________________________________ 74. REASONING AND SOLUTION In 12 minutes the sloth travels a distance of 60 s xs = vs t = (0.037 m/s)(12 min) = 27 m 1 min

while the tortoise travels a distance of 60 s xt = vt t = (0.076 m/s)(12 min) = 55 m 1 min

The tortoise goes farther than the sloth by an amount that equals 55 m – 27 m = 28 m ______________________________________________________________________________ 75.

SSM REASONING The cart has an initial velocity of v0 = +5.0 m/s, so initially it is moving to the right, which is the positive direction. It eventually reaches a point where the displacement is x = +12.5 m, and it begins to move to the left. This must mean that the cart comes to a momentary halt at this point (final velocity is v = 0 m/s), before beginning to move to the left. In other words, the cart is decelerating, and its acceleration must point opposite to the velocity, or to the left. Thus, the acceleration is negative. Since the initial

88

KINEMATICS IN ONE DIMENSION

(

velocity, the final velocity, and the displacement are known, Equation 2.9 v 2 = v02 + 2ax

)

can be used to determine the acceleration. SOLUTION Solving Equation 2.9 for the acceleration a shows that

a=

v 2 − v02 2x

=

( 0 m/s )2 − ( +5.0 m/s )2 2 ( +12.5 m )

= −1.0 m/s2

76. REASONING The initial velocity and the elapsed time are given in the problem. Since the rock returns to the same place from which it was thrown, its displacement is zero (y = 0 m). Using this information, we can employ Equation 2.8 y = v0t + 1 a t 2 to determine the

(

2

)

acceleration a due to gravity. SOLUTION Solving Equation 2.8 for the acceleration yields 2 ( y − v0 t )

2 0 m − ( +15 m / s )( 20.0 s ) = = −1.5 m / s 2 2 t ( 20.0 s ) ______________________________________________________________________________ a=

2

77. REASONING Since the average speed of the impulse is equal to the distance it travels divided by the elapsed time (see Equation 2.1), the elapsed time is just the distance divided by the average speed.

SOLUTION The time it takes for the impulse to travel from the foot to the brain is

Distance 1.8 m (2.1) = = 1.6 ×10−2 s Average speed 1.1×102 m/s ______________________________________________________________________________ Time =

78. REASONING Since the velocity and acceleration of the motorcycle point in the same direction, their numerical values will have the same algebraic sign. For convenience, we will choose them to be positive. The velocity, acceleration, and the time are related by Equation 2.4: v = v0 + at . SOLUTION

a. Solving Equation 2.4 for t we have t=

v − v0 a

=

(+31 m/s) – (+21 m/s) = 4.0 s +2.5 m/s 2

89

Chapter 2 Problems

b. Similarly,

v − v0

(+61 m/s) – (+51 m/s) = 4.0 s a +2.5 m/s 2 ______________________________________________________________________________ t=

=

79. REASONING At a constant velocity the time required for Secretariat to run the final mile is given by Equation 2.2 as the displacement (+1609 m) divided by the velocity. The actual time required for Secretariat to run the final mile can be determined from Equation 2.8, since the initial velocity, the acceleration, and the displacement are given. It is the difference between these two results for the time that we seek. SOLUTION According to Equation 2.2, with the assumption that the initial time is t0 = 0 s, the run time at a constant velocity is ∆t = t − t 0 = t =

(

∆x +1609 m = = 97.04 s v +16.58 m/s

)

Solving Equation 2.8 x = v0t + 12 at 2 for the time shows that

t=

=

( 12 a ) ( − x ) 2 ( 12 a )

−v0 ± v02 − 4

−16.58 m/s ±

( +16.58 m/s )2 − 4 ( 12 ) ( +0.0105 m/s2 ) ( −1609 m ) 2

( 12 ) ( +0.0105 m/s2 )

= 94.2 s

We have ignored the negative root as being unphysical. The acceleration allowed Secretariat to run the last mile in a time that was faster by 97.04 s − 94.2 s = 2.8 s

80. REASONING AND SOLUTION The distance covered by the cab driver during the two phases of the trip must satisfy the relation x1 + x2 = 2.00 km

(1)

where x1 and x2 are the displacements of the acceleration and deceleration phases of the trip,

(

)

respectively. The quantities x1 and x2 can be calculated from Equation 2.9 v 2 = v02 + 2ax :

90

KINEMATICS IN ONE DIMENSION

x1 =

v12 − ( 0 m/s ) 2a1

2

=

v12

and

2a1

x2

2 2 0 m/s ) − v02 v2 ( = = − 02

2a2

2a2

with v02 = v1 and a2 = −3a1 . Thus, x1 x2

so that

=

v12 /(2a1 ) −v12 /(−6a1 )

=3

x1 = 3 x2

(2)

Combining (1) and (2), we have, 3 x2 + x2 = 2.00 km

Therefore, x2 = 0.50 km , and from Equation (1), x1 = 1.50 km . Thus, the length of the acceleration phase of the trip is x1 = 1.50 km , while the length of the deceleration phase is x2 = 0.50 km . ______________________________________________________________________________ 81.

SSM REASONING AND SOLUTION a. The total displacement traveled by the bicyclist for the entire trip is equal to the sum of the displacements traveled during each part of the trip. The displacement traveled during each part of the trip is given by Equation 2.2: ∆x = v ∆t . Therefore,

60 s ∆x1 = (7.2 m/s)(22 min) = 9500 m 1 min 60 s ∆x2 = (5.1 m/s)(36 min) = 11 000 m 1 min 60 s ∆x3 = (13 m/s)(8.0 min) = 6200 m 1 min

The total displacement traveled by the bicyclist during the entire trip is then ∆x = 9500 m + 11 000 m + 6200 m =

2.67 × 104 m

Chapter 2 Problems

91

b. The average velocity can be found from Equation 2.2. ∆x 2.67 × 104 m 1min = = 6.74 m/s, due north ∆t ( 22 min + 36 min + 8.0 min ) 60 s ______________________________________________________________________________ v=

82. REASONING The time ttrip to make the entire trip is equal to the time tcart that the golfer rides in the golf cart plus the time twalk that she walks; ttrip = tcart + twalk. Therefore, the time that she walks is twalk = ttrip − tcart (1) The average speed vtrip for the entire trip is equal to the total distance, xcart + xwalk, she travels divided by the time to make the entire trip (see Equation 2.1);

vtrip =

xcart + xwalk ttrip

Solving this equation for ttrip and substituting the resulting expression into Equation 1 yields

twalk =

xcart + xwalk vtrip

− tcart

(2)

The distance traveled by the cart is xcart = vcart tcart , and the distance walked by the golfer is xwalk = vwalk t walk . Substituting these expressions for xcart and xwalk into Equation 2 gives

twalk =

vcart tcart + vwalk twalk vtrip

− tcart

The unknown variable twalk appears on both sides of this equation. Algebraically solving for this variable gives vcart tcart − vtriptcart twalk = vtrip − vwalk SOLUTION The time that the golfer spends walking is

twalk =

vcart tcart − vtriptcart vtrip − vwalk

=

( 3.10 m/s ) ( 28.0 s ) − (1.80 m/s )( 28.0 s ) = 73 s (1.80 m/s ) − (1.30 m/s )

______________________________________________________________________________

92

83.

KINEMATICS IN ONE DIMENSION

SSM REASONING AND SOLUTION The stone will reach the water (and hence the

log) after falling for a time t, where t can be determined from Equation 2.8: y = v0t + 12 at 2 .

Since the stone is dropped from rest, v0 = 0 m/s. Assuming that downward is positive and solving for t, we have 2y 2(75 m) = = 3.9 s t= a 9.80 m/s 2 During that time, the displacement of the log can be found from Equation 2.8. Since the log moves with constant velocity, a = 0 m/s2, and v0 is equal to the velocity of the log. x = v0t = (5.0 m/s)(3.9 s) = 2.0 ×101 m Therefore, the horizontal distance between the log and the bridge when the stone is released is 2.0 × 101 m . ______________________________________________________________________________ 84. REASONING The stopping distance is the sum of two parts. First, there is the distance the car travels at 20.0 m/s before the brakes are applied. According to Equation 2.2, this distance is the magnitude of the displacement and is the magnitude of the velocity times the time. Second, there is the distance the car travels while it decelerates as the brakes are applied. This distance is given by Equation 2.9, since the initial velocity, the acceleration, and the final velocity (0 m/s when the car comes to a stop) are given.

SOLUTION With the assumption that the initial position of the car is x0 = 0 m, Equation 2.2 gives the first contribution to the stopping distance as

∆x1 = x1 = vt1 = ( 20.0 m/s )( 0.530 s ) Solving Equation 2.9

( v2 = v02 + 2ax )

for x shows that the second part of the stopping

distance is x2 =

v 2 − v02 2a

=

( 0 m/s )2 − ( 20.0 m/s )2

(

2 −7.00 m/s 2

)

Here, the acceleration is assigned a negative value, because we have assumed that the car is traveling in the positive direction, and it is decelerating. Since it is decelerating, its acceleration points opposite to its velocity. The stopping distance, then, is

Chapter 2 Problems

xStopping = x1 + x2 = ( 20.0 m/s )( 0.530 s )

2 2 0 m/s ) − ( 20.0 m/s ) ( +

(

2 −7.00 m/s 2

)

93

= 39.2 m

85. REASONING We choose due north as the positive direction. Our solution is based on the fact that when the police car catches up, both cars will have the same displacement, relative to the point where the speeder passed the police car. The displacement of the speeder can be obtained from the definition of average velocity given in Equation 2.2, since the speeder is moving at a constant velocity. During the 0.800-s reaction time of the policeman, the police car is also moving at a constant velocity. Once the police car begins to accelerate, its displacement can be expressed as in Equation 2.8

( x = v0t + 12 at 2 ) , because the initial

velocity v0 and the acceleration a are known and it is the time t that we seek. We will set the displacements of the speeder and the police car equal and solve the resulting equation for the time t. SOLUTION Let t equal the time during the accelerated motion of the police car. Relative to the point where he passed the police car, the speeder then travels a time of t + 0.800 s before the police car catches up. During this time, according to the definition of average velocity given in Equation 2.2, his displacement is xSpeeder = vSpeeder ( t + 0.800 s ) = ( 42.0 m/s )( t + 0.800 s )

The displacement of the police car consists of two contributions, the part due to the constant-velocity motion during the reaction time and the part due to the accelerated motion. Using Equation 2.2 for the contribution from the constant-velocity motion and Equation 2.9 for the contribution from the accelerated motion, we obtain

xPolice car = v0, Police car ( 0.800 s ) + v0, Police car t + 12 at 2 Constant velocity motion, Equation 2.2

Accelerated motion, Equation 2.8

(

)

= (18.0 m/s )( 0.800 s ) + (18.0 m/s ) t + 12 5.00 m/s 2 t 2 Setting the two displacements equal we obtain

( 42.0 m/s )( t + 0.800 s ) = (18.0 m/s )( 0.800 s ) + (18.0 m/s ) t + 12 ( 5.00 m/s2 ) t 2 Displacement of speeder

Displacement of police car

Rearranging and combining terms gives this result in the standard form of a quadratic equation:

94

KINEMATICS IN ONE DIMENSION

( 2.50 m/s2 ) t 2 − ( 24.0 m/s ) t − 19.2 m = 0 Solving for t shows that

t=

− ( −24.0 m/s ) ±

( −24.0 m/s )2 − 4 ( 2.50 m/s2 ) ( −19.2 m )

(

2 2.50 m/s 2

)

= 10.3 s

We have ignored the negative root, because it leads to a negative value for the time, which is unphysical. The total time for the police car to catch up, including the reaction time, is 0.800 s + 10.3 s = 11.1 s

86. REASONING AND SOLUTION The balls pass at a time t when both are at a position y above the ground. Applying Equation 2.8 to the ball that is dropped from rest, we have

y = 24 m + v01t + 12 at 2 = 24 m + ( 0 m/s ) t + 12 at 2

(1)

Note that we have taken into account the fact that y = 24 m when t = 0 s in Equation (1). For the second ball that is thrown straight upward, y = v02t + 12 at 2

(2)

Equating Equations (1) and (2) for y yields 24 m + 12 at 2 = v02t + 12 at 2

or

24 m = v02t

Thus, the two balls pass at a time t, where

t=

24 m v02

The initial speed v02 of the second ball is exactly the same as that with which the first ball hits the ground. To find the speed with which the first ball hits the ground, we take upward

(

)

as the positive direction and use Equation 2.9 v 2 = v02 + 2ay . Since the first ball is dropped

from rest, we find that v02 = v = 2ay = 2(–9.80 m/s 2 )(–24 m) = 21.7 m/s

Chapter 2 Problems

95

Thus, the balls pass after a time t=

24 m = 1.11 s 21.7 m/s

At a time t = 1.11 s, the position of the first ball according to Equation (1) is 1 2

y = 24 m + (–9.80 m/s 2 )(1.11 s) 2 = 24 m – 6.0 m

which is 6.0 m below the top of the cliff . ______________________________________________________________________________ 87.

SSM REASONING Since the car is moving with a constant velocity, the displacement

of the car in a time t can be found from Equation 2.8 with a = 0 m/s2 and v0 equal to the velocity of the car: xcar = vcar t . Since the train starts from rest with a constant acceleration, the displacement of the train in a time t is given by Equation 2.8 with v0 = 0 m/s:

xtrain = 12 atrain t 2 At a time t1 , when the car just reaches the front of the train, xcar = Ltrain + xtrain , where Ltrain is the length of the train. Thus, at time t1 ,

vcar t1 = Ltrain + 12 atrain t12

(1)

At a time t2, when the car is again at the rear of the train, xcar = xtrain . Thus, at time t2

vcar t2 = 12 atrain t22

(2)

Equations (1) and (2) can be solved simultaneously for the speed of the car vcar and the acceleration of the train atrain . SOLUTION a. Solving Equation (2) for atrain we have atrain =

2vcar t2

Substituting this expression for atrain into Equation (1) and solving for vcar , we have

(3)

96

KINEMATICS IN ONE DIMENSION

vcar =

Ltrain t t1 1 − 1 t2

=

92 m = 13 m/s 14 s (14 s) 1 − 28 s

b. Direct substitution into Equation (3) gives the acceleration of the train: atrain =

2vcar

=

2 (13 m/s) = 0.93 m/s 2 28 s

t2 ______________________________________________________________________________

88. REASONING AND SOLUTION During the first phase of the acceleration,

a1 =

v t1

During the second phase of the acceleration, v = (3.4 m/s) – (1.1 m/s2)(1.2 s) = 2.1 m/s Then 2.1 m/s = 1.4 m/s 2 1.5 s ______________________________________________________________________________ a1 =

CHAPTER 3 KINEMATICS IN

TWO DIMENSIONS ANSWERS TO FOCUS ON CONCEPTS QUESTIONS 1.

(a) The horizontal component vx of the projectile’s velocity remains constant throughout the motion, since the acceleration ax in the horizontal direction is zero (ax = 0 m/s2). The vertical component vy, however, changes as the projectile moves. This component is greatest at point 1, decreases to zero at point 2 at the top of the trajectory, and then increases to a magnitude less than that at point 1 as the projectile approaches point 3.

2.

(b) The minimum speed of the projectile occurs when it is at the top of its trajectory. At this point the vertical component of its velocity is zero (vy = 0 m/s). Since there is no 2

acceleration in the x direction (ax = 0 m/s ), the x component of the projectile’s velocity remains constant at vx = +30 m/s throughout the motion. Thus, the minimum speed is 30 m/s. 3.

(c) The acceleration due to gravity is the same for both balls, despite the fact that they have different velocities.

4.

(d) The acceleration of a projectile is the same at all points on the trajectory. It points 2 downward, toward the earth, and has a magnitude of 9.80 m/s .

5.

(c) Since there is no acceleration in the x direction (ax = 0 m/s ) for projectile motion, the x component vx of the velocity is constant throughout the motion. And, the acceleration due to

2

2

gravity, ay = −9.80 m/s (“downward” is the negative direction), also remains constant. 6.

(c) The time for a projectile to reach the ground depends only on the y component (or vertical component) of its variables, i.e., y, v0y, and ay. These variables are the same for both balls. The fact that Ball 1 is moving horizontally at the top of its trajectory does not play a role in the time it takes for it to reach the ground.

7.

(a) Using y = −19.6 m, ay = −9.80 m/s , and v0y = 0 m/s, Equation 3.5b y = v0 y t + 12 a y t 2

2

(

can be used to calculate the time t. 8.

(b) The time a projectile is in the air is equal to twice the time it takes to fall from its maximum height. Both have the same maximum height, so the time of fall is the same. Therefore, both projectiles are in the air for the same amount of time.

)

98 KINEMATICS IN TWO DIMENSIONS

9.

(a) The time a projectile is in the air is equal to twice the time it takes to fall from its maximum height. Projectile 1 reaches the greater height, so it spends the greater amount of time in the air.

10. (c) The vertical component (or y component) of the final velocity depends on the y components of the kinematic variables ( y, v0y, and ay) and the time t. These variables are the same for both balls, so they have the same vertical component of the velocity. 11. (a) A person standing on the ground sees a car traveling at +25 m/s. When the driver of the car looks out the window, she sees the person moving in the opposite direction with the same speed, or with a velocity of −25 m/s. 12. (d) The velocity vBC of the bus relative to the car is given by the relation vBC = vBG + vGC = vBG + (−vCG) = +16 m/s + (−12 m/s) = +4 m/s. 13. (d) This answer is arrived at by using the relation vPG = vPB + vBG = +2 m/s + 16 m/s = +18 m/s. 14. (c) The velocity vPC of the passenger relative to the car is given by vPC = vPB + vBC, according to the subscripting method discussed in Section 3.4. However, the last term on the right of this equation is given by vBC = vBG + vGC. So, vPC = vPB + vBG + vGC = +2 m/s + 16 m/s + (−12 m/s) = +6 m/s. 15. (b) The velocity of the jeep relative to you is zero. Thus, the horizontal component of the tire’s velocity relative to you is also zero. Since this component of the velocity never changes as the tire falls, the car cannot hit the tire, regardless of how close the car is to the jeep. 16. The magnitude vAB of the velocity of car A relative to car B is vAB = 34.2 m/s. The angle that the velocity vAB makes with respect to due east is θ = 37.9° south of east.

Chapter 3 Problems

99

CHAPTER 3 KINEMATICS IN

TWO DIMENSIONS PROBLEMS 1.

SSM REASONING The displacement is a vector drawn from the initial position to the final position. The magnitude of the displacement is the shortest distance between the positions. Note that it is only the initial and final positions that determine the displacement. The fact that the squirrel jumps to an intermediate position before reaching his final position is not important. The trees are perfectly straight and both growing perpendicular to the flat horizontal A ground beneath them. Thus, the distance between the 2.5 m trees and the length of the trunk of the second tree below the squirrel’s final landing spot form the two perpendicular sides of a right triangle, as the drawing shows. To this triangle, we can apply the Pythagorean theorem and determine the magnitude A of the 1.3 m displacement vector A.

SOLUTION According to the Pythagorean theorem, we have A=

2.

(1.3 m )2 + ( 2.5 m )2

= 2.8 m

REASONING AND SOLUTION The horizontal and vertical components of the plane's velocity are related to the speed of the plane by the Pythagorean theorem: v 2 = vh2 + vv2 . Solving for vh we have v h = v 2 − v v2 = (245 m / s) 2 − ( 40.6 m / s) 2 = 242 m / s

3.

REASONING To determine the horizontal and vertical components of the launch velocity, we will use trigonometry. To do so, however, we need to know both the launch angle and the magnitude of the launch velocity. The launch angle is given. The magnitude of the launch velocity can be determined from the given acceleration and the definition of acceleration given in Equation 3.2.

100 KINEMATICS IN TWO DIMENSIONS

SOLUTION According to Equation 3.2, we have a=

v − v0

or

t − t0

340 m / s 2 =

v−0 m/s 0.050 s

or

c

v = 340 m / s 2

hb0.050 sg

Using trigonometry, we find the components to be

c hb0.050 sgcos 51° = 11 m / s = v sin 51° = c 340 m / s h b0.050 sgsin 51° = 13 m / s

v x = v cos 51° = 340 m / s 2 vy

4.

2

REASONING The meteoroid’s speed is the magnitude of its velocity vector, here described in terms of two perpendicular components, one directed toward the east and one directed vertically downward. Let east be the +x direction, and up be the +y direction. Then the components of the meteoroid’s velocity are vx = +18.3 km/s and vy = −11.5 km/s. The meteoroid’s speed v is related to these components by the Pythagorean theorem (Equation

1.7): v2 = vx2 + v2y . SOLUTION From the Pythagorean theorem,

v = vx2 + v2y =

( +18.3 km/s )2 + ( −11.5 km/s )2 =

21.6 km/s

It’s important to note that the negative sign for vy becomes a positive sign when this quantity is squared. Forgetting this fact would yield a value for v that is smaller than vx, but the magnitude of a vector cannot be smaller than either of its components.

5.

SSM REASONING AND SOLUTION

x = r cos θ = (162 km) cos 62.3° = y = r sin θ = (162 km) sin 62.3° =

75.3 km 143 km

____________________________________________________________________________________________

6.

REASONING AND SOLUTION The horizontal displacement is x = 19 600 m – 11 200 m = 8400 m

The vertical displacement is

Chapter 3 Problems

101

y = 4900 m – 3200 m = 1700 m The magnitude of the displacement is therefore, 2 2 ∆r = x 2 + y 2 = ( 8400 m ) + (1700 m ) = 8600 m

____________________________________________________________________________________________

7.

SSM REASONING The displacement of the elephant seal has two components; 460 m due east and 750 m downward. These components are mutually perpendicular; hence, the Pythagorean theorem can be used to determine their resultant.

460 m

R

750 m

SOLUTION From the Pythagorean theorem, R 2 = (460 m) 2 + (750 m) 2 Therefore, R = ( 460 m) 2 + ( 750 m) 2 = 8.8 × 10 2 m

8.

REASONING Consider first the shopper’s ride up the escalator. Let the diagonal length of the escalator be L, the height of the upper floor be H, and the angle that the escalator makes with respect to the horizontal be θ (see the diagram). Because L is the hypotenuse of the right triangle and H is opposite the angle θ, the three quantities are related by the inverse sine s function: L H h H −1 o −1 θ = sin = sin (1.4) θ L D L h Up the escalator

Now consider the entire trip from the bottom to the top of the escalator (a distance L), and then Entire view from the top of the escalator to the store entrance (a distance s). The right turn between these two parts of the trip means that they are perpendicular (see the diagram). The shopper’s total displacement has a magnitude D, and this serves as the hypotenuse of a right triangle with L and s. From the Pythagorean theorem, the three sides are related as follows: D 2 = L2 + s 2 .

SOLUTION Solving D 2 = L2 + s 2 for the length L of the escalator gives L = D 2 − s 2 . We H now use this result and the relation θ = sin −1 to obtain the angle θ: L

102 KINEMATICS IN TWO DIMENSIONS

θ = sin

9.

−1

H L

H −1 = sin 2 2 D −s

−1 = sin

6.00 m

(16.0 m )2 − ( 9.00 m )2

= 27.0

SSM WWW REASONING a. We designate the direction down and parallel to the ramp as the +x direction, and the table shows the variables that are known. Since three of the five kinematic variables have values, one of the equations of kinematics can be employed to find the acceleration ax.

x-Direction Data x

ax

vx

v0x

+12.0 m

?

+7.70 m/s

0 m/s

b. The acceleration vector points down and parallel to the ramp, and the angle of the ramp is 25.0° relative to the ground (see the drawing). Therefore, trigonometry can be used to determine the component aparallel of the acceleration that is parallel to the ground.

t

ax 25.0°

aparallel

SOLUTION a. Equation 3.6a ( vx2 = v02x + 2ax x ) can be used to find the acceleration in terms of the three known variables. Solving this equation for ax gives ax =

vx2 − v02x 2x

( +7.70 m/s ) − ( 0 m/s ) = 2 ( +12.0 m ) 2

2

= 2.47 m/s 2

b. The drawing shows that the acceleration vector is oriented 25.0° relative to the ground. The component aparallel of the acceleration that is parallel to the ground is

aparallel = ax cos 25.0° = ( 2.47 m/s 2 ) cos 25.0° = 2.24 m/s 2

10. REASONING AND SOLUTION a. Average speed is defined as the total distance d covered divided by the time ∆t required to cover the distance. The total distance covered by the earth is one-fourth the circumference of its circular orbit around the sun: 1

d = 4 x 2π (1.50 x 1011 m) = 2.36 x 1011 m

v=

d 2 .36 × 1011 m = = 2.99 × 10 4 m / s 6 ∆t 7.89 × 10 s

Chapter 3 Problems

103

b. The average velocity is defined as the displacement divided by the elapsed time. In moving one-fourth of the distance around the sun, the earth completes the displacement shown in the figure at the right. From the Pythagorean theorem, the magnitude of this displacement is

∆r r r

∆r =

r2

+

r2

=

2r

Thus, the magnitude of the average velocity is v=

∆r = ∆t

2 × 1.50 × 10 11 m = 2.69 × 10 4 m / s 7.89 × 10 6 s

11. REASONING The component method can be used to determine the magnitude and direction of the bird watcher's displacement. Once the displacement is known, Equation 3.1 can be used to find the average velocity.

SOLUTION The following table gives the components of the individual displacements of the bird watcher. The last entry gives the components of the bird watcher's resultant displacement. Due east and due north have been chosen as the positive directions.

Displacement

East/West Component

North/South Component

A B C

0.50 km 0 –(2.15 km) cos 35.0° = –1.76 km

0 –0.75 km (2.15 km) sin 35.0° = 1.23 km

r=A+B+C

–1.26 km

0.48 km

a. From the Pythagorean theorem, we have r=

North

(–1.26 km) 2 + (0.48 km) 2 = 1.35 km

The angle θ is given by

θ = tan −1

0.48 km I F G H1.26 km JK=

r 0.48 km

θ

1.26 km 21° , north of west

b. From Equation 3.1, the average velocity is

East

104 KINEMATICS IN TWO DIMENSIONS

v=

∆r 1.35 km = = 0.540 km/h, 21° north of west ∆t 2.50 h

Note that the direction of the average velocity is, by definition, the same as the direction of the displacement.

12. REASONING AND SOLUTION The vertical motion consists of the ball rising for a time t stopping and returning to the ground in another time t. For the upward portion

t=

v0 y − v y g

=

v0 y g

Note: vy = 0 m/s since the ball stops at the top. Now

V0y = v0 sin θ0 = (25.0 m/s) sin 60.0° = 21.7 m/s t = (21.7 m/s)/(9.80 m/s2) = 2.21 s The required "hang time" is 2t = 4.42 s .

13. SSM REASONING AND SOLUTION As shown in Example 3, the time required for the package to hit the ground is given by t = 2 y / a y and is independent of the plane’s horizontal velocity. Thus, the time needed for the package to hit the ground is still 14.6 s .

14. REASONING The magnitude v of the puck’s velocity is related to its x and y velocity components (vx and vy ) by the Pythagorean theorem (Equation 1.7); v = v x2 + v 2y . The relation vx = v0 x + axt (Equation 3.3a) may be used to find vx, since v0x, ax, and t are known. Likewise, the relation v y = v0 y + a y t (Equation 3.3b) may be employed to determine vy, since v0y, ay, and t are known. Once vx and vy are determined, the angle θ that the velocity makes with respect to the x axis can be found by using the inverse tangent

(

)

function (Equation 1.6); θ = tan −1 v y / v x .

Chapter 3 Problems

105

SOLUTION Using Equations 3.3a and 3.3b, we find that

( ) v y = v0 y + a y t = +2.0 m/s + ( −2.0 m/s2 ) ( 0.50 s ) = +1.0 m/s v x = v0 x + a x t = +1.0 m/s + 2.0 m/s2 ( 0.50 s ) = +2.0 m/s

The magnitude v of the puck’s velocity is

v = v x2 + v 2y =

( 2.0 m/s )2 + (1.0 m/s )2

= 2.2 m/s

The angle θ that the velocity makes with respect to the +x axis is vy −1 1.0 m/s = tan = 27° above the +x axis 2.0 m/s vx

θ = tan −1

_____________________________________________________________________________________________

15. REASONING Trigonometry indicates that the x and y components of the dolphin’s velocity are related to the launch angle θ according to tan θ = vy /vx.

SOLUTION Using trigonometry, we find that the y component of the dolphin’s velocity is

b

g

v y = v x tan θ = v x tan 35° = 7 .7 m / s tan 35° = 5.4 m / s

16. REASONING a. Here is a summary of the data for the skateboarder, using v0 = 6.6 m/s and θ = 58°: y-Direction Data

y

ay

vy

v0y

?

−9.80 m/s2

0 m/s

+(6.6 m/s)sin 58° = +5.6 m/s

t

x-Direction Data

x

ax

?

0 m/s2

vx

v0x

t

+(6.6 m/s)cos 58° = +3.5 m/s

We will use the relation v2y = v02y + 2a y y (Equation 3.6b) to find the skateboarder’s vertical displacement y above the end of the ramp. When the skateboarder is at the highest point, his vertical displacement y1 above the ground is equal to his initial height y0 plus his vertical displacement y above the end of the ramp: y1 = y0 + y. Next we will use v y = v0 y + a y t

106 KINEMATICS IN TWO DIMENSIONS

(Equation 3.3b) to calculate the time t from the launch to the highest point, which, with x = v0 xt + 12 axt 2 (Equation 3.5a), will give us his horizontal displacement x. SOLUTION a. Substituting vy = 0 m/s into Equation 3.6b and solving for y, we find

( 0 m/s )

2

=

v02 y

+ 2a y y

or

y=

−v02 y 2a y

=

− ( 5.6 m/s )

(

2

2 −9.80 m/s2

Therefore, the highest point y1 reached by the y1 = y0 + y = 1.2 m + 1.6 m = +2.8 m above the ground.

)

= +1.6 m

skateboarder

occurs

when

b. We now turn to the horizontal displacement, which requires that we first find the elapsed time t. Putting vy = 0 m/s into Equation 3.3b and solving for t yields 0 m/s = v0 y + a y t

or

t=

−v0 y ay

(1)

Given that ax = 0 m/s2, the relation x = v0 xt + 12 axt 2 (Equation 3.5a) reduces to x = v0 x t . Substituting Equation (1) for t then gives the skateboarder’s horizontal displacement: −v0 y − v0 x v0 y − ( +3.5 m/s )( +5.6 m/s ) = = +2.0 m x = v0 x t = v0 x = ay −9.80 m/s ay

17. SSM REASONING The upward direction is chosen as positive. Since the ballast bag is released from rest relative to the balloon, its initial velocity relative to the ground is equal to the velocity of the balloon relative to the ground, so that v0 y = 3.0 m/s . Time required for the ballast to reach the ground can be found by solving Equation 3.5b for t. SOLUTION Using Equation 3.5b, we have 1 2

a y t 2 + v0 y t − y = 0

or

1 2

(–9.80 m / s 2 ) t 2 + ( 3.0 m / s) t − ( −9 .5 m) = 0

This equation is quadratic in t, and t may be found from the quadratic formula. Using the quadratic formula, suppressing the units, and discarding the negative root, we find −3.0 ± (3.0) 2 − 4 ( −4 .90 )( 9 .5) t= = 1.7 s 2 ( −4 .90 )

Chapter 3 Problems

107

____________________________________________________________________________________________

18. REASONING a. The maximum possible distance that the ball can travel occurs when it is launched at an angle of 45.0°. When the ball lands on the green, it is at the same elevation as the tee, so the vertical component (or y component) of the ball's displacement is zero. The time of flight is given by the y variables, which are listed in the table below. We designate "up" as the +y direction. y-Direction Data y

ay

0m

−9.80 m/s

v0y

t

+(30.3 m/s) sin 45.0° = +21.4 m/s

?

vy 2

Since three of the five kinematic variables are known, we can employ one of the equations of kinematics to find the time t that the ball is in the air. b. The longest hole in one that the golfer can make is equal to the range R of the ball. This distance is given by the x variables and the time of flight, as determined in part (a). Once again, three variables are known, so an equation of kinematics can be used to find the range of the ball. The +x direction is taken to be from the tee to the green. x-Direction Data x

ax

R=?

0 m/s

vx 2

v0x

t

+(30.3 m/s) cos 45.0° = +21.4 m/s

from part a

SOLUTION a. We will use Equation 3.5b to find the time, since this equation involves the three known variables in the y direction:

(

)

y = v0 y t + 12 a y t 2 = v0 y + 12 a y t t 0 m = +21.4 m/s +

1 2

( −9.80 m/s ) t t 2

Solving this quadratic equation yields two solutions, t = 0 s and t = 4.37 s. The first solution represents the situation when the golf ball just begins its flight, so we discard this one. Therefore, t = 4.37 s . b. With the knowledge that t = 4.37 s and the values for ax and v0x (see the x-direction data table above), we can use Equation 3.5a to obtain the range R of the golf ball.

x = v0 xt + 12 axt 2 = ( +21.4 m/s )( 4.37 s ) + 12 ( 0 m/s 2 ) ( 4.37 s ) = 93.5 m 2

=R

108 KINEMATICS IN TWO DIMENSIONS

______________________________________________________________________________ 19. REASONING AND SOLUTION The maximum vertical displacement y attained by a projectile is given by Equation 3.6b ( v 2y = v02y + 2a y y ) with vy = 0:

y=–

v02y 2a y

In order to use Equation 3.6b, we must first estimate his initial speed v0y . When Jordan has reached his maximum vertical displacement, vy = 0, and t = 1.00 s. Therefore, according to Equation 3.3b ( v y = v0 y + a y t ), with upward taken as positive, we find that v0 y = – a y t = – (–9.80 m/s 2 ) (1.00 s) = 9.80 m/s Therefore, Jordan's maximum jump height is y=–

(9.80 m/s)2 = 4.90 m 2(–9.80 m/s 2 )

This result far exceeds Jordan’s maximum jump height, so the claim that he can remain in the air for two full seconds is false. ____________________________________________________________________________________________

20. REASONING The magnitude (or speed) v of the ball’s velocity is related to its x and y velocity

components

(vx and vy )

by

the

Pythagorean

theorem;

v = vx2 + v 2y

(Equation 1.7). The horizontal component vx of the ball’s velocity never changes during the flight, since, in the absence of air resistance, there is no acceleration in the x direction (ax = 0 m/s2). Thus, vx is equal to the horizontal component v0x of the initial velocity, or v x = v0 x = v0 cos 40.0 . Since v0 is known, vx can be determined. The vertical component vy of the ball’s velocity does change during the flight, since the acceleration in the y direction is that due to gravity (ay = −9.80 m/s2). The relation v 2y = v02 y + 2a y y (Equation 2.6b) may be used to find v 2y , since ay, y, and v0y are known

(v0y = v0 sin 40.0º).

Chapter 3 Problems

109

SOLUTION The speed v of the golf ball just before it lands is

v = v x2 + v 2y =

( v0 cos 40.0°)2 + v02y + 2a y y v2y

=

( v0 cos 40.0°) + ( v0 sin 40.0°) 2

2

+ 2a y y

(

)

= (14.0 m/s ) cos 40.0° + (14.0 m/s ) sin 40.0° + 2 −9.80 m/s2 ( 3.00 m ) = 11.7 m/s 2

2

21. SSM REASONING The time that the ball spends in the air is determined by its vertical motion. The time required for the ball to reach the lake can be found by solving Equation 3.5b for t. The motion of the golf ball is characterized by constant velocity in the x direction and accelerated motion (due to gravity) in the y direction. Thus, the x component of the velocity of the golf ball is constant, and the y component of the velocity at any time t can be found from Equation 3.3b. Once the x and y components of the velocity are known for a particular time t, the speed can be obtained from v = v x2 + v y2 . SOLUTION a. Since the ball rolls off the cliff horizontally, v0y = 0. If the origin is chosen at top of the cliff and upward is assumed to be the positive direction, then the vertical component of the ball's displacement is y = – 15.5 m. Thus, Equation 3.5b gives t=

2y = ay

2( −15.5 m) = 1.78 s (–9.80 m / s 2 )

b. Since there is no acceleration in the x direction, v x = v0 x = 11.4 m/s . The y component of the velocity of the ball just before it strikes the water is, according to Equation 3.3b,

v y = v0 y + a y t = 0 + (–9.80 m/s2 )(1.78 s) = –17.4 m/s The speed of the ball just before it strikes the water is, therefore,

v = v x2 + v y2 = (11.4 m / s) 2 + ( −17.4 m / s) 2 = 20.8 m / s ____________________________________________________________________________________________

110 KINEMATICS IN TWO DIMENSIONS

22. REASONING The vehicle’s initial velocity is in the +y direction, and it accelerates only in the +x direction. Therefore, the y component of its velocity remains constant at vy = +21.0 m/s. Initially, the x component of the vehicle’s velocity is zero (v0x = 0 m/s), but it increases at a rate of

vx +y +x vy

v

θ

ax = +0.320 m/s2, reaching a final value vx given by the relation v x = v0 x + a x t (Equation 3.3a) when the pilot shuts off the RCS thruster. Once we have found vx, we will use the right triangle

shown in the drawing to find the magnitude v and direction θ of the vehicle’s final velocity.

SOLUTION a. During the 45.0-second thruster burn, the x component of the vehicle’s velocity increases from zero to v x = v0 x + a x t = ( 0 m/s ) + 0.320 m/s 2 ( 45.0 s ) = 14.4 m/s

(

)

Applying the Pythagorean theorem to the right triangle in the drawing, we find the magnitude of the vehicle’s velocity to be:

v = vx2 + v 2y =

(14.4 m/s )2 + ( 21.0 m/s )2 =

25.5 m/s

b. Referring again to the drawing, we see that the x and y components of the vehicle’s velocity are related by the tangent of the angle θ. Therefore, the angle of the vehicle’s final velocity is v 14.4 m/s θ = tan −1 x = tan −1 = 34.4 vy 21.0 m/s

23. SSM REASONING

Since the magnitude of the velocity of the fuel tank is given by

v = v x2 + v y2 , it is necessary to know the velocity components vx and v y just before impact. At the instant of release, the empty fuel tank has the same velocity as that of the plane. Therefore, the magnitudes of the initial velocity components of the fuel tank are given by v0 x = v0 cos θ and v0 y = v0 sin θ , where v0 is the speed of the plane at the instant of release. Since the x motion has zero acceleration, the x component of the velocity of the plane remains equal to v0x for all later times while the tank is airborne. The y component of the velocity of the tank after it has undergone a vertical displacement y is given by Equation 3.6b.

Chapter 3 Problems

111

SOLUTION a. Taking up as the positive direction, the velocity components of the fuel tank just before it hits the ground are v x = v0 x = v cos θ = (135 m/s) cos 15° = 1.30 × 102 m/s

From Equation 3.6b, we have v y = – v 02 y + 2 a y y = – ( v 0 sin θ ) 2 + 2 a y y =–

( 135 m / s) sin 15.0 °

2

+ 2 ( −9 .80 m / s 2 )( −2.00 × 10 3 m) = –201 m / s

Therefore, the magnitude of the velocity of the fuel tank just before impact is

v = v x2 + v y2 = (1.30 × 10 2 m / s) 2 + ( 201 m / s) 2 = 239 m / s The velocity vector just before impact is inclined at an angle φ above the horizontal. This angle is 201 m / s φ = tan −1 = 57.1° 1.30 × 10 2 m / s

F G H

IJ K

b. As shown in Conceptual Example 10, once the fuel tank in part a rises and falls to the same altitude at which it was released, its motion is identical to the fuel tank in part b. Therefore, the velocity of the fuel tank in part b just before impact is 239 m/s at an angle of 57.1° above the horizontal . ____________________________________________________________________________________________

24. REASONING In the absence of air resistance, the horizontal velocity component never changes from its initial value of v0x. Therefore the horizontal distance D traveled by the criminal (which must equal or exceed the distance between the two buildings) is the initial velocity times the travel time t, or D = v0x t . The time t can be found by noting that the motion of the criminal between the buildings is that of a projectile whose acceleration in the y direction is that due to gravity (ay = −9.80 m/s2, assuming downward to be the negative direction). The relation y = v0 y t + 12 a y t 2 (Equation 3.5b) allows us to determine the time, since y, v0y, and ay are

known. Since the criminal is initially running in the horizontal direction, v0y = 0 m/s. Setting v0y = 0 m/s and solving the equation above for t yields t = 2 y / a y . In this result, y = −2.0 m, since downward is the negative direction.

112 KINEMATICS IN TWO DIMENSIONS

SOLUTION

The horizontal distance traveled after launch is D = v0x t . Substituting

t = 2 y / a y into this relation gives 2 ( −2.0 m ) 2y = ( 5.3 m/s ) = 3.4 m ay −9.8 m/s2

D = v0 x t = v0 x

25. REASONING The magnitude and direction of the initial velocity v0 can be obtained using the Pythagorean theorem and trigonometry, once the x and y components of the initial velocity v0x and v0y are known. These components can be calculated using Equations 3.3a and 3.3b. SOLUTION Using Equations 3.3a and 3.3b, we obtain the following results for the velocity components:

c hb565 sg= 893.5 m / s − a t = 4816 m / s − c 7 .30 m / s h b565 sg= 691.5 m / s

v 0 x = v x − a x t = 3775 m / s − 5.10 m / s 2 v0 y = v y

2

y

Using the Pythagorean theorem and trigonometry, we find

b893.5 m / sg+ b691.5 m / sg = Fv I = tan F 691.5 m / s I G J G Hv K H893.5 m / s JK= 37.7 ° 2

v 0 = v 02 x + v 02 y =

θ = tan −1

0y

2

1130 m / s

−1

0x

____________________________________________________________________________________________

26. REASONING We begin by considering the flight time of the ball on the distant planet. Once the flight time is known, we can determine the maximum height and the range of the ball. The range of a projectile is proportional to the time that the projectile is in the air. Therefore, the flight time on the distant plant 3.5 times larger than on earth. The flight time can be found from Equation 3.3b ( v y = v0 y + a y t ). When the ball lands, it is at the same level as the tee; therefore, from the symmetry of the motion v y = – v0 y . Taking upward and to the right as the positive directions, we find that the flight time on earth would be t=

v y − v0 y ay

=

−2v0 y ay

=

−2v0 sin θ ay

=

−2(45 m/s) sin 29° –9.80 m/s 2

= 4.45 s

Chapter 3 Problems

113

Therefore, the flight time on the distant planet is 3.5 × (4.45 s) = 15.6 s. From the symmetry of the problem, we know that this is twice the amount of time required for the ball to reach its maximum height, which, consequently, is 7.80 s. SOLUTION a. The height y of the ball at any instant is given by Equation 3.4b as the product of the average velocity component in the y direction

1 2

ev

0y

j

+ v y and the time t: y =

1 2

ev

0y

j

+ vy t .

Since the maximum height H is reached when the final velocity component in the y direction is zero (vy = 0 m/s), we find that

H = 21 v 0 y t = 21 v 0 (sin 29 ° ) t = 21 (45 m / s) (sin 29 ° ) (7.80 s) =

85 m

b. The range of the ball on the distant planet is x = v 0 x t = v 0 (cos 29 ° ) t = ( 45 m / s) (cos 29 ° ) (15.6 s) = 610 m

27. SSM REASONING AND SOLUTION The water exhibits projectile motion. The x component of the motion has zero acceleration while the y component is subject to the acceleration due to gravity. In order to reach the highest possible fire, the displacement of the hose from the building is x, where, according to Equation 3.5a (with ax = 0 m/s2), x = v0 xt = (v0 cos θ )t

with t equal to the time required for the water the reach its maximum vertical displacement. The time t can be found by considering the vertical motion. From Equation 3.3b, v y = v0 y + a y t

When the water has reached its maximum vertical displacement, vy = 0 m/s. Taking up and to the right as the positive directions, we find that t=

−v0 y ay

=

−v0 sin θ ay

and

−v sin θ x = (v0 cos θ ) 0 ay

Therefore, we have x=−

v02 cos θ sin θ ay

=−

(25.0 m/s) 2cos 35.0° sin 35.0° = 30.0 m –9.80 m/s 2

114 KINEMATICS IN TWO DIMENSIONS

28. REASONING When the ball is thrown straight up with an initial speed v0, the maximum height y that it reaches can be found by using with the relation v 2y = v02 y + 2 a y y (Equation 3.6b). Since the ball is thrown straight up, v0y = v0, where v0 is the initial speed of the ball. Also, the speed of the ball is momentarily zero at its maximum height, so vy = 0 m/s at that point. The acceleration ay is that due to gravity, so the only unknown besides y is the initial speed v0 of the ball. To determine v0 we will employ Equation 3.6b a second time, but now it will be applied to the case where the ball is thrown upward at an angle of 52º above the horizontal. In this case the maximum height reached by the ball is y1 = 7.5 m, the initial speed in the y direction is v0y = v0 sin 52º, and the y-component of the speed at the maximum height is vy = 0 m/s. SOLUTION We will start with the relation v 2y = v02 y + 2 a y y (Equation 3.6b) to find the maximum height y that the ball attains when it is thrown straight up. Solving this equation for y, and substituting in v0y = v0 and vy = 0 m/s gives y=−

v02 2a y

(1)

To determine v0, we now apply the equation v 2y = v02 y + 2 a y y to the situation where the ball is thrown upward at an angle of 52º relative to the horizontal. In this case we note that vy = v0 sin 52º, vy = 0 m/s, and y = y1 (the maximum height of 7.5 m reached by the ball). Solving for v02 , we find

v 2y = ( v0 sin 52° ) + 2a y y1 2

or

v02 =

−2a y y1 sin 2 52°

Substituting this expression for v02 into Equation (1) gives y=−

v02

−2a y y1

2 y 7.5 m = − sin 52° = 2 1 = = 12 m 2a y 2a y sin 52° sin 2 52°

___________________________________________________________________________________________

Chapter 3 Problems

115

29. REASONING The vertical displacement y of the ball depends on the time that it is in the air before being caught. These variables depend on the y-direction data, as indicated in the table, where the +y direction is "up." y-Direction Data y

ay

?

−9.80 m/s2

v0y

t

0 m/s

?

vy

Since only two variables in the y direction are known, we cannot determine y at this point. Therefore, we examine the data in the x direction, where +x is taken to be the direction from the pitcher to the catcher. x-Direction Data

x

ax

+17.0 m

0 m/s

vx 2

v0x

t

+41.0 m/s

?

Since this table contains three known variables, the time t can be evaluated by using an equation of kinematics. Once the time is known, it can then be used with the y-direction data, along with the appropriate equation of kinematics, to find the vertical displacement y. SOLUTION Using the x-direction data, Equation 3.5a can be employed to find the time t that the baseball is in the air: x = v0 x t + 12 ax t 2 = v0 x t Solving for t gives

t=

( since a

x

= 0 m/s 2 )

x +17.0 m = = 0.415 s v0 x +41.0 m/s

The displacement in the y direction can now be evaluated by using the y-direction data table and the value of t = 0.415 s. Using Equation 3.5b, we have

y = v0 y t + 12 a y t 2 = ( 0 m/s )( 0.415 s ) + 12 ( −9.80 m/s 2 ) ( 0.415 s ) = −0.844 m 2

The distance that the ball drops is given by the magnitude of this result, so Distance = 0.844 m . ______________________________________________________________________________

116 KINEMATICS IN TWO DIMENSIONS

30. REASONING The data for the problem are summarized below. In the tables, we use the symbol v0 to denote the speed with which the ball is thrown and choose upward and to the right as the positive directions. v 30.0° 0 v0y x-Direction Data v0x v0 x t ax vx v0x

183 m

0 m/s2

v0 cos 30.0°

30.0°

?

183 m

y-Direction Data

y

ay

0m

−9.80 m/s2

vy

v0y

t

v0 sin 30.0°

Same as for x direction

Note that ax = 0 m/s2, because air resistance is being ignored. In addition, note that y = 0 m, because the football rises and then returns to the same vertical level from which it was launched. Finally, we have used trigonometry to express the components v0x and v0y of the initial velocity in terms of the speed v0 and the 30.0° launch angle. The key here is to remember that the horizontal and vertical parts of the football’s motion can be treated separately, the time for the motion being the same for each. Since air resistance is being ignored, we can apply the equations of kinematics separately to the motions in the x and y directions. SOLUTION Since there is no acceleration in the horizontal direction, motion in that direction is constant-velocity motion, and the horizontal displacement x is simply the initial velocity component v0x times the time t: x = v0 x t = ( v0 cos 30.0° ) t

An expression for t can be obtained by considering the motion in the vertical direction. Thus, we use Equation 3.5b from the equations of kinematics and recognize that the displacement y is zero and v0y = v0 sin 30.0°: 1 2

y = v0 y t + a y t 2

or

0 m = ( v0 sin 30.0° ) t + a y t 2 1 2

or

Substituting this result for the time into the expression for x gives

t=

−2v0 sin 30.0° ay

Chapter 3 Problems

117

−2v sin 30.0° 2v 2 cos 30.0° sin 30.0° 0 =− 0 x = ( v0 cos 30.0° ) t = ( v0 cos 30.0° ) ay ay − xa y

v0 =

2 cos 30.0° sin 30.0°

=

(

− (183 m ) −9.80 m/s 2 2 cos 30.0° sin 30.0°

) = 45.5 m/s

31. SSM WWW REASONING Once the diver is airborne, he moves in the x direction with constant velocity while his motion in the y direction is accelerated (at the acceleration due to gravity). Therefore, the magnitude of the x component of his velocity remains constant at 1.20 m/s for all times t. The magnitude of the y component of the diver's velocity after he has fallen through a vertical displacement y can be determined from Equation 3.6b: v 2y = v02y + 2a y y . Since the diver runs off the platform horizontally, v0 y = 0 m/s. Once the x

and y components of the velocity are known for a particular vertical displacement y, the speed of the diver can be obtained from v = v x2 + v y2 . SOLUTION For convenience, we will take downward as the positive y direction. After the diver has fallen 10.0 m, the y component of his velocity is, from Equation 3.6b,

v y = v 02 y + 2 a y y = 0 2 + 2(9.80 m / s 2 )(10.0 m) = 14 .0 m / s Therefore,

v = v x2 + v y2 = (1.20 m / s) 2 + (14.0 m / s) 2 = 14.1 m / s ____________________________________________________________________________________________

32. REASONING Since we know the launch angle θ = 15.0°, the launch speed v0 can be obtained using trigonometry, which gives the y component of the launch velocity as v0y = v0 sin θ. Solving this equation for v0 requires a value for v0y, which we can obtain from the vertical height of y = 13.5 m by using Equation 3.6b from the equations of kinematics. SOLUTION From Equation 3.6b we have

v y2 = v 02 y + 2 a y y

b0 m / sg= v 2

2 0y

hb

c

+ 2 −9 .80 m / s 2 13.5 m

g or

c

Using trigonometry, we find

v0 =

v0 y sin 15.0 °

=

c

hb

2 9 .80 m / s 2 13.5 m sin 15.0 °

hb

v 0 y = 2 9 .80 m / s 2 13.5 m

g=

62.8 m / s

g

118 KINEMATICS IN TWO DIMENSIONS

____________________________________________________________________________________________

y

33. REASONING The drawing at the right shows the velocity vector v of the water at a point below the top of the falls. The components of the velocity are also shown. The angle θ is given by tan θ = v y / v x , so that

x θ

vy

v θ

v y = – vx tan θ = –(2.7 m/s) tan 75°

vx

Here we have used the fact that the horizontal velocity component vx remains unchanged at its initial value of 2.7 m/s as the water falls. Knowing the y component of the velocity, we can use Equation 3.6b, ( v 2y = v02y + 2a y y ) to find the vertical distance y. SOLUTION

Taking v0 y = 0 m/s, and taking upward as the positive direction, we have

from Equation 3.5b that y=

v 2y 2a y

=

–( 2 .7 m / s) tan 75°

2

2(–9.80 m / s 2 )

= –5.2 m

Therefore, the velocity vector of the water points downward at a 75° angle below the horizontal at a vertical distance of 5.2 m below the edge. ____________________________________________________________________________________________

34. REASONING In the absence of air resistance, there is no acceleration in the x direction (ax = 0 m/s2), so the range R of a projectile is given by R = v0x t, where v0x is the horizontal component of the launch velocity and t is the time of flight. We will show that v0x and t are each proportional to the initial speed v0 of the projectile, so the range is proportional to v02 . Since v0x = v0 cos θ, where θ is the launch angle, we see that v0x is proportional to v0. To show that the time of flight t is also proportional to the launch speed v0, we use the fact that, for a projectile that is launched from and returns to ground level, the vertical displacement is y = 0 m. Using the relation y = v0 y t + 12 a y t 2 (Equation 3.5b), we have 0 m = v0 y t + 12 a y t 2

or

0 m = v0 y + 12 a y t

or

t=

−2v0 y ay

Thus, the flight time is proportional to the vertical component of the launch velocity v0y, which, in turn, is proportional to the launch speed v0 since v0y = v0 sin θ.

Chapter 3 Problems

119

SOLUTION The range of a projectile is given by R = v0x t and, since both v0x and t are proportional to v0, the range is proportional to v02 . The given range is 23 m. When the launch speed doubles, the range increases by a factor of 22 = 4, since the range is proportional to the square of the speed. Thus, the new range is

b g

R = 4 23 m = 92 m

35. REASONING The rocket will clear the top of the wall by an amount that is the height of the rocket as it passes over the wall minus the height of the wall. To find the height of the rocket as it passes over the wall, we separate the rocket’s projectile motion into its horizontal and vertical parts and treat each one separately. From the horizontal part we will obtain the time of flight until the rocket reaches the location of the wall. Then, we will use this time along with the acceleration due to gravity in the equations of kinematics to determine the height of the rocket as it passes over the wall. SOLUTION velocity

We begin by finding the horizontal and vertical components of the launch

b g sin 60.0 ° = b 75.0 m / s g sin 60.0 °

v 0 x = v 0 cos 60.0 ° = 75.0 m / s cos 60.0 ° v0 y = v0

Using v0x, we can obtain the time of flight, since the distance to the wall is known to be 27.0 m: 27 .0 m 27 .0 m t= = = 0.720 s 75.0 m / s cos 60.0 ° v0 x

b

g

The height of the rocket as it clears the wall can be obtained from Equation 3.5b, in which we take upward to be the positive direction. The amount by which the rocket clears the wall can then be obtained: y = v 0 y t + 21 a y t 2

b

gb

gb

gc

y = 75.0 m / s sin 60.0 ° 0.720 s + 21 −9 .80 m / s 2

hb0.720 sg= 44.2 m 2

clearance = 44 .2 m − 11.0 m = 33.2 m _____________________________________________________________________________________________

120 KINEMATICS IN TWO DIMENSIONS

36. REASONING We will treat the horizontal and vertical y parts of the motion separately. The directions upward v0x and to the right are chosen as the positive directions in the drawing. Ignoring air resistance, we can apply the equations of kinematics to the vertical part of the motion. The data for the vertical motion are x summarized in the following tables. Note that the initial velocity component v0y is zero, because the bullets are fired horizontally. The vertical component y of the bullets’ displacements are entered in the tables with minus signs, because the bullets move downward in the negative y direction. The times of flight tA and tB have been identified in the tables as a matter of convenience. y-Direction Data, First Shot

y

ay

−HA

−9.80 m/s2

y-Direction Data, Second Shot

v0y

t

y

ay

0 m/s

tA

−HB

−9.80 m/s2

vy

vy

v0y

t

0 m/s

tB

With these data, Equation 3.5b gives y = v0 y t + a y t 2 = ( 0 m/s ) t + a y t 2 = a y t 2 1 2

1 2

1 2

(1)

SOLUTION Applying Equation (1) to both shots, we find that

yB yA

=

−HB

−H A

=

1 a t2 2 y B 1 a t2 2 y A

or

HB HA

=

tB2

(2)

tA2

To use this result to calculate the ratio HB/HA, it is necessary to determine the times tA and tB. To do this, we consider the horizontal part of the motion and note that there is no acceleration in the horizontal direction. Therefore, the horizontal component v0x of the bullet’s initial velocity remains unchanged throughout the motion, and the horizontal component x of the displacement is simply v0x times the time t during which the motion occurs. We have that

xA = v0 x tA and xB = v0 xtB

or

Substituting these results into Equation (2) gives

tA =

xA v0 x

and tB =

xB v0 x

(3)

Chapter 3 Problems

xB H B tB2 ( xB / v0 x ) = 2 = = H A tA ( x / v )2 xA A 0x 2

121

2

It is given that xB = 2xA, so that we find

HB HA

37.

x = B x A

2

2 xA = xA

2

= 4

SSM REASONING a. The drawing shows the initial velocity v0 of the package when it is released. The initial speed of the package is 97.5 m/s. The component of its displacement along the ground is labeled as x. The data for the x direction are indicated in the data table below.

+y

v0

+x

50.0°

y

x x-Direction Data

x

ax

?

0 m/s2

vx

v0x

t

+(97.5 m/s) cos 50.0° = +62.7 m/s

Since only two variables are known, it is not possible to determine x from the data in this table. A value for a third variable is needed. We know that the time of flight t is the same for both the x and y motions, so let’s now look at the data in the y direction. y-Direction Data

y

ay

−732 m

−9.80 m/s

vy 2

v0y

t

+(97.5 m/s) sin 50.0° = +74.7 m/s

?

Note that the displacement y of the package points from its initial position toward the ground, so its value is negative, i.e., y = −732 m. The data in this table, along with the appropriate equation of kinematics, can be used to find the time of flight t. This value for t can, in turn, be used in conjunction with the x-direction data to determine x.

122 KINEMATICS IN TWO DIMENSIONS +y

b.

+x

The drawing at the right shows the velocity of the package just before impact. The angle that the velocity makes with respect to the ground can be found from the inverse

(

)

tangent function as θ = tan −1 v y / vx . Once the time has been found in part (a), the values of vy and vx can be determined from the data in the tables and the appropriate equations of kinematics.

vx θ

vy

v SOLUTION a. To determine the time that the package is in the air, we will use Equation 3.5b

(y =v

0y

)

t + 12 a y t 2 and the data in the y-direction data table. Solving this quadratic equation

for the time yields

t=

t=

( a )(− y) 2( a )

−v0 y ± v02y − 4 1 2

− ( 74.7 m/s ) ±

1 2

y

y

( 74.7 m/s ) − 4 ( 12 ) ( −9.80 m/s 2 ) ( 732 m ) = 2 ( 12 ) ( −9.80 m/s 2 ) 2

− 6.78 s and 22.0 s

We discard the first solution, since it is a negative value and, hence, unrealistic. The displacement x can be found using t = 22.0 s, the data in the x-direction data table, and Equation 3.5a: x = v0 x t + 12 ax t 2 = ( +62.7 m/s )( 22.0 s ) +

1 2

( 0 m/s ) ( 22.0 s ) 2

2

= +1380 m

=0

b. The angle θ that the velocity of the package makes with respect to the ground is given by

(

)

θ = tan −1 v y / vx . Since there is no acceleration in the x direction (ax = 0 m/s2), vx is the same as v0x, so that vx = v0x = +62.7 m/s. Equation 3.3b can be employed with the y-direction data to find vy : v y = v0 y + a y t = +74.7 m/s + ( −9.80 m/s 2 ) ( 22.0 s ) = −141 m/s Therefore, vy −1 −141 m/s = tan = −66.0° + v 62.7 m/s x

θ = tan −1

where the minus sign indicates that the angle is 66.0° below the horizontal . ______________________________________________________________________________

Chapter 3 Problems

123

38. REASONING We can obtain an expression for the car’s initial velocity v0 by starting with the relation x = v0 x t + 12 a x t 2 (Equation 3.5a). Here x is the horizontal component of the car’s displacement, v0x is the horizontal component of the car’s initial velocity (v0x = v0 for a horizontal launch), and ax is its acceleration in the horizontal direction (ax = 0 m/s2 for projectile motion). Solving for the initial velocity, we find that

(

)

x = v0t + 12 0 m/s2 t 2 = v0t

or

v0 =

x t

(1)

The time t in Equation (1) is known, but x is not. To find x, we note that at t = 1.1 s, the car’s displacement has a magnitude of r = 7.0 m. The displacement r of the car has a horizontal component x and a vertical component y. The magnitude of the displacement is related to x and y by the Pythagorean theorem: (2) ( ∆r )2 = x 2 + y 2 Solving Equation (2) for x and substituting the result into Equation (1) gives v0 =

x = t

( ∆r )2 − y 2

(3)

t

To determine y, we turn to the relation y = v0 yt + 12 a yt 2 (Equation 3.5b). Setting v0y = 0 m/s (again, for a horizontal launch), we find that Equation (3) becomes

y = ( 0 m/s ) t + 12 a yt 2 = 12 a yt 2 SOLUTION Substituting Equation (4) into Equation (3) gives

v0 =

( ∆r )2 − y 2 t

=

( ∆r )2 − 14 a 2y t 4 t

Thus, the car’s speed just as it drives off the end of the dock is

( 7.0 m )2 − 14 ( −9.80 m/s2 ) (1.1 s )4 2

v0 =

1.1 s

= 3.4 m/s

(4)

124 KINEMATICS IN TWO DIMENSIONS

39. REASONING The speed v of the soccer ball just before the goalie catches it is given by v = v x2 + v y2 , where vx and vx are the x and y components of the final velocity of the ball.

The data for this problem are (the +x direction is from the kicker to the goalie, and the +y direction is the “up” direction): x-Direction Data x

ax

+16.8 m

0 m/s

2

t

vx

v0x

?

+(16.0 m/s) cos 28.0° = +14.1 m/s y-Direction Data

y

ay −9.80 m/s

2

t

vy

v0y

?

+(16.0 m/s) sin 28.0° = +7.51 m/s 2

Since there is no acceleration in the x direction (ax = 0 m/s ), vx remains the same as v0x, so vx = v0x = +14.1 m/s. The time t that the soccer ball is in the air can be found from the x-direction data, since three of the variables are known. With this value for the time and the y-direction data, the y component of the final velocity can be determined. SOLUTION Since ax = 0 m/s2, the time can be calculated from Equation 3.5a as x +16.8 m t= = = 1.19 s . The value for vy can now be found by using Equation 3.3b v0 x +14.1 m/s with this value of the time and the y-direction data: v y = v0 y + a y t = +7.51 m/s + ( −9.80 m/s 2 ) (1.19 s ) = −4.15 m/s

The speed of the ball just as it reaches the goalie is v = vx2 + v y2 =

( +14.1 m/s ) + ( −4.15 m/s ) 2

2

= 14.7 m/s

______________________________________________________________________________ 40. REASONING AND SOLUTION On impact vx = v cos 75.0° = (8.90 m/s) cos 75.0° = 2.30 m/s and 2

2

2

2

2

v0y = vy + 2gy = (8.90 m/s) sin 75.0° + 2(9.80 m/s )(–3.00 m) so

v0y = 3.89 m/s

Chapter 3 Problems

125

The magnitude of the diver's initial velocity is 2 2 v0 = ( 2.30 m/s ) + ( 3.89 m/s ) = 4.52 m/s

The angle the initial velocity vector makes with the horizontal is

θ0 = tan−1 (voy /vox) = 59.4° _____________________________________________________________________________________________

41. REASONING As discussed in Conceptual Example 5, the horizontal velocity component of the bullet does not change from its initial value and is equal to the horizontal velocity of the car. The same thing is true here for the tomato. In other words, regardless of its vertical position relative to the ground, the tomato always remains above you as you travel in the convertible. From the symmetry of free fall motion, we know that when you catch the tomato, its velocity will be 11 m/s straight downward. The time t required to catch the tomato can be found by solving Equation 3.3b ( v y = v0 y + a y t ) with v y = – v0 y . Once t is known, the distance that the car moved can be found from x = vxt . SOLUTION Taking upward as the positive direction, we find the flight time of the tomato to be v y – v0 y –2v0 y –2(11 m/s) t= = = = 2.24 s ay ay –9.80 m/s 2 Thus, the car moves through a distance of

x = vx t = (25 m/s )(2.24 s) = 56 m

42. REASONING As shown in the drawing, the angle that the velocity vector makes with the horizontal is given by vy tan θ = vx

y v

θ vx

where, from Equation 3.3b, v y = v0 y + a y t = v0 sin θ 0 + a y t

and, from Equation 3.3a (since ax = 0 m/s2),

v x = v0 x = v0 cos θ 0

vy x

126 KINEMATICS IN TWO DIMENSIONS

Therefore,

tan θ =

vy vx

=

v0 sin θ 0 + a y t v0 cos θ0

SOLUTION Solving for t, we find t=

v0 ( cos θ 0 tan θ – sin θ0 ) ay

=

( 29 m/s )( cos 36° tan 18° – sin 36° ) = –9.80 m/s 2

0.96 s

____________________________________________________________________________________________

43. SSM WWW REASONING The horizontal distance covered by stone 1 is equal to the distance covered by stone 2 after it passes point P in the following diagram. Thus, the distance ∆x between the points where the stones strike the ground is equal to x2, the horizontal distance covered by stone 2 when it reaches P. In the diagram, we assume up and to the right are positive. x 2

2 θ

P θ

θ

1

∆x

SOLUTION If tP is the time required for stone 2 to reach P, then x2 = v0 xtP = (v0 cos θ )tP

For the vertical motion of stone 2, v y = v0 sin θ + a y t . Solving for t gives t=

v y − v0 sin θ ay

Chapter 3 Problems

127

When stone 2 reaches P, v y = −v0 sin θ , so the time required to reach P is tP =

−2v0 sin θ ay

Then,

−2v sin θ 0 x2 = v0 x tP = (v0 cos θ ) ay x2 =

−2v02 sin θ cos θ ay

=

−2(13.0 m/s)2 sin 30.0° cos 30.0° = 14.9 m –9.80 m/s 2

v0x Since the vertical 44. REASONING vx height is asked for, we will begin with the vertical part of the motion, 65° treating it separately from the horizontal part. The directions vy H upward and to the right are chosen as the positive directions in the drawing. The data for the vertical motion are vx summarized in the following table. Note that the initial velocity component v0y is zero, because the vy marble is thrown horizontally. The vertical component y of the marble’s displacement is entered in the table as −H, where H is the height we seek. The minus sign is included, because y-Direction Data the marble moves downward in the negative y direction. The vertical y t ay vy v0y component vy of the final velocity is checked as an important variable in 0 m/s −H √ −9.80 m/s2 the table, because we are given the angle that the final velocity makes with respect to the horizontal. Ignoring air resistance, we apply the equations of kinematics. With the data indicated in the table, Equation 3.6b becomes v 2y 2 v 2y = v02y + 2a y y = ( 0 m/s ) + 2a y ( − H ) or H = − (1) 2a y

128 KINEMATICS IN TWO DIMENSIONS

SOLUTION To use Equation (1), we need to determine the vertical component vy of the final velocity. We are given that the final velocity makes an angle of 65° with respect to the horizontal, as the inset in the drawing shows. Thus, from trigonometry, it follows that

−v y

tan 65° =

v y = −vx tan 65°

or

vx

or

v y = −v0 x tan 65°

where the minus sign is included, because vy points downward in the negative y direction. In the absence of air resistance, there is no acceleration in the x direction, and the horizontal component vx of the final velocity is equal to the initial value v0x. Substituting this result into Equation (1) gives H =−

v 2y 2a y

2 − (15 m/s ) tan 65° −v0 x tan 65° ) ( =− =−

(

2 −9.80 m/s 2

2a y

)

2

= 53 m

45. REASONING Using the data given in the problem, we can find the maximum flight time t 1 2

of the ball using Equation 3.5b ( y = v0 y t + a y t 2 ). Once the flight time is known, we can use the definition of average velocity to find the minimum speed required to cover the distance x in that time. SOLUTION Equation 3.5b is quadratic in t and can be solved for t using the quadratic formula. According to Equation 3.5b, the maximum flight time is (with upward taken as the positive direction)

t=

=

– v0 y ± v02y – 4 2

( )a

( )a 1 2

1 2

y

(– y)

y

– (15.0 m/s ) sin 50.0° ±

= 0.200 s

and

=

– v0 y ± v02y + 2a y y ay

(15.0 m/s ) sin 50.0° +2(–9.80 m/s 2 ) (2.10 m) –9.80 m/s 2 2

2.145 s

where the first root corresponds to the time required for the ball to reach a vertical displacement of y = +2.10 m as it travels upward, and the second root corresponds to the time required for the ball to have a vertical displacement of y = +2.10 m as the ball travels upward and then downward. The desired flight time t is 2.145 s. During the 2.145 s, the horizontal distance traveled by the ball is

x = vx t = (v0 cos θ ) t = [ (15.0 m/s) cos 50.0°] (2.145 s) = 20.68 m

Chapter 3 Problems

129

Thus, the opponent must move 20.68 m –10.0 m = 10.68 m in 2.145 s – 0.30 s = 1.845 s . The opponent must, therefore, move with a minimum average speed of

v min =

10.68 m = 5.79 m / s 1.845 s

46. REASONING We will treat the horizontal and vertical parts of the motion separately. The range R is the product of the horizontal component of the initial velocity v0x and the time of flight t. The time of flight can be obtained from the vertical part of the motion by using Equation 3.5b

ey = v

0y

t + 21 a y t 2

j and the fact that the displacement y in the vertical

direction is zero, since the projectile is launched from and returns to ground level. The expression for the range obtained in this way can then be applied to obtain the desired launch angle for doubling the range. SOLUTION The range of the projectile is

c

h

R = v 0 x t = v 0 cos θ t

Using Equation 3.5b, we obtain the time of flight as

y = 0 = v 0 y t + 21 a y t 2

or

t=−

2v0 y ay

=−

2 v 0 sin θ ay

Substituting this expression for t into the range expression gives

F−2 v sin θ I = − 2 v R=c v cos θ h G H a JK 0

2 0

cos θ sin θ

0

ay

y

=−

v 02 sin 2θ ay

where we have used the fact that 2 cos θ sin θ = sin 2θ. We can now apply this expression for the range to the initial range R1 for θ1 and the range R2 for θ2: R1 = −

v 02 sin 2θ 1 ay

or

R2 = −

v 02 sin 2θ 2 ay

Dividing these two expressions gives

R2 R1

=

c h v sin 2θ h −c /a

− v 02 sin 2θ 2 / a y 2 0

1

y

=

sin 2θ 2 sin 2θ 1

=2

where we have used the fact that R2/R1 = 2. Since θ1 = 12.0°, we find that

130 KINEMATICS IN TWO DIMENSIONS

b g

sin 2θ 2 = 2 .00 sin 2θ 1 = 2 .00 sin 2 12 .0 ° = 0.813

θ2 =

sin −1 0.813 = 27 .2 ° 2 .00

47. SSM REASONING AND SOLUTION In the absence of air resistance, the bullet exhibits projectile motion. The x component of the motion has zero acceleration while the y component of the motion is subject to the acceleration due to gravity. The horizontal distance traveled by the bullet is given by Equation 3.5a (with ax = 0 m/s2): x = v0 xt = (v0 cos θ )t

with t equal to the time required for the bullet to reach the target. The time t can be found by considering the vertical motion. From Equation 3.3b, v y = v0 y + a y t

When the bullet reaches the target, v y = −v0 y . Assuming that up and to the right are the positive directions, we have

t=

−2v0 y ay

=

−2v0 sin θ

−2v sin θ 0 x = (v0 cos θ ) ay

and

ay

Using the fact that 2sin θ cos θ = sin 2θ , we have x=−

2v02 cos θ sin θ ay

=−

v02 sin 2θ ay

Thus, we find that

sin 2θ = − and

x ay v02

=−

(91.4 m) (–9.80 m/s 2 ) = 4.91× 10−3 2 (427 m/s)

2θ = 0.281° or 2θ = 180.000° − 0.281° = 179.719°

Therefore,

θ = 0.141° and 89.860° _____________________________________________________________________________________________

Chapter 3 Problems

131

48. REASONING The angle θ is the angle that the balloon’s initial velocity v0 makes with the horizontal, and can be found from the horizontal and vertical components of the initial velocity: v0 y v0 y (1) tan θ = or θ = tan −1 v0 x v0 x (Note: because the balloon follows a curved trajectory, θ is not related in this fashion to the horizontal and vertical components x and y of the balloon’s displacement.) We will use x = v0 x t + 12 a x t 2 (Equation 3.5a) and y = v0 yt + 12 ayt 2 (Equation 3.5b) to find expressions

for the horizontal and vertical components (v0x and v0y) of the balloon’s initial velocity, and then combine those results with Equation (1) to find θ. For the initial horizontal velocity component, with ax = 0 m/s2 since air resistance is being ignored, Equation 3.5a gives

(

)

x = v0 x t + 12 0 m/s 2 t 2 = v0 x t

v0 x =

or

x t

(2)

For the initial vertical velocity component, we obtain from Equation 3.5b that

y = v0 y t +

1 a t2 2 y

or

v0 y =

y − 12 a y t 2 t

(3)

In part b, the initial speed v0 of the second balloon can be found by first noting that it is related to the x component v0x of the balloon’s initial velocity by v0 x = v0 cos θ . Since there is no acceleration in the x direction (ax = 0 m/s2), v0x is equal to the x component of the balloon’s displacement (x = +35.0 m) divided by the time t that the second balloon is in the air, or v0x = x/t. Thus, the relation v0 x = v0 cos θ can be written as

x = v0 cosθ t

(4)

In Equation (4), x = +35.0 m and θ is known from the result of part (a). The time of flight t is related to the y component of the balloon’s displacement by y = v0 yt + 12 ayt 2 (Equation 3.5b). Since v0 y = v0 sin θ , Equation 3.5b can be expressed as

y = v0 yt + 12 a y t 2 = ( v0 sin θ ) t + 12 a yt 2

(5)

Equations (4) and (5) provide everything necessary to find the initial speed v0 of the water balloon for the second launch.

132 KINEMATICS IN TWO DIMENSIONS

SOLUTION a. Substituting Equations (2) for v0x and (3) for v0y into Equation (1) gives an expression for the balloon’s initial direction θ :

y − 1 a yt 2 2 y − 1 a yt 2 v t y 0 2 −1 −1 −1 θ = tan = tan = tan x v x 0x t

(5)

The balloon leaves the roof of Jackson, 15.0 m above the ground, and hits halfway up Walton, a point 12 (22.0 m) = 11.0 m above the ground. The balloon’s height at impact is therefore 15.0 m − 11.0 m = 4.0 m below its launch height. Taking up as the positive direction, the vertical displacement of the balloon is therefore y = −4.0 m, and its horizontal displacement is x = +35.0 m, the distance between the buildings. Therefore, the angle at which the first balloon is launched is

(

)

−4.0 m − 1 −9.80 m/s 2 ( 2.0 s )2 2 = 24 θ = tan −1 35.0 m b. The angle θ is that found in part (a), and the vertical displacement is the difference in heights between the Walton & Jackson dorms: y = 22.0 m − 15.0 m = +7.0 m. Solving Equation (4) for the elapsed time t and substituting the result into Equation (5) gives

y = ( v0 sin θ ) t +

1 a t2 2 y

x 1 x = ( v0 sin θ ) + 2 ay v0 cos θ v0 cos θ

2

Solving this equation for the initial speed v0 of the second balloon gives x v0 = cos θ

(

)

1 −9.80 m/s2 35.0 m 2 = = 29 m/s sin θ cos 24° 7.0 m − 35.0 m sin 24° y−x ( ) cosθ cos 24° 1a 2 y

49. SSM REASONING Since the horizontal motion is not accelerated, we know that the x component of the velocity remains constant at 340 m/s. Thus, we can use Equation 3.5a (with ax = 0 m/s2) to determine the time that the bullet spends in the building before it is embedded in the wall. Since we know the vertical displacement of the bullet after it enters the building, we can use the flight time in the building and Equation 3.5b to find the y

Chapter 3 Problems

133

component of the velocity of the bullet as it enters the window. Then, Equation 3.6b can be used (with v0 y = 0 m/s) to determine the vertical displacement y of the bullet as it passes between the buildings. We can determine the distance H by adding the magnitude of y to the vertical distance of 0.50 m within the building. Once we know the vertical displacement of the bullet as it passes between the buildings, we can determine the time t1 required for the bullet to reach the window using Equation 3.4b. Since the motion in the x direction is not accelerated, the distance D can then be found from D = v0 xt1 . SOLUTION Assuming that the direction to the right is positive, we find that the time that the bullet spends in the building is (according to Equation 3.5a)

t=

x v0 x

=

6.9 m = 0.0203 s 340 m/s

The vertical displacement of the bullet after it enters the building is, taking down as the negative direction, equal to –0.50 m. Therefore, the vertical component of the velocity of the bullet as it passes through the window is, from Equation 3.5b, v0 y (window) =

1 2

y − a yt 2 t

=

y 1 –0.50 m 1 − a yt = − (−9.80 m/s 2 )(0.0203 s) = –24.5 m/s 2 t 0.0203 s 2

The vertical displacement of the bullet as it travels between the buildings is (according to Equation 3.6b with v0 y = 0 m/s)

y=

v 2y 2a y

=

(−24.5 m/s)2 = −30.6 m 2(–9.80 m/s 2 )

Therefore, the distance H is H = 30.6 m + 0.50 m = 31 m The time for the bullet to reach the window, according to Equation 3.4b, is t1 =

2y 2 y 2(–30.6 m) = = = 2.50 s v0 y + v y v y (–24.5 m/s)

134 KINEMATICS IN TWO DIMENSIONS

Hence, the distance D is given by

D = v0 xt1 = (340 m/s)(2.50 s) = 850 m

50. REASONING AND SOLUTION Let H be the initial height of the can above the muzzle of the rifle. Relative to its initial position the vertical coordinate of the bullet at time, t, is y = v0 y t − 12 gt 2

Relative to the can’s initial position the vertical coordinate of the can at the same time is 2 y' = (1/2)gt . NOTE: y = H – y' if the bullet hits the can. Then y = v0yt – y'. The horizontal distance traveled by the bullet in time t is x = v0xt. Solving for t and substituting gives y = (v0y/v0x)x – y' Now v0y/v0x = tan θ and it is seen from the figure in the text that H = x tan θ if the bullet is to hit the can so y = H – y'. Hence, both objects are at the same place at the same time, and the bullet will always strike the can.

51. REASONING The drawing shows the trajectory of the ball, along with its initial speed v0, horizontal displacement x, and vertical displacement y. The angle that the initial velocity of the ball makes with the horizontal is θ. The known data are shown in the tables below:

+y +x

v0 y

θ x

x-Direction Data

x

ax

+26.9 m

0 m/s

vx 2

v0x +(19.8 m/s) cos θ

t

135

Chapter 3 Problems

y-Direction Data y

ay

+2.74 m

−9.80 m/s2

vy

t

v0y +(19.8 m/s) sin θ

There are only two known variables in each table, so we cannot directly use the equations of kinematics to find the angle θ. Our approach will be to first use the x direction data and obtain an expression for the time of flight t in terms of x and v0x. We will then enter this expression for t into the y-direction data table. The four variables in this table, y, ay, v0y, and t, can be related by using the appropriate equation of kinematics. This equation can then be solved for the angle θ. SOLUTION Using the x-direction data, Equation 3.5a can be employed to find the time t that the ball is in the air:

( since a

x = v0 x t + 12 ax t 2 = v0 x t

Solving for t gives

t=

x

= 0 m/s 2 )

x +26.9 m = v0 x + (19.8 m/s ) cos θ

Using the expression above for the time t and the data in the y-direction data table, the displacement in the y direction can be written with the aid of Equation 3.5b: y = v0 y t + 12 a y t 2 +26.9 m +2.74 m = 19.8 m/s sin θ 19.8 m/s cos θ

(

)

(

)

+

+26.9 m 1 − 9.80 m/s ( ) 2 (19.8 m/s ) cos θ 2

=t

2

= t2

Evaluating the numerical factors and using the fact that sin θ /cos θ = tan θ , the equation above becomes ( −9.04 m ) +2.74 m = ( +26.9 m ) tan θ + cos 2 θ 1 Using = 1 + tan 2 θ , this equation can be rearranged and placed into a quadratic form: 2 cos θ

( −9.04 m ) tan 2 θ + ( 26.9 m ) tan θ − 11.8 m = 0 The solutions to this quadratic equation are

136 KINEMATICS IN TWO DIMENSIONS

tan θ =

−26.9 m ±

( 26.9 m ) − 4 ( −9.04 m )( −11.8 m ) = 2 ( −9.04 m ) 2

0.535 and 2.44

The two angles are θ1 = tan −1 0.535 = 28.1° and θ 2 = tan −1 2.44 = 67.7° ______________________________________________________________________________ 52. REASONING Since car A is moving faster, it will eventually catch up with car B. Each car is traveling at a constant velocity, so the time t it takes for A to catch up with B is equal to the displacement between the two cars (x = +186 m) divided by the velocity vAB of A relative to B. (If the relative velocity were zero, A would never catch up with B). We can find the velocity of A relative to B by using the subscripting technique developed in Section 3.4 of the text. vAB = velocity of car A relative to car B vAG = velocity of car A relative to the Ground = +24.4 m/s vBG = velocity of car B relative to the Ground = +18.6 m/s We have chosen the positive direction for the displacement and velocities to be the direction in which the cars are moving. The velocities are related by vAB = vAG + vGB SOLUTION The velocity of car A relative to car B is vAB = vAG + vGB = +24.4 m/s + (−18.6 m/s) = +5.8 m/s, where we have used the fact that vGB = − vBG = −18.6 m/s. The time it takes for car A to catch car B is x +186 m t= = = 32.1 s v AB +5.8 m/s ______________________________________________________________________________

Chapter 3 Problems

53. SSM REASONING The velocity vSG of the swimmer relative to the ground is the vector sum of the velocity vSW of the swimmer relative to the water and the velocity v WG of the water relative to the ground as shown at the right: vSG = vSW + v WG .

137

vWG = 0.91 m/s

vSW = 1.4 m/s

Direction of current

vSG

The component of vSG that is parallel to the width of the river determines how fast the swimmer is moving across the river; this parallel component is vSW . The time for the swimmer to cross the river is equal to the width of the river divided by the magnitude of this velocity component. The component of vSG that is parallel to the direction of the current determines how far the swimmer is carried down stream; this component is vWG . Since the motion occurs with constant velocity, the distance that the swimmer is carried downstream while crossing the river is equal to the magnitude of vWG multiplied by the time it takes for the swimmer to cross the river. SOLUTION a. The time t for the swimmer to cross the river is

t=

width 2.8 × 103 m = = 2.0 ×103 s vSW 1.4 m/s

b. The distance x that the swimmer is carried downstream while crossing the river is x = vWG t = (0.91 m/s)(2.0 × 103 s) = 1.8 × 103 m

54. REASONING There are three velocities involved: v NB = velocity of Neil relative to Barbara v ΝG = velocity of Neil relative to the Ground (3.2 m/s, due west) v BG = velocity of Barbara relative to the Ground (4.0 m/s, due south)

Ordering the vectors vNB , vNG and vGB by their subscripts in the manner discussed in Section 3.4 of the text, we see that v NB = v NG + v GB . Note that this ordering involves v GB , the velocity of the ground relative to Barbara. According to the discussion in Section 3.4,

138 KINEMATICS IN TWO DIMENSIONS

v GB is related to v BG by v GB = − v BG . Thus, the three vectors listed above are related by

v NB = v NG + ( − v BG ) . The drawing shows this relationship.

North

Since the three vectors form a right triangle, the magnitudes of the vectors are related by the Pythagorean theorem, which we will use to obtain Neil’s speed relative to Barbara. Trigonometry can be used to determine the angle θ. SOLUTION Using the Pythagorean theorem, we find that the speed vNB of Neil relative to Barbara is 2 v NB = v NG + ( − vBG ) = 2

v NB

− vBG

θ

West

( 3.2 m/s )2 + ( −4.0 m/s )2

v NG = 5.1 m/s

The angle θ can be found from trigonometry: v NG −1 3.2 m/s = cos = 51° ( north of west ) 5.1 m/s v NB

θ = cos −1

55. REASONING As indicated by Equation 3.7, the velocity of the student relative to the ground is vSG = v SE + v EG where vSE is the velocity of the student relative to the escalator and vEG is the velocity of the escalator relative to the ground. Thus, we have vSE = v SG – v EG SOLUTION Taking the direction in which the student runs as the positive direction, we find that 30.0 m vSE = vSG – vEG = – ( –1.8 m/s ) = 4.5 m/s 11 s

The student must exceed this speed of 4.5 m/s to beat the record.

Chapter 3 Problems

139

56. REASONING There are three velocities involved: v TG = the velocity of the Truck relative to the Ground v TP = the velocity of the Truck relative to the Police car v PG = the velocity of the Police car relative to the Ground

Of these three vectors, we know the directions of v PG (north) and v TG (west), and the magnitudes of v PG (29 m/s) and v TP (48 m/s). Ordering these vectors by their subscripts in the manner discussed in Section 3.4 of the text, we see that v TG = v TP + v PG . The triangle formed by this vector sum is a right triangle, because two of the vectors ( v TG and v PG ) are perpendicular to one another. The third vector, v TP , must be the triangle’s hypotenuse. The magnitudes of the three vectors are thus related to one another by the Pythagorean theorem, which we will use to obtain the truck’s speed relative to the ground. SOLUTION a. To represent the vector sum v TG = v TP + v PG graphically, the vectors being added together, v TP

North

and v PG , must be drawn tail-to-head (see the drawing). The resultant vector v TG runs from the tail of v TP

vTG

West

v PG v TP

to the head of v PG . The

direction of v TP shows that, relative to the police car, the truck is moving both west and south. b. As the drawing shows, v TP is the hypotenuse of the vector right triangle. Therefore, the Pythagorean theorem gives 2 2 2 vTP = vTG + vPG

or

2 2 vTG = vTP − vPG

The speed of the truck relative to the ground is 2 2 vTG = vTP − vPG =

This is equivalent to 85 mph.

( 48 m/s )2 − ( 29 m/s )2

= 38 m/s

140 KINEMATICS IN TWO DIMENSIONS

57. SSM REASONING

The velocity v AB of train A relative to train B is the vector sum of

the velocity v AG of train A relative to the ground and the velocity v GB of the ground relative to train B, as indicated by Equation 3.7: v AB = v AG + v GB . The values of v AG and v BG are given in the statement of the problem. We must also make use of the fact that v GB = − v BG .

SOLUTION a. Taking east as the positive direction, the velocity of A relative to B is, according to Equation 3.7,

v AB = v AG + v GB = v AG − v BG = (+13 m/s) – (–28 m/s) = +41 m/s The positive sign indicates that the direction of v AB is due east . b. Similarly, the velocity of B relative to A is v BA = v BG + v GA = v BG − v AG = (–28 m/s) – (+13 m/s) = –41 m/s The negative sign indicates that the direction of v BA is due west .

58. REASONING The time it takes for the passenger to walk the distance on the boat is the distance divided by the passenger’s speed vPB relative to the boat. The time it takes for the passenger to cover the distance on the water is the distance divided by the passenger’s speed vPW relative to the water. The passenger’s velocity relative to the boat is given. However, we need to determine the passenger’s velocity relative to the water. SOLUTION a. In determining the velocity of the passenger relative to the water, we define the following symbols:

vPW = Passenger’s velocity relative to the water vPB = Passenger’s velocity relative to the boat vBW = Boat’s velocity relative to the water The passenger’s velocity relative to the water is

b

gb

g

v PW = v PB + v BW = 1.5 m / s, north + 5.0 m / s, south = 3.5 m / s, south b. The time it takes for the passenger to walk a distance of 27 m on the boat is

Chapter 3 Problems

t=

141

27 m 27 m = = 18 s vPB 1.5 m/s

c. The time it takes for the passenger to cover a distance of 27 m on the water is t=

27 m 27 m = = 7.7 s vPW 3.5 m/s

____________________________________________________________________________________________

59. REASONING Let v HB represent the velocity of the hawk relative to the balloon and v BG represent the velocity of the balloon relative to the ground. Then, as indicated by Equation 3.7, the velocity of the hawk relative to the ground is v HG = v HB + v BG . Since the vectors v HB and v BG are at right angles to each other, the vector addition can be carried out using the Pythagorean theorem. SOLUTION Using the drawing at the right, we have from the Pythagorean theorem,

NORTH

vHG

2 2 v HG = v HB + v BG

θ

= (2.0 m / s) 2 + (6.0 m / s) 2 = 6.3 m / s The angle θ is

θ = tan −1

Fv I = tan F 2.0 m / s I J= G G J H 6.0 m / s K Hv K −1

HB

vBG

vHB EAST

18 ° , north of east

BG

60. REASONING Using subscripts to make the relationships among the three relative velocities clear, we have the following: vPG = the velocity of the Plane relative to the Ground (unknown speed, due west) vPA = the velocity of the Plane relative to the Air (245 m/s, unknown direction) vAG = the velocity of the Air relative to the Ground (38.0 m/s, due north)

Arranging the subscripts as shown in Section 3.4 of the text, we find that the velocity of the plane relative to the ground is the vector sum of the other two velocities: vPG = vPA + vAG. This vector sum may be illustrated as follows: vPG vAG

θ

vPA

North East

142 KINEMATICS IN TWO DIMENSIONS

As the three relative velocity vectors form a right triangle, we will apply trigonometry to find the direction θ that the pilot should aim the plane relative to due west. SOLUTION The magnitudes of the two known sides of the vector right triangle are vPA (245 m/s) and vAG (38.0 m/s) which are, respectively, the hypotenuse and the side opposite the angle θ (see the drawing). The sine of the angle θ is therefore the ratio of vAG to vPA, so that the direction the pilot should head the plane is vAG −1 38.0 m/s = sin = 8.92 , south of west 245 m/s vPA

θ = sin −1

61. REASONING AND SOLUTION by

The velocity of the raindrops relative to the train is given vRT = vRG + vGT

where vRG is the velocity of the raindrops relative to the ground and vGT is the velocity of the ground relative to the train. Since the train moves horizontally, and the rain falls vertically, the velocity vectors are related as shown in the figure at the right. Then

vRT

25°

vRG

vGT = vRG tan θ = (5.0 m/s) (tan 25°) = 2.3 m/s The train is moving at a speed of 2.3 m/s

vGT

____________________________________________________________________________________________

62. REASONING The three relative velocities in this situation are vTG = the velocity of the Truck relative to the Ground (unknown) vTC = the velocity of the Truck relative to the Car (24.0 m/s, 52.0° north of east) vCG = the velocity of the Car relative to the Ground (16.0 m/s, due north)

The velocity of the truck relative to the ground may be expressed as the vector sum of the other two velocities: vTG = vTC + vCG . Note the fashion in which the “middle” subscripts on the right side of the equals sign are matched. See the diagram for an illustration of this vector sum. Because the vectors do not form a right triangle, we will utilize the component method of vector addition to determine the eastward and northward components of the resultant vector

Chapter 3 Problems

vTG. For convenience, we will take east as the +x direction and north as the +y direction. Once we know vTG,x and vTG,y, we will use the Pythagorean theorem to calculate the magnitude of the truck’s velocity relative to the ground, which, from the diagram, we expect to be larger than vTC = 24.0 m/s. SOLUTION The vector vCG points due north, and thus has no x component. Therefore the x component of vTG is equal to the x component of vTC:

143

North

vCG

vTG vTC 52.0°

East

vTG,x = vTC,x + vCG,x = vTC cos 52.0 + 0 m/s = ( 24.0 m/s ) cos 52.0 = 14.8 m/s

Next, we find the y component of the truck’s velocity relative to the ground. Noting that the vector vCG points due north, so that vCG,y = vCG, we have

vTG,y = vTC,y + vCG,y = vTC sin 52.0 + vCG = ( 24.0 m/s ) sin 52.0 + 16.0 m/s = 34.9 m/s The magnitude of vTG is found from the Pythagorean theorem: 2 2 vTG = vTG, x + vTG,y =

(14.8 m/s )2 + ( 34.9 m/s )2 = 37.9 m/s

63. SSM REASONING The velocity v PM of the puck relative to Mario is the vector sum of the velocity v PI of the puck relative to the ice and the velocity v IM of the ice relative to Mario as indicated by Equation 3.7: v PM = v PI + v IM . The values of v MI and v PI are given in the statement of the problem. In order to use the data, we must make use of the fact that v IM = − v MI , with the result that v PM = v PI − v MI . SOLUTION The first two rows of the following table give the east/west and north/south components of the vectors v PI and –v MI . The third row gives the components of their

resultant v PM = v PI − v MI . Due east and due north have been taken as positive.

144 KINEMATICS IN TWO DIMENSIONS

Vector

East/West Component

North/South Component

v PI

–(11.0 m/s) sin 22° = –4.1 m/s

–(11.0 m/s) cos 22° = –10.2 m/s

–v MI

0

+7.0 m/s

v PM = v PI – v MI

–4.1 m/s

–3.2 m/s

Now that the components of v PM are known, the Pythagorean theorem can be used to find the magnitude. v PM =

(–4.1 m / s) 2 + (–3.2 m / s) 2 = 5.2 m / s

4.1 m / s I F G H3.2 m / s JK=

64. REASONING AND SOLUTION

φ

3.2 m/s

4.1 m/s

The direction of v PM is found from

φ = tan −1

v PM

52 ° west of south

While flying west, the airplane has a ground speed of 2

vPG = 2.40 × 10 m/s – 57.8 m/s = 182 m/s and requires time tW = x/(182 m/s) to reach the turn-around point. While flying east the airplane has a ground speed of 2

vPG = 2.40 × 10 m/s + 57.8 m/s = 298 m/s and requires tE = x/(298 m/s) to return home. Now the total time for the trip is 4 t = tW + tE = 6.00 h = 2.16 × 10 s, so

x/182 + x/298 = 2.16 × 10

4

or

x = 2.44 × 106 m = 2440 km

145

Chapter 3 Problems

65. SSM WWW

REASONING

The velocity v OW of the object relative to the water is

the vector sum of the velocity v OS of the object relative to the ship and the velocity vSW of the ship relative to the water, as indicated by Equation 3.7: v OW = v OS + vSW . The value of vSW is given in the statement of the problem. We can find the value of v OS from the fact that we know the position of the object relative to the ship at two different times. The initial position is rOS1, and the final position is rOS2. Since the object moves with constant velocity, ∆r −r r v OS = OS = OS2 OS1 (1) ∆t ∆t SOLUTION The first two rows of the following table give the east/west and north/south components of the vectors rOS2 and −rOS1 . The third row of the table gives the components of ∆rOS = rOS2 − rOS1 . Due east and due north have been taken as positive.

Vector

East/West Component

North/South Component

rOS2

–(1120 m) cos 57.0° = −6.10 × 10 2 m

–(1120 m) sin 57.0° = −9.39 × 10 2 m

−rOS1

–(2310 m) cos 32.0° = – 1.96 × 103 m

+(2310 m) sin 32.0° = 1.22 × 103 m

−2.57 × 103 m

2.81× 10 2 m

∆rOS = rOS2 − rOS1

Now that the components of ∆rOS are known, the Pythagorean theorem can be used to find the magnitude.

∆r

OS

2

∆rOS = (–2.57 × 10 m ) + ( 2 .81 × 10 m ) = 2.59 × 10 m 3

2

2

3

2

The direction of ∆rOS is found from

φ = tan −1 Therefore, from Equation (1),

F 2.81 × 10 m I G H2.57 × 10 m JK= 6.24 ° 2

3

2.81 × 10 m

φ

2.57 × 103 m

146 KINEMATICS IN TWO DIMENSIONS

v OS =

∆rOS ∆t

=

rOS 2 − rOS1 ∆t

=

2.59 × 10 3 m = 7 .19 m / s, 6.24 ° north of west 360 s

Now that v OS is known, we can find v OW , as indicated by Equation 3.7: v OW = v OS + vSW . The following table summarizes the vector addition:

Vector

East/West Component

North/South Component

–(7.19 m/s) cos 6.24° = –7.15 m/s (7.19 m/s) sin 6.24° = 0.782 m/s

v OS vSW

+4.20 m/s

0 m/s

v OW = v OS + vSW

–2.95 m/s

0.782 m/s

Now that the components of v OW are known, the Pythagorean theorem can be used to find the magnitude.

v OW 0.782 m/s

v OW = (–2.95 m / s) 2 + ( 0.782 m / s) 2 = 3.05 m / s

φ 2.95 m/s

The direction of v OW is found from

φ = tan −1

0.782 m / s I F G H2.95 m / s JK=

14.8 ° north of west

____________________________________________________________________________________________

66. REASONING The data given in the problem are summarized as follows: x-Direction Data x

ax

4.11 × 106 m

?

vx

y-Direction Data

v0x

t

y

ay

4370 m/s

684 s

6.07 × 106 m

?

vy

v0y

t

6280 m/s

684 s

With these data, we can use Equations 3.5a and 3.5b to determine the acceleration component ax and ay.

Chapter 3 Problems

147

SOLUTION Using Equations 3.5a and 3.5b, we can solve for the acceleration components and find that 1 2

x = v0 xt + a xt 2 or a x = ax =

(

2 x − 2v0 xt t

2

(3.5a)

)

2 4.11×106 m − 2 ( 4370 m/s )( 684 s )

( 684 s )

1 2

y = v0 y t + a y t 2 or a y = ay =

2

= 4.79 m/s 2

(

2 y − 2v0 y t t2

(3.5b)

)

2 6.07 × 106 m − 2 ( 6280 m/s )( 684 s )

( 684 s )2

= 7.59 m/s 2

67. REASONING The vertical component of the ball’s velocity v0 changes as the ball approaches the opposing player. It changes due to the acceleration of gravity. However, the horizontal component does not change, assuming that air resistance can be neglected. Hence, the horizontal component of the ball’s velocity when the opposing player fields the ball is the same as it was initially. SOLUTION Using trigonometry, we find that the horizontal component is

b

g

v x = v 0 cosθ = 15 m / s cos 55° = 8 . 6 m / s

68. REASONING Since the spider encounters no appreciable air resistance during its leap, it can be treated as a projectile. The thickness of the magazine is equal to the spider’s vertical displacement y during the leap. The relevant data are as follows (assuming upward to be the +y direction): y-Direction Data y

ay

?

−9.80 m/s2

vy

v0y

t

(0.870 m/s) sin 35.0° = +0.499 m/s

0.0 770 s

SOLUTION We will calculate the spider’s vertical displacement directly from

y = v0 yt + 12 a yt 2 (Equation 3.5b):

(

)

y = ( 0.499 m/s )( 0.0770 s ) + 12 −9.80 m/s2 ( 0.0770 s ) = 0.0094 m 2

To express this result in millimeters, we use the fact that 1 m equals 1000 mm:

148 KINEMATICS IN TWO DIMENSIONS

1000 mm Thickness = 0.0094 m = 9.4 mm 1 m

(

)

69. REASONING AND SOLUTION The components of the initial velocity are v0x = v0 cos θ = (22 m/s) cos 40.0° = 17 m/s v0y = v0 sin θ = (22 m/s) sin 40.0° = 14 m/s

a. Solving Equation 3.6b for y gives y=

v 2y − v 02 y 2a y

When the football is at the maximum height y = H, and the football is momentarily at rest, so vy = 0 m/s. Thus, H=

0 − v 02 y

=

2a y

0 − (14 m/s ) 2

= 6.0 × 101 m

2 ( −1.62 m/s 2 )

b. When the ball strikes the ground, y = 0 m; therefore, the time of flight can be determined from Equation 3.5b with y = 0 m. 1

y = v0yt + 2 ayt2 or 0 = [(14 m/s) +

1 2

( −1.62 m / s 2 ) t] t

t = 17 s

The range is x = R = v0xt = (17 m/s)(17 s) =

290 m

____________________________________________________________________________________________

70. REASONING When the skier leaves the ramp, she exhibits projectile motion. Since we know the maximum height attained by the skier, we can find her launch speed v0 using Equation 3.6b , v 2y = v02y + 2a y y , where v0 y = v0 sin 63° . SOLUTION At the highest point in her trajectory, vy = 0. Solving Equation 3.6b for v0y we obtain, taking upward as the positive direction,

v 0 y = v 0 sin 63° =

–2 a y y

or

v0 =

–2 a y y sin 63°

=

–2(–9.80 m / s 2 ) (13 m) = 18 m / s sin 63°

____________________________________________________________________________________________

Chapter 3 Problems

149

71. SSM REASONING The speed of the fish at any time t is given by v = v x2 + v y2 , where

vx and vy are the x and y components of the velocity at that instant. Since the horizontal

motion of the fish has zero acceleration, vx = v0 x for all times t. Since the fish is dropped by

the eagle, v0x is equal to the horizontal speed of the eagle and v0 y = 0 . The y component of the velocity of the fish for any time t is given by Equation 3.3b with v0 y = 0 . Thus, the speed at any time t is given by v = v 02 x + ( a y t ) 2 . SOLUTION 2 2 2 2 a. The initial speed of the fish is v 0 = v 0 x + v 0 y = v 0 x + 0 = v 0 x . When the fish's speed

doubles, v = 2v0 x . Therefore,

2 v 0 x = v 02 x + ( a y t ) 2

or

4 v 02 x = v 02 x + ( a y t ) 2

Assuming that downward is positive and solving for t, we have t= 3

v0 x ay

= 3

6.0 m / s I F G H9.80 m / s JK= 2

1.1 s

b. When the fish's speed doubles again, v = 4v0 x . Therefore, 4 v 0 x = v 02 x + ( a y t ) 2 Solving for t, we have t = 15

v0 x ay

= 15

or

16 v 02 x = v 02 x + ( a y t ) 2

6.0 m / s I F G H9.80 m / s JK= 2.37 s 2

Therefore, the additional time for the speed to double again is ( 2 .4 s) – (1.1 s) = 1.3 s . ____________________________________________________________________________________________

72. REASONING We will treat the horizontal and vertical parts v0x of the motion separately. The directions upward and to the right are chosen as the positive directions in the drawing. y Ignoring air resistance, we note that there is no acceleration in the horizontal direction. Thus, the horizontal component x v0x of the ball’s initial velocity remains unchanged throughout the motion, and the horizontal component x of the displacement is simply v0x times the time t during which the motion occurs. This is true for both bunted balls. Since the value of x is the same for both, we have that

150 KINEMATICS IN TWO DIMENSIONS

t (1) v0 x,B = v0 x,A A t B To use this result to calculate v0x,B, it is necessary to determine the times tA and tB. We can accomplish this by applying the equations of kinematics to the vertical part of the motion for each ball. The data for the vertical motion are summarized as follows: xA = xB

v0 x,A tA = v0 x,BtB

or

or

y-Direction Data, Ball A

yA

ay,A

−1.2 m

−9.80 m/s2

vy,A

y-Direction Data, Ball B

v0y,A

tA

yB

ay,B

0 m/s

?

−1.5 m

−9.80 m/s2

vy,B

v0y,B

tB

0 m/s

?

Note that the initial velocity components v0y are zero, because the balls are bunted horizontally. With these data, Equation 3.5b gives y = v0 y t + a y t 2 = ( 0 m/s ) t + a y t 2 1 2

1 2

or

t=

2y ay

(2)

SOLUTION Using Equation (2) for each ball and substituting the expressions for tA and tB into Equation (1) gives 2 yA / a y ,A t v0 x,B = v0 x,A A = v0 x,A t 2 yB / a y ,B B

= v0 x,A

yA 1.2 m = (1.9 m/s ) = 1.7 m/s yB 1.5 m

Note that the accelerations ay,A and ay,B both equal the acceleration due to gravity and are eliminated algebraically from the calculation.

73. REASONING The drawing shows the trajectory of the ball, along with its initial speed v0 and vertical displacement y. The angle that the initial velocity of the ball makes with the ground is 35.0°. The known data are shown in the table below:

+y +x

v0 35.0°

y

Chapter 3 Problems

151

y-Direction Data y

ay

+5.50 m

−9.80 m/s2

v0y

t

+(46.0 m/s) sin 35.0° = +26.4 m/s

?

vy

Since three of the kinematic variables are known, we will employ the appropriate equation of kinematics to determine the time of flight. SOLUTION

Equation 3.5b

(y =v

0y

t + 12 a y t 2

)

relates the time t to the three known

variables. The terms in this equation can be rearranged to as to place it in a standard quadratic form: 12 a y t 2 + v0 y t − y = 0 . The solution of this quadratic equation is

t=

t=

( a )(− y) 2( a )

−v0 y ± v02y − 4 1 2

1 2

y

y

− ( +26.4 m/s ) ±

( +26.4 m/s )

2

− 4 ( 12 ) ( −9.80 m/s 2 ) ( −5.50 m )

−9.80 m/s 2

= 0.217 s or 5.17 s

The first solution (t = 0.217 s) corresponds to the situation where the ball is moving upward and has a displacement of y = +5.50 m. The second solution represents the later time when the ball is moving downward and its displacement is also y = +5.50 m (see the drawing). This is the solution we seek, so t = 5.17 s . ______________________________________________________________________________ 74. REASONING There are three velocities involved: v CR = the initial velocity of the Car relative to the Road v CT = the velocity of the Car relative to the first Truck v TR = the velocity of the first Truck relative to the Road (speed = 11 m/s)

Ordering these vectors by their subscripts in the manner discussed in Section 3.4 of the text, we see that vCR = vCT + vTR. The velocity v TR is known, and we will use the equations of kinematics to find vCT. The two trucks have the same velocity relative to the road, and thus have zero velocity relative to one another. Therefore, we will analyze the sports car’s jump across the 15-m gap as if both trucks were stationary. Neglecting air resistance, we treat the car as a projectile and find the initial speed v0 it must have, relative to the trucks, in order to reach the flat trailer. (This speed is, in fact, the speed vCT that we need to find.) We will use the relation

152 KINEMATICS IN TWO DIMENSIONS

x = v0 x t + 12 a x t 2 (Equation 3.5a) to find the elapsed time. With ax = 0 m/s2, Equation 3.5a

(

)

becomes x = v0 xt + 12 0 m/s 2 t 2 = v0 x t . Thus the car is in the air for t seconds, where t=

x v0 x

x v0 cos θ

=

(1)

In Equation (1), x = 15 m and θ = 16º, but v0 and t are not known. The heights of the trailer and the ramp are the same, so the car’s vertical displacement is zero. To use this fact, we turn

to

y = v0 yt + 12 ayt 2

(Equation

3.5b)

and

substitute

both

y=0m

and

v0y = v0 sin θ: 0 = v0 y t + 12 a y t 2

or

v0 y = − 12 a y t

v0 sin θ = − 12 a y t

or

(2)

To eliminate the time, we substitute t from Equation (1) into Equation (2) and find that

x v0 sin θ = − 21 a y v0 cos θ

or

v02 =

−a y x 2 cos θ sin θ

or

v0 =

−a y x 2 cos θ sin θ

(3)

SOLUTION Taking up as the positive direction, we use Equation (3) to calculate the speed vCT = v0 of the car relative to the trucks:

vCT = v0 =

−a y x 2 cos θ sin θ

=

(

)

− −9.80 m/s 2 (15 m ) 2 cos16° sin16

= 17 m/s

Since the ramp alters the direction of the car’s velocity but not its magnitude, the initial jump speed v0 is also the magnitude of the car’s velocity vCT relative to the truck before the car reaches the ramp: v0 = vCT. The sports car must overtake the truck at a speed of at least 17 m/s relative to the truck, so that the car’s minimum required speed vCR relative to the road is vCR = vCT + vTR = 17 m/s + 11 m/s = 28 m/s

Chapter 3 Problems

75. SSM

153

REASONING The angle θ can be found from 2400 m x

θ = tan −1

(1)

where x is the horizontal displacement of the flare. Since ax = 0 m/s2, it follows that

x = (v0 cos 30.0°)t . The flight time t is determined by the vertical motion. In particular, the time t can be found from Equation 3.5b. Once the time is known, x can be calculated.

SOLUTION From Equation 3.5b, assuming upward is the positive direction, we have

y = –(v0 sin 30.0°) t + 12 a y t 2 which can be rearranged to give the following equation that is quadratic in t: 1 a t2 2 y

− (v0 sin 30.0°) t − y = 0

Using y = –2400 m and a y = −9.80 m/s 2 and suppressing the units, we obtain the quadratic equation 4.9t 2 + 120t − 2400 = 0

Using the quadratic formula, we obtain t = 13 s. Therefore, we find that x = ( v 0 cos 30 .0 ° ) t = ( 240 m / s ) (cos 30.0 ° ) ( 13 s) = 2700 m

Equation (1) then gives 2400 m = 42° 2700 m

θ = tan −1

76. REASONING The relative velocities in this problem are: vPW = velocity of the Passenger relative to the Water vPB = velocity of the Passenger relative to the Boat (2.50 m/s, due east) vBW = velocity of the Boat relative to the Water (5.50 m/s, at 38.0° north of east)

The velocities are shown in the drawing are related by the subscripting method discussed in Section 3.4: vPW = vPB + vBW

154 KINEMATICS IN TWO DIMENSIONS

+y (North)

vPW

vPW

+x (East)

5.50 m/s

vBW θ

θ

vPB

2.50 m/s

38.0°

We will determine the magnitude and direction of vPW from the equation above by using the method of scalar components. SOLUTION The table below lists the scalar components of the three vectors. Vector

x Component

y Component

vPB

+2.50 m/s

0 m/s

vBW vPW = vPB + vBW

+(5.50 m/s) cos 38.0° = +4.33 m/s +(5.50 m/s) sin 38.0° = +3.39 m/s +2.50 m/s + 4.33 m/s = +6.83 m/s

0 m/s + 3.39 m/s = +3.39 m/s

The magnitude of vPW can be found by applying the Pythagorean theorem to the x and y components: vPW =

( 6.83 m/s ) + ( 3.39 m/s ) 2

2

= 7.63 m/s

The angle θ (see the drawings) that vPW makes with due east is +3.39 m/s = 26.4° north of east +6.83 m/s ______________________________________________________________________________

θ = tan −1

77. REASONING The drawings show the initial and final velocities of the ski jumper and their scalar components. The initial speed of the ski jumper is given by v0 = v02x + v02 y , and the v angle that the initial velocity makes with the horizontal is θ = tan −1 0 y . The scalar v 0x components v0x and v0y can be determined by using the equations of kinematics and the data in the following tables. (The +x direction is in the direction of the horizontal displacement of the skier, and the +y direction is “up.”)

Chapter 3 Problems

v0

155

vx

v0y

43.0°

θ

v

v0x Initial velocity

vy

Final velocity

x-Direction Data x

ax

vx

v0x

+51.0 m

0 m/s2

+(23.0 m/s) cos 43.0° = +16.8 m/s

?

t

y-Direction Data y

ay −9.80 m/s

2

vy

v0y

−(23.0 m/s) sin 43.0° = −15.7 m/s

?

t

Since there is no acceleration in the x direction (ax = 0 m/s2), v0x is the same as vx, so we have that v0x = vx = +16.8 m/s. The time that the skier is in the air can be found from the x-direction data, since three of the variables are known. With the value for the time and the y-direction data, the y component of the initial velocity can be determined. 2

SOLUTION Since ax = 0 m/s , the time can be determined from Equation 3.5a as x +51.0 m t= = = 3.04 s . The value for v0y can now be found by using Equation 3.3b v0 x +16.8 m/s with this value of the time and the y-direction data: v0 y = v y − a y t = −15.7 m/s − ( −9.80 m/s 2 ) ( 3.04 s ) = +14.1 m/s

The speed of the skier when he leaves the end of the ramp is v0 = v02x + v02y =

( +16.8 m/s ) + ( +14.1 m/s ) 2

2

= 21.9 m/s

The angle that the initial velocity makes with respect to the horizontal is v0 y −1 +14.1 m/s = tan = 40.0° 16.8 m/s + v 0x ______________________________________________________________________________

θ = tan −1

156 KINEMATICS IN TWO DIMENSIONS

78. REASONING The relative velocities in this problem are: vPS = velocity of the Passenger relative to the Shore vP2 = velocity of the Passenger relative to Boat 2 (1.20 m/s, due east) v2S = velocity of Boat 2 relative to the Shore v21 = velocity of Boat 2 relative to Boat 1 (1.60 m/s, at 30.0° north of east) v1S = velocity of Boat 1 relative to the Shore (3.00 m/s, due north) The velocity vPS of the passenger relative to the shore is related to vP2 and v2S by (see the method of subscripting discussed in Section 3.4): vPS = vP2 + v2S But v2S, the velocity of Boat 2 relative to the shore, is related to v21 and v1S by +y (North)

v2S = v21 + v1S

v1S vPS

3.00 m/s

Substituting this expression for v2S into the first equation yields vPS = vP2 + v21 + v1S 1.60 m/s

This vector sum is shown in the diagram. We will determine the magnitude of vPS from the equation above by using the method of scalar components.

1.20 m/s

v21

30.0°

vP2

+x (East)

SOLUTION The table below lists the scalar components of the four vectors in the drawing. Vector

x Component

y Component

vP2

+1.20 m/s

0 m/s

v21 v1S vPS = vP2 + v21 + v1S

+(1.60 m/s) cos 30.0° = +1.39 m/s +(1.60 m/s) sin 30.0° = +0.80 m/s 0 m/s

+3.00 m/s

+1.20 m/s + 1.39 m/s = +2.59 m/s +0.80 m/s +3.00 m/s = +3.80 m/s

The magnitude of vPS can be found by applying the Pythagorean theorem to its x and y components:

vPS = ( 2.59 m/s ) + ( 3.80 m/s ) = 4.60 m/s ______________________________________________________________________________ 2

2

Chapter 3 Problems

157

79. REASONING We can use information about the motion of clown A and the collision to determine the initial velocity components for clown B. Once the initial velocity components are known, the launch speed v0B and the launch angle θ B for clown B can be determined.

e

j

1 2 we can find the time of flight until SOLUTION From Equation 3.5b y = v 0 y t + 2 a y t the collision. Taking upward as positive and noting for clown A that v0y = (9.00 m/s) sin 75.0° = 8.693 m/s, we have

b

g c

h

1.00 m = 8.693 m / s t + 21 –9.80 m / s 2 t 2 Rearranging this result and suppressing the units gives 4.90t 2 – 8.693t + 1.00 = 0

The quadratic equation reveals that

t=

8.693 ±

b–8.693g– 4 b4.90gb1.00g= 1.650 s 2b 4 .90 g 2

or

0.1237 s

Using these values for t with the magnitudes v0xA and v0xB of the horizontal velocity components for each clown, we note that the horizontal distances traveled by each clown must add up to 6.00 m. Thus, v0 xA t + v0 xB t = 6.00 m

or

v0 xB =

6.00 m – v0 xA t

Using v0xA = (9.00 m/s) cos 75.0° = 2.329 m/s, we find v0 xB =

6.00 m – 2.329 m/s = 1.307 m/s 1.650 s

or

v0 xB =

6.00 m – 2.329 m/s = 46.175 m/s 0.1237 s

The vertical component of clown B’s velocity is v0yB and must be the same as that for clown A, since each clown has the same vertical displacement of 1.00 m at the same time. Hence, v0yB = 8.693 m/s (see above). The launch speed of clown B, finally, is

v 0 B = v 02 xB + v 02 yB . Thus, we find v0 B = or

b1.307 m / sg+ b8.693 m / sg = 8.79 m / s 2

2

158 KINEMATICS IN TWO DIMENSIONS

v0 B =

b46.175 m / sg+ b8.693 m / sg = 47.0 m / s 2

2

For these two possible launch speeds, we find the corresponding launch angles using the following drawings, neither of which is to scale: +y

+y

v0B

8.693 m/s

θB 1.307 m/s

θB

+x

8.693 m/s = 81.5° 1.307 m/s

θ = tan –1

v0B

8.693 m/s

46.175 m/s

+x

8.693 m/s = 10.7° 46.175 m/s

θ = tan –1

Since the problem states that θB > 45°, the solution is v0 B = 8.79 m/s and θ B = 81.5° .

CHAPTER 4 FORCES AND NEWTON'S LAWS

OF MOTION ANSWERS TO FOCUS ON CONCEPTS QUESTIONS

1. (b) If only one force acts on the object, it is the net force; thus, the net force must be nonzero. Consequently, the velocity would change, according to Newton’s first law, and could not be constant. 2. (d) This situation violates the first law, which predicts that the rabbit’s foot tends to remain in place where it was when the car begins accelerating. The car would leave the rabbit’s foot behind. That is, the rabbit’s foot would swing away from, not toward, the windshield. 3. (e) Newton’s first law states that an object continues in a state of rest or in a state of motion at a constant speed along a straight line, unless compelled to change that state by a net force. All three statements are consistent with the first law. 4. (a) Newton’s second law with a net force of 7560 N – 7340 N = 220 N due north gives the answer directly. 5. (c) Newton’s second law gives the answer directly, provided the net force is calculated by vector addition of the two given forces. The direction of the net force gives the direction of the acceleration. 6. (e) Newton’s second law gives the answer directly. One method is to determine the total acceleration by vector addition of the two given components. The net force has the same direction as the acceleration. 7. (e) Answers a and b are false, according to the third law, which states that whenever one body exerts a force on a second body, the second body exerts an oppositely directed force of equal magnitude on the first body. It does not matter whether one of the bodies is stationary or whether it collapses. Answer c is false, because according to the third law, Sam and his sister experience forces of equal magnitudes during the push-off. Since Sam has the greater mass, he flies off with the smaller acceleration, according to the second law. Answer d is false, because in catching and throwing the ball each astronaut applies a force to it, and, according to the third law, the ball applies an oppositely directed force of equal magnitude to each astronaut. These reaction forces accelerate the astronauts away from each other, so that the distance between them increases. 8. (b) Newton’s third law indicates that Paul and Tom apply forces of equal magnitude to each other. According to Newton’s second law, the magnitude of each of these forces is the mass times the magnitude of the acceleration. Thus, we have mPaulaPaul = mTomaTom, or mPaul/mTom = aTom/aPaul.

160 FORCES AND NEWTON'S LAWS OF MOTION

9. (e) Newton’s law of gravitation gives the answer directly. According to this law the weight is directly proportional to the mass of the planet, so twice the mass means twice the weight. However, this law also indicates that the weight is inversely proportional to the square of the planet’s radius, so three times the radius means one ninth the weight. Together, these two factors mean that the weight on the planet is 2/9 or 0.222 times your earth-weight. 10. (c) Newton’s law of gravitation gives the answer, provided that the distance between the centers of the spheres is used for r (r = 0.50 m + 1.20 m + 0.80 m), rather than the distance between the surfaces of the spheres. 11. (a) The answer follows directly from the fact that weight W is given by W = mg, where m is the mass and g is the acceleration due to the earth’s gravity. Thus, m = (784 N)/(9.80 m/s2) = 80.0 kg. The mass is the same on Mercury as on Earth, because mass is an intrinsic property of matter. 12. (d) What matters is the direction of the elevator’s acceleration. When the acceleration is upward, the apparent weight is greater than the true weight. When the acceleration is downward, the apparent weight is less than the true weight. In both possibilities the acceleration points upward. 13. (b) According to Newton’s third law, the pusher and the wall exert forces of equal magnitude but opposite directions on each other. The normal force is the component of the wall’s force that is perpendicular to the wall. Thus, it has the same magnitude as the component of the pusher’s force that is perpendicular to the wall. As a result, the normal forces are ranked in the same order as the perpendicular components of the pusher’s forces. The smallest perpendicular component is in B, and the largest is in C. 14. (a) The static frictional force is balancing the component of the block’s weight that points down the slope of the incline. This component is smallest in B and greatest in A. 15. (b) The static frictional force that blocks A and B exert on each other has a magnitude f. The force that B exerts on A is directed to the right (the positive direction), while the force that A exerts on B is directed to the left. Blocks B and C also exert static frictional forces on each other, but these forces have a magnitude 2f, because the normal force pressing B and C together is twice the normal force pressing A and B together. The force that C exerts on B is directed to the right, while the force that B exerts on C is directed to the left. In summary, then, block A experiences a single frictional force +f, which is the net frictional force; block B experiences two frictional forces, −f and +2f, the net frictional force being −f +2f = +f; block C experiences a single frictional force +2f, which is the net frictional force. It follows that fs, A = fs, B = fs, C/2. 16. (c) The magnitude of the kinetic frictional force is proportional to the magnitude of the normal force. The normal force is smallest in B, because the vertical component of F compensates for part of the block’s weight. In contrast, the normal force is greatest in C, because the vertical component of F adds to the weight of the block.

Chapter 4 Answers to Focus on Concepts Questions

161

17. (d) Acceleration is inversely proportional to mass, according to Newton’s second law. This law also indicates that acceleration is directly proportional to the net force. The frictional force is the net force acting on a block, and its magnitude is directly proportional to the magnitude of the normal force. However, in each of the pictures the normal force is directly proportional to the weight and, thus, the mass of a block. The inverse proportionality of the acceleration to mass and the direct proportionality of the net force to mass offset each other. The result is that the deceleration is the same in each case. 18. (e) In B the tension T is the smallest, because three rope segments support the weight W of the block, with the result that 3T = W, or T = W/3. In A the tension is the greatest, because only one rope segment supports the weight of the block, with the result that T = W. 19. (c) Since the engines are shut down and since nothing is nearby to exert a force, the net force acting on the probe is zero, and its acceleration must be zero, according to Newton’s second law. With zero acceleration the probe is in equilibrium. 20. (a) The hallmark of an object in equilibrium is that it has no acceleration. Therefore, an object in equilibrium need not be at rest. It can be moving with a constant velocity. 21. (b) Since the object is not in equilibrium, it must be accelerating. Newton’s second law, in turn, implies that a net force must be present to cause the acceleration. Whether the net forces arises from a single force, two perpendicular forces, or three forces is not important, because only the net force appears in the second law. 22. (d) The block is at rest and, therefore, in equilibrium. According to Newton’s second law, then, the net force acting on the block in a direction parallel to the inclined surface of the incline must be zero. This means that the force of static friction directed up the incline must balance the component of the block’s weight directed down the incline [(8.0 kg)(9.8 m/s2) sin 22º = 29 N]. 23. (b) Since the boxes move at a constant velocity, they have no acceleration and are, therefore, in equilibrium. According to Newton’s second law, the net force acting on each box must be zero. Thus, Newton’s second law applied to each box gives two equations in two unknowns, the magnitude of the tension in the rope between the boxes and the kinetic frictional force that acts on each box. Note that the frictional forces acting on the boxes are identical, because the boxes are identical. Solving these two equations shows that the tension is one-half of the applied force. 24. 31 kg·m/s2 25. 517 N

162 FORCES AND NEWTON'S LAWS OF MOTION

CHAPTER 4 FORCES AND NEWTON'S LAWS

OF MOTION PROBLEMS ______________________________________________________________________________ 1.

REASONING AND SOLUTION According to Newton’s second law, the acceleration is a = ΣF/m. Since the pilot and the plane have the same acceleration, we can write

ΣF ΣF = m PILOT m PLANE

ΣF m PLANE

( ΣF )PILOT = mPILOT

or

Therefore, we find 3.7 × 104 N = 93 N 4 × 3.1 10 kg ______________________________________________________________________________

( ΣF )PILOT = ( 78 kg )

2.

REASONING Suppose the bobsled is moving along the +x direction. There are two forces acting on it that are parallel to its motion; a force +Fx propelling it forward and a force of –450 N that is resisting its motion. The net force is the sum of these two forces. According to Newton’s second law, Equation 4.2a, the net force is equal to the mass of the bobsled times its acceleration. Since the mass and acceleration are known, we can use the second law to determine the magnitude of the propelling force. SOLUTION a. Newton’s second law states that + Fx − 450 N = ma x

(4.2a)

ΣFx

Solving this equation for Fx gives

(

)

Fx = max + 450 N = ( 270 kg ) 2.4 m/s2 + 450 N = 1100 N b. The magnitude of the net force that acts on the bobsled is

(

)

ΣFx = max = ( 270 kg ) 2.4 m/s2 = 650 N

(4.2a)

____________________________________________________________________________________________

3.

REASONING In each case, we will apply Newton’s second law. Remember that it is the net force that appears in the second law. The net force is the vector sum of both forces.

Chapter 4 Problems

163

SOLUTION a. We will use Newton’s second law, ΣFx = max, to find the force F2. Taking the positive x direction to be to the right, we have

F1 + F2 = max

so

F2 = max − F1

ΣFx F2 = (3.0 kg)(+5.0 m/s2) − (+9.0 N) = +6 N

b. Applying Newton’s second law again gives F2 = max − F1 = (3.0 kg)(−5.0 m/s2) − (+9.0 N) = −24 N

c. An application of Newton’s second law gives F2 = max − F1 = (3.0 kg)(0 m/s2) − (+9.0 N) = −9.0 N

______________________________________________________________________________ 4.

REASONING According to Newton’s second law, Equation 4.1, the average net force ΣF is equal to the product of the object’s mass m and the average acceleration a . The average acceleration is equal to the change in velocity divided by the elapsed time (Equation 2.4), where the change in velocity is the final velocity v minus the initial velocity v0. SOLUTION The average net force exerted on the car and riders is

v − v0

)

(

45 m/s − 0 m/s = 5.5 × 103 kg = 3.5 × 104 N t − t0 7.0 s ______________________________________________________________________________ ∑ F = ma = m

5.

SSM REASONING The net force acting on the ball can be calculated using Newton's second law. Before we can use Newton's second law, however, we must use Equation 2.9 from the equations of kinematics to determine the acceleration of the ball.

SOLUTION According to Equation 2.9, the acceleration of the ball is given by

a=

v 2 − v02 2x

164 FORCES AND NEWTON'S LAWS OF MOTION

Thus, the magnitude of the net force on the ball is given by

v 2 − v02 (45 m/s)2 – (0 m/s)2 ∑ F = ma = m = (0.058 kg) = 130 N 2x 2(0.44 m) ______________________________________________________________________________ 6.

REASONING AND SOLUTION The acceleration is obtained from x = v0t +

1 2 at 2

where v0 = 0 m/s. So a = 2x/t2

Newton’s second law gives 2 (18 m ) 2x = 2900 N ΣF = ma = m 2 = ( 72 kg ) ( 0.95 s )2 t

______________________________________________________________________________ 7.

SSM REASONING AND SOLUTION The acceleration required is

a=

v 2 − v02 2x

− (15.0 m/s ) = = −2.25 m/s 2 2 ( 50.0 m ) 2

Newton's second law then gives the magnitude of the net force as F = ma = (1580 kg)(2.25 m/s2) = 3560 N ____________________________________________________________________________________________

8.

REASONING We do not have sufficient information to calculate the average net force applied to the fist by taking the vector sum of the individually applied forces. However, we have the mass m of the fist, as well as its initial velocity (v0 = 0 m/s, since the fist starts from rest), final velocity (v = 8.0 m/s), and the elapsed time ( t = 0.15 s). Therefore we can use ∆v v − v0 = Equation 2.4 a = to determine the average acceleration a of the fist and then ∆t ∆t use Equation 4.1 (Newton’s second law, ΣF = ma ) to find the average net force ΣF applied to the fist. SOLUTION Inserting the relation a = average net force applied to the fist:

∆v v − v0 = into Newton’s second law yields the ∆t ∆t

Chapter 4 Problems

165

v − v0 8.0 m/s − 0 m/s ΣF = ma = m = ( 0.70 kg ) = 37 N 0.15 s ∆t

9.

SSM WWW REASONING Let due east be chosen as the positive direction. Then, when both forces point due east, Newton's second law gives

FA + FB = ma1

(1)

ΣF

where a1 = 0.50 m/s 2 . When FA points due east and FB points due west, Newton's second law gives FA – FB = ma2

(2)

ΣF

where a2 = 0.40 m/s 2 . These two equations can be used to find the magnitude of each force. SOLUTION a. Adding Equations 1 and 2 gives

FA =

m ( a1 + a2 ) 2

=

(8.0 kg ) ( 0.50 m / s2 + 0.40 m / s 2 ) 2

= 3.6 N

b. Subtracting Equation 2 from Equation 1 gives FB =

m ( a1 − a2 ) 2

=

(8.0 kg ) ( 0.50 m / s2 − 0.40 m / s 2 ) 2

= 0.40 N

____________________________________________________________________________________________

10. REASONING From Newton’s second law, we know that the net force ΣF acting on the electron is directly proportional to its acceleration, so in part a we will first find the electron’s acceleration. The problem text gives the electron’s initial velocity (v0 = +5.40×105 m/s) and final velocity (v = +2.10×106 m/s), as well as its displacement (x = +0.038 m) during the interval of acceleration. The elapsed time is not known, so we

(

)

will use Equation 2.9 v 2 = v02 + 2ax to calculate the electron’s acceleration. Then we will find the net force acting on the electron from Equation 4.1 ( ΣF = ma ) and the electron’s mass. Because F1 points in the +x direction and F2 points in the −x direction, the net force

166 FORCES AND NEWTON'S LAWS OF MOTION

acting on the electron is ΣF = F1 − F2 . In part b of the problem, we will rearrange this expression to obtain the magnitude of the second electric force. SOLUTION a. Solving Equation 2.9 for the electron’s acceleration, we find that

a=

v 2 − v02 2x

+2.10 × 106 m/s ) − ( +5.40 × 105 m/s ) ( = 2

2

2 ( +0.038 m )

= +5.42 × 1013 m/s 2

Newton’s 2nd law of motion then gives the net force causing the acceleration of the electron:

∑ F = ma = ( 9.11×10−31 kg )( +5.42 ×1013 m/s2 ) = +4.94 ×10−17 N b. The net force acting on the electron is ΣF = F1 − F2 , so the magnitude of the second electric force is F2 = F1 − ΣF , where ΣF is the net force found in part a:

F2 = F1 − ΣF = 7.50 × 10−17 N − 4.94 ×10−17 N = 2.56 × 10−17 N

11. REASONING Newton’s second law gives the acceleration as a = (ΣF)/m. Since we seek only the horizontal acceleration, it is the x component of this equation that we will use; ax = (ΣFx)/m. For completeness, however, the free-body diagram will include the vertical forces also (the normal force FN and the weight W). SOLUTION The free-body diagram is shown at the right, where

+y

F1 = 59.0 N F2 = 33.0 N θ = 70.0°

When F1 is replaced by its x and y components, we obtain the free body diagram in the following drawing.

FN

F2

+x

θ W

F1

Chapter 4 Problems

Choosing right to be the positive direction, we have ΣF F cos θ − F2 ax = x = 1 m m ax

( 59.0 N ) cos 70.0° − ( 33.0 N ) = = 7.00 kg

−1.83 m/s

167

+y

FN

2

F2

The minus sign indicates that the horizontal acceleration points to the left .

F1cos θ

+x

F1sin θ W

12. REASONING The net force ΣF has a horizontal component ΣFx and a vertical component ΣFy. Since these components are perpendicular, the Pythagorean theorem applies (Equation 1.7), and the magnitude of the net force is ΣF =

( ΣFx )

2

( )

+ ΣFy

2

. Newton’s second law

allows us to express the components of the net force acting on the ball in terms of its mass and the horizontal and vertical components of its acceleration: ΣFx = ma x , ΣFy = ma y (Equations 4.2a and 4.2b).

SOLUTION Combining the Pythagorean theorem with Newton’s second law, we obtain the magnitude of the net force acting on the ball: ΣF =

( ΣFx )

2

= ( 0.430 kg )

13.

( )

+ ΣFy

2

=

( max )

2

(

+ ma y

)

2

(810 m/s2 ) + (1100 m/s2 ) 2

2

= m ax2 + a 2y = 590 N

SSM REASONING To determine the acceleration we will use Newton’s second law ΣF = ma. Two forces act on the rocket, the thrust T and the rocket’s weight W, which is mg = (4.50 × 105 kg)(9.80 m/s2) = 4.41 × 106 N. Both of these forces must be considered when determining the net force ΣF. The direction of the acceleration is the same as the direction of the net force.

SOLUTION In constructing the free-body diagram for the rocket we choose upward and to the right as the positive directions. The free-body diagram is as follows:

168 FORCES AND NEWTON'S LAWS OF MOTION

The x component of the net force is

ΣFx = T cos 55.0° = 7.50 ×106 N cos 55.0° = 4.30 ×106 N

(

T

+y

Ty

)

55.0º

The y component of the net force is W

(

= T sin 55.0º +x

Tx = T cos 55.0º

)

ΣFy = T sin 55.0° − W = 7.50 × 106 N sin 55.0° − 4.41×106 N = 1.73 × 106 N The magnitudes of the net force and of the acceleration are

ΣF =

a=

( ΣFx )

2

( )

+ ΣFy

2

2

( )

+ ΣFy

2

( 4.30 ×106 N ) + (1.73 ×106 N ) 2

=

m

( ΣFx )

2

4.50 ×10 kg 5

= 10.3 m/s 2

The direction of the acceleration is the same as the direction of the net force. Thus, it is directed above the horizontal at an angle of 6 ΣFy −1 1.73 × 10 N = tan = 21.9° 6 4.30 × 10 N ΣFx

θ = tan −1

14. REASONING The acceleration of the sky diver can be obtained directly from Newton’s second law as the net force divided by the sky diver’s mass. The net force is the vector sum of the sky diver’s weight and the drag force. SOLUTION From Newton’s second law, ΣF = ma (Equation 4.1), the sky diver’s acceleration is a=

f

ΣF m

The free-body diagram shows the two forces acting on the sky diver, his weight W and the drag force f. The net force is ΣF = f − W . Thus, the acceleration can be written as f −W a= m

+ −

W Free-body diagram

Chapter 4 Problems

169

The acceleration of the sky diver is a=

f − W 1027 N − 915 N = = +1.20 m/s 2 93.4 kg m

Note that the acceleration is positive, indicating that it points upward . ____________________________________________________________________________________________

15. REASONING Newton’s second law, ΣF = ma, states that a net force of ΣF must act on an object of mass m in order to impart an acceleration a to the object. In the impact shock test the box is subjected to a large deceleration and, hence, a correspondingly large net force. To determine the net force we will determine the deceleration in a kinematics calculation and use it in Newton’s second law.

SOLUTION According to Newton’s second law, the net force is ΣF = ma, where the acceleration can be determined with the aid of Equation 2.4 (v = v0 + at). According to this equation v − v0 a= t Substituting this result for the acceleration into the second law gives

v − v0 ΣF = ma = m t Since the initial velocity (v0 = +220 m/s), final velocity (v = 0 m/s), and the duration of the collision (t = 6.5 × 10−3 s) are known, we find

v − v0 0 m/s − 220 m/s 6 ΣF = m = ( 41 kg ) = −1.39 × 10 N − 3 6.5 ×10 s t The minus sign indicates that the net force points opposite to the direction in which the box is thrown, which has been assumed to be the positive direction. The magnitude of the net force is 1.39 × 106 N , which is over three hundred thousand pounds.

16. REASONING Since there is only one force acting on the man in the horizontal direction, it is the net force. According to Newton’s second law, Equation 4.1, the man must accelerate under the action of this force. The factors that determine this acceleration are (1) the magnitude and (2) the direction of the force exerted on the man, and (3) the mass of the man.

170 FORCES AND NEWTON'S LAWS OF MOTION

When the woman exerts a force on the man, the man exerts a force of equal magnitude, but opposite direction, on the woman (Newton’s third law). It is the only force acting on the woman in the horizontal direction, so, as is the case with the man, she must accelerate. The factors that determine her acceleration are (1) the magnitude and (2) the direction of the force exerted on her, and (3) the her mass. SOLUTION a. The acceleration of the man is, according to Equation 4.1, equal to the net force acting on him divided by his mass. aman =

ΣF 45 N = = 0.55 m / s 2 (due east) m 82 kg

b. The acceleration of the woman is equal to the net force acting on her divided by her mass.

awoman =

ΣF 45 N = = 0.94 m / s 2 (due west) m 48 kg

______________________________________________________________________________________

17. REASONING Equations 3.5a

( x = v0 xt + 12 axt 2 )

and 3.5b

( y = v0 yt + 12 ayt 2 )

give the

displacements of an object under the influence of constant accelerations ax and ay. We can add these displacements as vectors to find the magnitude and direction of the resultant displacement. To use Equations 3.5a and 3.5b, however, we must have values for ax and ay. We can obtain these values from Newton’s second law, provided that we combine the given forces to calculate the x and y components of the net force acting on the duck, and it is here that our solution begins. SOLUTION Let the directions due east and due north, respectively, be the +x and +y directions. Then, the components of the net force are ΣFx = 0.10 N + ( 0.20 N ) cos 52° = 0.2231 N ΣFy = – ( 0.20 N ) sin 52° = –0.1576 N

According to Newton’s second law, the components of the acceleration are ax = ay =

ΣFx m ΣFy m

=

0.2231 N = 0.08924 m/s 2 2.5 kg

=

–0.1576 N = –0.06304 m/s 2 2.5 kg

Chapter 4 Problems

171

From Equations 3.5a and 3.5b, we now obtain the displacements in the x and y directions:

(

)

x = v0 xt + 12 axt 2 = ( 0.11 m/s )( 3.0 s ) + 12 0.08924 m/s 2 ( 3.0 s ) = 0.7316 m

(

2

)

y = v0 y t + 12 a y t 2 = ( 0 m/s )( 3.0 s ) + 12 –0.06304 m/s2 ( 3.0 s ) = –0.2837 m 2

The magnitude of the resultant displacement is r = x2 + y 2 =

( 0.7316 m )2 + ( –0.2837 m )2

= 0.78 m

The direction of the resultant displacement is 0.2837 m = 21° south of east 0.7316 m

θ = tan –1

____________________________________________________________________________________________

(

)

18. REASONING For both the tug and the asteroid, Equation 2.8 x = v0t + 12 at 2 applies with

v0 = 0 m/s, since both are initially at rest. In applying this equation, we must be careful and use the proper acceleration for each object. Newton’s second law indicates that the acceleration is given by a = ΣF/m. In this expression, we note that the magnitudes of the net forces acting on the tug and the asteroid are the same, according to Newton’s action-reaction law. The masses of the tug and the asteroid are different, however. Thus, the distance traveled for either object is given by, where we use for ΣF only the magnitude of the pulling force ΣF 2 x = v0t + 12 at 2 = 12 t m SOLUTION Let L be the initial distance between the tug and the asteroid. When the two objects meet, the distances that each has traveled must add up to equal L. Therefore,

L = xT + x A = 12 aT t 2 + 12 a At 2 ΣF 2 1 ΣF 2 1 1 1 2 + L = 12 t + 2 t = 2 ΣF t m T mA mT mA Solving for the time t gives

172 FORCES AND NEWTON'S LAWS OF MOTION

t=

2L 1 1 ΣF + mT m A

=

2 ( 450 m ) 1 1 + ( 490 N ) 3500 kg 6200 kg

= 64 s

____________________________________________________________________________________________

19.

SSM WWW REASONING We first determine the acceleration of the boat. Then, using Newton's second law, we can find the net force ∑ F that acts on the boat. Since two of the three forces are known, we can solve for the unknown force FW once the net force

∑ F is known. SOLUTION Let the direction due east be the positive x direction and the direction due north be the positive y direction. The x and y components of the initial velocity of the boat are then v0 x = (2.00 m/s) cos 15.0° = 1.93 m/s v0 y = (2.00 m/s) sin 15.0° = 0.518 m/s

Thirty seconds later, the x and y velocity components of the boat are vx = (4.00 m/s) cos 35.0° = 3.28 m/s v y = (4.00 m/s) sin 35.0° = 2.29 m/s

Therefore, according to Equations 3.3a and 3.3b, the x and y components of the acceleration of the boat are v −v 3.28 m/s – 1.93 m/s ax = x 0 x = = 4.50 × 10−2 m/s 2 t 30.0 s

ay =

v y − v0 y t

=

2.29 m/s – 0.518 m/s = 5.91× 10−2 m/s 2 30.0 s

Thus, the x and y components of the net force that act on the boat are ∑ Fx = max = (325 kg) (4.50 × 10 –2 m/s 2 ) = 14.6 N

∑ Fy = ma y = (325 kg) (5.91×10 –2 m/s 2 ) = 19.2 N The following table gives the x and y components of the net force ∑ F and the two known forces that act on the boat. The fourth row of that table gives the components of the unknown force FW .

Chapter 4 Problems

173

Force

x-Component

y-Component

∑F

14.6 N

19.2 N

F1

(31.0 N) cos 15.0° = 29.9 N

(31.0 N) sin 15.0° = 8.02 N

F2

–(23.0 N ) cos 15.0° = –22.2 N

–(23.0 N) sin 15.0° = –5.95 N

FW = ∑ F − F1 − F2

14.6 N – 29.9 N + 22.2 N = 6.9 N

19.2 N – 8.02 N + 5.95 N = 17.1 N

The magnitude of FW is given by the Pythagorean theorem as FW = (6.9 N) 2 + (17.1N ) 2 = 18.4 N

The angle θ that FW makes with the x axis is

17.1 N = 68° 6.9 N

θ = tan −1

17.1 N

θ

Therefore, the direction of FW is 68°, north of east .

6.9 N

____________________________________________________________________________________________

20. REASONING The gravitational force acting on each object is specified by Newton’s law of universal gravitation. The acceleration of each object when released can be determined with the aid of Newton’s second law. We recognize that the gravitational force is the only force acting on either object, so that it is the net force to use when applying the second law. SOLUTION a. The magnitude of the gravitational force exerted on the rock by the earth is given by Equation 4.3 as

Frock = =

Gmearth mrock

(

2 rearth 6.67 × 10−11 N ⋅ m 2 / kg 2 5.98 × 1024 kg ( 5.0 kg )

(

)(

6.38 × 106 m

)

)

2

= 49 N

The magnitude of the gravitational force exerted on the pebble by the earth is

174 FORCES AND NEWTON'S LAWS OF MOTION

Fpebble = =

Gmearth mpebble

(

2 rearth 6.67 × 10−11 N ⋅ m 2 / kg 2 5.98 × 1024 kg 3.0 × 10−4 kg

)(

( 6.38 × 10 m ) 6

)(

2

)=

2.9 × 10−3 N

b. According to the second law, the magnitude of the acceleration of the rock is equal to the gravitational force exerted on the rock divided by its mass. arock =

Frock mrock

Gmearth

= =

(

2 rearth 6.67 × 10−11 N ⋅ m 2 / kg 2 5.98 × 1024 kg

)( 2 ( 6.38 × 106 m )

)=

9.80 m /s 2

According to the second law, the magnitude of the acceleration of the pebble is equal to the gravitational force exerted on the pebble divided by its mass. apebble =

Fpebble mpebble

=

Gmearth 2 rearth

6.67 × 10−11 N ⋅ m 2 / kg 2 )( 5.98 × 1024 kg ) ( = = 2 6 6.38 × 10 m ( )

9.80 m /s 2

______________________________________________________________________________ 21. REASONING AND SOLUTION a. According to Equation 4.5, the weight of the space traveler of mass

m = 115 kg

on

earth is W = mg = (115 kg) (9.80 m/s 2 ) = 1.13 × 103 N

b. In interplanetary space where there are no nearby planetary objects, the gravitational force exerted on the space traveler is zero and g = 0 m/s2. Therefore, the weight is W = 0 N . Since the mass of an object is an intrinsic property of the object and is

independent of its location in the universe, the mass of the space traveler is still m = 115 kg . ____________________________________________________________________________________________

Chapter 4 Problems

22. REASONING With air resistance neglected, only two forces act on the bungee jumper at this instant (see the free-body diagram): the bungee cord pulls up on her with a force B, and the earth pulls down on her with a gravitational force mg. Because we know the jumper’s mass and acceleration, we can apply Newton’s second law to this free-body diagram and solve for B.

175

B + − mg

SOLUTION We will take the direction of the jumper’s acceleration (downward) as negative. Then, the net force acting on the bungee jumper is ΣF = B − mg. With Newton’s second law (ΣF = ma), this becomes ma = B − mg. We now solve for B:

Free-body diagram of jumper

(

)

B = ma + mg = m ( a + g ) = ( 55 kg ) −7.6 m/s2 + 9.80 m/s2 = 120 N As indicated in the free-body diagram, the direction of the force applied by the bungee cord is upward .

23. REASONING The earth exerts a gravitational force on the raindrop, and simultaneously the raindrop exerts a gravitational force on the earth. This gravitational force is equal in magnitude to the gravitational force that the earth exerts on the raindrop. The forces that the raindrop and the earth exert on each other are Newton’s third law (action–reaction) forces. Newton’s law of universal gravitation specifies the magnitude of both forces. SOLUTION a. The magnitude of the gravitational force exerted on the raindrop by the earth is given by Equation 4.3: Fraindrop = =

Gmearth mraindrop

(

2 rearth 6.67 × 10−11 N ⋅ m 2 / kg 2 5.98 × 1024 kg 5.2 × 10−7 kg

)(

( 6.38 × 10 m ) 6

)(

2

)=

5.1 × 10−6 N

b. The magnitude of the gravitational force exerted on the earth by the raindrop is Fearth = =

Gmearth mraindrop

(

2 rearth 6.67 × 10−11 N ⋅ m 2 / kg 2 5.98 × 1024 kg 5.2 × 10−7 kg

)(

( 6.38 × 10 m ) 6

)(

2

)=

5.1 × 10−6 N

______________________________________________________________________________

176 FORCES AND NEWTON'S LAWS OF MOTION

24. REASONING Newton’s law of universal gravitation indicates that the gravitational force that each uniform sphere exerts on the other has a magnitude that is inversely proportional to the square of the distance between the centers of the spheres. Therefore, the maximum gravitational force between two uniform spheres occurs when the centers of the spheres are as close together as possible, and this occurs when the surfaces of the spheres are touching. Then, the distance between the centers of the spheres is the sum of the two radii. SOLUTION When the bowling ball and the billiard ball are touching, the distance between their centers is r = rBowling + rBilliard. Using this expression in Newton’s law of universal gravitation gives

F=

GmBowling mBilliard

GmBowling mBilliard

=

( rBowling + rBilliard ) 6.67 ×10−11 N ⋅ m 2 / kg 2 ) ( 7.2 kg )( 0.38 kg ) ( = = 9.6 × 10−9 N r2

2

( 0.11 m + 0.028 m )2

25.

SSM REASONING AND SOLUTION a. Combining Equations 4.4 and 4.5, we see that the acceleration due to gravity on the surface of Saturn can be calculated as follows: gSaturn = G

M Saturn 2 rSaturn

(

= 6.67 ×10

–11

N ⋅ m /kg 2

2

(5.67 ×1026 kg ) =

) (6.00 ×107 m)2

10.5 m/s 2

b. The ratio of the person’s weight on Saturn to that on earth is

WSaturn Wearth

=

mgSaturn mg earth

=

gSaturn gearth

=

10.5 m/s 2 = 1.07 9.80 m/s 2

____________________________________________________________________________________________

26. REASONING As discussed in Conceptual Example 7, the same net force is required on the moon as on the earth. This net force is given by Newton’s second law as ΣF = ma, where the mass m is the same in both places. Thus, from the given mass and acceleration, we can calculate the net force. On the moon, the net force comes about due to the drive force and the opposing frictional force. Since the drive force is given, we can find the frictional force. SOLUTION Newton’s second law, with the direction of motion taken as positive, gives

ΣF = ma

or

(1430 N ) – f

(

)(

= 5.90 × 103 kg 0.220 m/s 2

)

177

Chapter 4 Problems

Solving for the frictional force f , we find

(

)(

)

f = (1430 N ) – 5.90 × 103 kg 0.220 m/s 2 = 130N ____________________________________________________________________________________________

27.

SSM REASONING AND SOLUTION According to Equations 4.4 and 4.5, the weight of an object of mass m at a distance r from the center of the earth is mg =

GM E m r2

In a circular orbit that is 3.59 ×107 m above the surface of the earth ( radius = 6.38 ×106 m , mass = 5.98 ×1024 kg ), the total distance from the center of the earth is

r = 3.59 ×107 m + 6.38 ×106 m . Thus the acceleration g due to gravity is g=

GM E r

2

=

(6.67 × 10−11N ⋅ m 2 /kg 2 )(5.98 ×1024 kg) = (3.59 × 107 m + 6.38 ×106 m) 2

0.223 m/s 2

____________________________________________________________________________________________

28. REASONING AND SOLUTION The magnitude of the net force acting on the moon is found by the Pythagorean theorem to be

F

SM

Moon

F Sun

F=

2 FSM

+

F

EM

2 FEM

Earth

Newton's law of gravitation applied to the sun-moon (the units have been suppressed)

FSM =

GmSmM 2 rSM

=

( 6.67 ×10−11 )(1.99 ×1030 )( 7.35 ×1022 ) = 4.34 ×1020 N 2 11 (1.50 ×10 )

A similar application to the earth-moon gives

FEM =

GmE mM 2 rEM

=

( 6.67 ×10−11 )( 5.98 ×1024 )( 7.35 ×1022 ) = 1.98 ×1020 N 2 8 ( 3.85 ×10 )

The net force on the moon is then F=

( 4.34 × 1020 N ) + (1.98 × 1020 N ) 2

2

= 4.77 × 1020 N

178 FORCES AND NEWTON'S LAWS OF MOTION

____________________________________________________________________________________________

29. REASONING The magnitude of the gravitational force exerted on the satellite by the earth is given by Equation 4.3 as F = Gmsatellite mearth / r 2 , where r is the distance between the satellite and the center of the earth. This expression also gives the magnitude of the gravitational force exerted on the earth by the satellite. According to Newton’s second law, the magnitude of the earth’s acceleration is equal to the magnitude of the gravitational force exerted on it divided by its mass. Similarly, the magnitude of the satellite’s acceleration is equal to the magnitude of the gravitational force exerted on it divided by its mass. SOLUTION a. The magnitude of the gravitational force exerted on the satellite when it is a distance of two earth radii from the center of the earth is

F=

Gmsatellite mearth r

2

=

( 6.67 × 10−11 N ⋅ m2 / kg2 ) ( 425 kg ) (5.98 × 1024 kg ) 2 ( 2 ) ( 6.38 × 106 m )

= 1.04 × 103 N

b. The magnitude of the gravitational force exerted on the earth when it is a distance of two earth radii from the center of the satellite is F=

Gmsatellite mearth r

2

=

( 6.67 × 10−11 N ⋅ m2 / kg2 ) ( 425 kg ) (5.98 × 1024 kg ) 2 ( 2 ) ( 6.38 × 106 m )

= 1.04 × 103 N

c. The acceleration of the satellite can be obtained from Newton’s second law. asatellite =

F msatellite

=

1.04 × 103 N = 425 kg

2.45 m / s 2

d. The acceleration of the earth can also be obtained from Newton’s second law.

aearth =

F mearth

=

1.04 × 103 N = 1.74 × 10−22 m / s 2 24 5.98 × 10 kg

____________________________________________________________________________________________

30. REASONING The weight of a person on the earth is the gravitational force Fearth that it exerts on the person. The magnitude of this force is given by Equation 4.3 as Fearth = G

mearth mperson 2 rearth

Chapter 4 Problems

179

where rearth is the distance from the center of the earth to the person. In a similar fashion, the weight of the person on another planet is

Fplanet = G

mplanet mperson 2 rplanet

We will use these two expressions to obtain the weight of the traveler on the planet. SOLUTION Dividing Fplanet by Fearth we have

Fplanet Fearth

G = G

mplanet mperson 2 rplanet

mearth mperson

mplanet = m earth

2 rearth

rearth rplanet

2

or Fplanet

Since we are given that

mplanet mearth

mplanet = Fearth m earth

= 3 and

rearth rplanet

=

rearth rplanet

2

1 , the weight of the space traveler on the 2

planet is 2

1 Fplanet = ( 540.0 N )( 3) = 405.0 N 2 ______________________________________________________________________________

31.

SSM REASONING According to Equation 4.4, the weights of an object of mass m on the surfaces of planet A (mass = MA, radius = R ) and planet B (mass = MB , radius = R ) are

WA =

GM A m R2

and

WB =

GM B m R2

The difference between these weights is given in the problem.

180 FORCES AND NEWTON'S LAWS OF MOTION

SOLUTION The difference in weights is WA – WB =

GM A m R

2

–

GM B m R

2

=

Gm (MA – MB ) R2

Rearranging this result, we find

MA – MB

(W – WB ) R2 = = A Gm

( 3620 N ) (1.33 × 107 m ) = (6.67 × 10−11N ⋅ m 2 /kg 2 ) ( 5450 kg ) 2

1.76 × 1024 kg

____________________________________________________________________________________________

32. REASONING AND SOLUTION The figure at the right shows the three spheres with sphere 3 being the sphere of unknown mass. Sphere 3 feels a force F31 due to the presence of sphere 1, and a force F32 due to the presence of sphere 2. The net force on sphere 3 is the resultant of F31 and F32. Note that since the spheres form an equilateral triangle, each interior angle is 60°. Therefore, both F31 and F32 make a 30° angle with the vertical line as shown.

3

F

F

31

32

1.20 m

30°

2

1

Furthermore, F31 and F32 have the same magnitude given by GMm3 F= r2 where M is the mass of either sphere 1 or 2 and m3 is the mass of sphere 3. The components of the two forces are shown in the following drawings:

F

31 30.0°

F sin θ

F cos θ

F cos θ

F

32

30.0°

F sin θ

Clearly, the horizontal components of the two forces add to zero. Thus, the net force on sphere 3 is the resultant of the vertical components of F31 and F32:

Chapter 4 Problems

F3 = 2 F cos θ = 2

GMm3 r2

181

cos θ

The acceleration of sphere 3 is given by Newton's second law: a3 =

F3 m3

=2

(6.67 ×10 GM cos θ = 2 2 r

−11

N ⋅ m 2 /kg 2 ) ( 2.80 kg )

(1.20 m )2

cos 30.0°

= 2.25 × 10 –10 m/s 2 ____________________________________________________________________________________________

33.

SSM WWW REASONING AND SOLUTION There are two forces that act on the balloon; they are, the combined weight of the balloon and its load, Mg, and the upward buoyant force FB . If we take upward as the positive direction, then, initially when the balloon is motionless, Newton's second law gives FB − Mg = 0 . If an amount of mass m is dropped overboard so that the balloon has an upward acceleration, Newton's second law for this situation is FB − ( M − m) g = ( M − m)a

But FB = mg , so that

Mg – ( M − m ) g = mg = ( M – m ) a

Solving for the mass m that should be dropped overboard, we obtain

(310 kg )(0.15 m/s 2 ) Ma = = 4.7 kg g + a 9.80 m/s 2 + 0.15 m/s 2 ______________________________________________________________________________ m=

34. REASONING AND SOLUTION acceleration, it follows that

Since both motions are characterized by constant yJ yE

=

1 a t2 2 J J 1 a t2 2 E E

where the subscripts designate those quantities that pertain to Jupiter and Earth. Since both objects fall the same distance (yJ = yE), the above ratio is equal to unity. Solving for the ratio of the times yields tJ tE

=

aE GM E / RE2 RJ = = aJ GM J / RJ2 RE

ME 1 = (11.2 ) = 0.628 MJ 318

____________________________________________________________________________________________

182 FORCES AND NEWTON'S LAWS OF MOTION

35. REASONING The gravitational force that the sun exerts on a person standing on the earth 2 is given by Equation 4.3 as Fsun = GM sun m / rsun-earth , where Msun is the mass of the sun, m is the mass of the person, and rsun-earth is the distance from the sun to the earth. Likewise, the gravitational force that the moon exerts on a person standing on the earth is given by 2 , where Mmoon is the mass of the moon and rmoon-earth is the Fmoon = GM moon m / rmoon-earth distance from the moon to the earth. These relations will allow us to determine whether the sun or the moon exerts the greater gravitational force on the person. SOLUTION Taking the ratio of Fsun to Fmoon, and using the mass and distance data from the inside of the text’s front cover, we find

GM sun m Fsun Fmoon

2 M rsun-earth = = sun M GM moon m moon 2 rmoon-earth

rmoon-earth rsun-earth

1.99 × 1030 kg 3.85 × 108 m = 11 7.35 × 1022 kg 1.50 × 10 m

2

2

= 178

Therefore, the sun exerts the greater gravitational force. ____________________________________________________________________________________________

36. REASONING The gravitational attraction between the planet and the moon is governed by Newton’s law of gravitation F = GMm/r2 (Equation 4.3), where M is the planet’s mass and m is the moon’s mass. Because the magnitude of this attractive force varies inversely with the square of the distance r between the center of the moon and the center of the planet, the maximum force Fmax occurs at the minimum distance rmin, and the minimum force Fmin at the maximum distance rmax. The problem text states that Fmax exceeds Fmin by 11%, or Fmax = 1.11 Fmin. This expression can be rearranged to give the ratio of the forces: Fmax/ Fmin = 1.11. We will use Equation 4.3 to compute the desired distance ratio in terms of this force ratio. SOLUTION From Equation 4.3, the ratio of the maximum gravitational force to the minimum gravitational force is

Fmax Fmin

=

GMm 2 rmin GMm 2 rmax

1 =

2 rmin

1 2 rmax

=

2 rmax 2 rmin

Chapter 4 Problems

183

Taking the square root of both sides of this expression and substituting the ratio of forces Fmax/ Fmin = 1.11 yields the ratio of distances:

rmax

Fmax

=

rmin

Fmin

= 1.11 = 1.05

The moon’s maximum distance from the center of the planet is therefore about 5% larger than its minimum distance.

37. REASONING We place the third particle (mass = m3) as shown in the following drawing: L D m3 m 2m The magnitude of the gravitational force that one particle exerts on another is given by Newton’s law of gravitation as F = Gm1m2/r2. Before the third particle is in place, this law

indicates that the force on each particle has a magnitude Fbefore = Gm2m/L2. After the third particle is in place, each of the first two particles experiences a greater net force, because the third particle also exerts a gravitational force on them. SOLUTION For the particle of mass m, we have Gmm3

Fafter Fbefore

=

Gm 2m 2 D L2 = L m3 + 1 Gm 2m 2mD 2 2 L 2

+

For the particle of mass 2m, we have

G 2mm3

Fafter Fbefore

=

( L – D)

2

+

Gm2m L2

Gm2m L2

=

L2 m3

m ( L – D)

2

+1

Since Fafter/Fbefore = 2 for both particles, we have

L2 m3 2mD

2

+1 =

L2 m3

m ( L – D)

2

+1

or

2D2 = ( L – D )

2

184 FORCES AND NEWTON'S LAWS OF MOTION

Expanding and rearranging this result gives D 2 + 2 LD − L2 = 0 , which can be solved for D using the quadratic formula:

D=

–2 L ±

( 2 L )2 – 4 (1) ( – L2 ) = 2 (1)

0.414 L or

– 2.414 L

The negative solution is discarded because the third particle lies on the +x axis between m and 2m. Thus, D = 0.414 L . ____________________________________________________________________________________________

38. REASONING In each case the object is in equilibrium. According to Equation 4.9b, ΣFy = 0, the net force acting in the y (vertical) direction must be zero. The net force is composed of the weight of the object(s) and the normal force exerted on them. SOLUTION a. There are three vertical forces acting on the crate: an upward normal force +FN that the floor exerts, the weight –m1g of the crate, and the weight –m2g of the person standing on the crate. Since the weights act downward, they are assigned negative numbers. Setting the sum of these forces equal to zero gives

FN + (−m1g ) + (−m2 g ) = 0 ΣFy The magnitude of the normal force is FN = m1g + m2g = (35 kg + 65 kg)(9.80 m/s2) = 980 N b. There are only two vertical forces acting on the person: an upward normal force +FN that the crate exerts and the weight –m2g of the person. Setting the sum of these forces equal to zero gives FN + (−m2 g ) = 0

ΣFy The magnitude of the normal force is FN = m2g = (65 kg)(9.80 m/s2) = 640 N ____________________________________________________________________________________________

Chapter 4 Problems

39.

185

SSM REASONING The book is kept from falling as long as the total static frictional force balances the weight of the book. The forces that act on the book are shown in the following free-body diagram, where P is the pressing force applied by each hand. fsMAX

fsMAX

P

P

W

In this diagram, note that there are two pressing forces, one from each hand. Each hand also applies a static frictional force, and, therefore, two static frictional forces are shown. The maximum static frictional force is related in the usual way to a normal force FN, but in this problem the normal force is provided by the pressing force, so that FN = P. SOLUTION Since the frictional forces balance the weight, we have 2 f sMAX = 2 ( µ s FN ) = 2 ( µ s P ) = W Solving for P, we find that

P=

W 31 N = = 39 N 2µ s 2 ( 0.40 )

____________________________________________________________________________________________

40. REASONING AND SOLUTION a. The apparent weight of the person is given by Equation 4.6 as FN = mg + ma = (95.0 kg)(9.80 m/s2 + 1.80 m/s2) = 1.10 × 103 N b.

FN = (95.0 kg)(9.80 m/s2) = 931 N

c.

FN = (95.0 kg)(9.80 m/s2 – 1.30 m/s2) = 808 N

____________________________________________________________________________________________

186 FORCES AND NEWTON'S LAWS OF MOTION

41. REASONING As the drawing shows, the normal FN force FN points perpendicular to the hill, while the weight W points vertically down. Since the car does not leave the surface of the hill, there is no acceleration in this perpendicular direction. Therefore, the magnitude of the θ perpendicular component of the weight W cos θ must equal W cos θ θ the magnitude of the normal force, FN = W cos θ . Thus, the W magnitude of the normal force is less than the magnitude of the weight. As the hill becomes steeper, θ increases, and cos θ decreases. Consequently, the normal force decreases as the hill becomes steeper. The magnitude of the normal force does not depend on whether the car is traveling up or down the hill. SOLUTION a. From the REASONING, we have that FN = W cos θ . The ratio of the magnitude of the normal force to the magnitude W of the weight is FN W

=

W cos θ = cos 15° = 0.97 W

b. When the angle is 35°, the ratio is W cos θ = cos 35° = 0.82 W W ______________________________________________________________________________ FN

=

42. REASONING The reading on the bathroom scale is Free-body diagrams of the man proportional to the normal force it exerts on the man. When he simply stands on the scale, his acceleration FN1 is zero, so the normal force pushing up on him balances the downward pull of gravity: FN1 = mg (see P FN2 the free-body diagram). Thus, the first reading on the scale is his actual mass m, the ratio of the normal force the scale exerts on him to the acceleration due to gravity: First reading = m = FN1/g = 92.6 kg. With the mg mg chin-up bar helping to support him, the normal force exerted on him by the scale decreases, and the second Standing Pulling down reading is the ratio of the reduced normal force FN2 to the acceleration due to gravity: Second reading = FN2/g = 75.1 kg. Lastly, we note that the magnitude P of the force the chin-up bar exerts on the man is exactly equal to the magnitude P of the force that the man exerts on the chin-up bar. This prediction is due to Newton’s third law. Therefore, it is a value for P that we seek. SOLUTION When the man is pulling down on the chin-up bar, there are two upward forces acting on him (see the second part of the drawing), and he is still at rest, so the sum of these

187

Chapter 4 Problems

two forces balances the downward pull of gravity: FN2 + P = mg, or P = mg − FN2 . Since the second reading on the scale is equal to FN2/g, the normal force the scale exerts on him is FN2 = (Second reading)g. Thus we obtain the magnitude P of the force the man exerts on the chin-up bar: P = mg − FN2 = mg − (Second reading) g = ( m − Second reading ) g

(

)

= ( 92.6 kg − 75.1 kg ) 9.80 m/s 2 = 172 N

43. REASONING As shown in the free-body diagram below, three forces act on the car: the static frictional force fs (directed up the hill), the normal force FN (directed perpendicular to the road), and its weight mg. As it sits on the hill, the car has an acceleration of zero 2 (ax = ay = 0 m/s ). Therefore, the net force acting on the car in the x direction must be zero

( ΣFx = 0 )

and the net force in the y direction must be zero ( ΣFy = 0 ) . These two relations

will allow us to find the normal force and the static frictional force. SOLUTION a. Applying Newton’s second ( ΣFy = 0) yields

+y

law to the y direction

FN

fs

ΣFy = + FN − mg cos15° = 0

(4.2b)

where the term −mg cos 15° is the y component of the car’s weight (negative, because this component points along the negative y axis). Solving for the magnitude FN of the normal force, we obtain

15º

+x

mg 15º

FN = mg cos15° = (1700 kg ) ( 9.80 m/s 2 ) cos15° = 1.6 × 104 N b. Applying Newton’s second law to the x direction ( ΣFx = 0 ) gives ΣFx = + mg sin15° − fs = 0

(4.2a)

where the term mg sin 15° is the x component of the car’s weight. Solving this expression for the static frictional force gives

fs = mg sin15° = (1700 kg ) ( 9.80 m/s2 ) sin15° = 4.3 ×103 N

188 FORCES AND NEWTON'S LAWS OF MOTION

44. REASONING a. Since the refrigerator does not move, the static frictional force must be equal in magnitude, but opposite in direction, to the horizontal pushing force that the person exerts on the refrigerator. b. The magnitude of the maximum static frictional force is given by Equation 4.7 as fsMAX = µs FN . This is also the largest possible force that the person can exert on the refrigerator before it begins to move. Thus, the factors that determine this force magnitude are the coefficient of static friction µs and the magnitude FN of the normal force (which is equal to the weight of the refrigerator in this case). SOLUTION a. Since the refrigerator does not move, it is in equilibrium, and the magnitude of the static frictional force must be equal to the magnitude of the horizontal pushing force. Thus, the magnitude of the static frictional force is 267 N . The direction of this force must be opposite to that of the pushing force, so the static frictional force is in the +x direction . b. The magnitude of the largest pushing force is given by Equation 4.7 as fsMAX = µs FN = µs mg = (0.65)(57 kg)(9.80 m/s 2 ) = 360 N ____________________________________________________________________________________________

45.

SSM REASONING AND SOLUTION Four forces act on the sled. They are the pulling force P, the force of kinetic friction f k , the weight mg of the sled, and the normal force FN exerted on the sled by the surface on which it slides. The following figures show free-body diagrams for the sled. In the diagram on the right, the forces have been resolved into their x and y components. y

P sin θ

y

FN

FN

P fk

P cos θ

fk

θ x

mg

x

mg

Chapter 4 Problems

189

Since the sled is pulled at constant velocity, its acceleration is zero, and Newton's second law in the direction of motion is (with right chosen as the positive direction) ∑ Fx = P cos θ − f k = ma x = 0 From Equation 4.8, we know that f k = µ k FN , so that the above expression becomes P cos θ − µk FN = 0

(1)

In the vertical direction, ∑ Fy = P sin θ + FN − mg = ma y = 0

(2)

Solving Equation (2) for the normal force, and substituting into Equation (1), we obtain

P cos θ − µk ( mg − P sin θ ) = 0 Solving for µ k , the coefficient of kinetic friction, we find

µk =

(80.0 N) cos 30.0° P cos θ = = 0.444 mg − P sin θ (20.0 kg) (9.80 m/s 2 ) − (80.0 N) sin 30.0°

____________________________________________________________________________________________

46. REASONING In each of the three cases under consideration the kinetic frictional force is given by fk = µkFN. However, the normal force FN varies from case to case. To determine the normal force, we use Equation 4.6 (FN = mg + ma) and thereby take into account the acceleration of the elevator. The normal force is greatest when the elevator accelerates upward (a positive) and smallest when the elevator accelerates downward (a negative).

SOLUTION a. When the elevator is stationary, its acceleration is a = 0 m/s2. Using Equation 4.6, we can express the kinetic frictional force as

f k = µ k FN = µ k ( mg + ma ) = µ k m ( g + a )

(

) (

)

= ( 0.360 )( 6.00 kg ) 9.80 m/s 2 + 0 m/s 2 = 21.2 N b. When the elevator accelerates upward, a = +1.20 m/s2. Then, f k = µ k FN = µ k ( mg + ma ) = µk m ( g + a )

(

) (

)

= ( 0.360 )( 6.00 kg ) 9.80 m/s 2 + 1.20 m/s 2 = 23.8 N

190 FORCES AND NEWTON'S LAWS OF MOTION

c. When the elevator accelerates downward, a = –1.20 m/s2. Then,

f k = µ k FN = µ k ( mg + ma ) = µ k m ( g + a )

(

) (

)

= ( 0.360 )( 6.00 kg ) 9.80 m/s 2 + –1.20 m/s 2 = 18.6 N ____________________________________________________________________________________________

47. REASONING The magnitude of the kinetic frictional force is given by Equation 4.8 as the coefficient of kinetic friction times the magnitude of the normal force. Since the slide into second base is horizontal, the normal force is vertical. It can be evaluated by noting that there is no acceleration in the vertical direction and, therefore, the normal force must balance the weight. To find the player’s initial velocity v0, we will use kinematics. The time interval for the slide into second base is given as t = 1.6 s. Since the player comes to rest at the end of the slide, his final velocity is v = 0 m/s. The player’s acceleration a can be obtained from Newton’s second law, since the net force is the kinetic frictional force, which is known from part (a), and the mass is given. Since t, v, and a are known and we seek v0, the appropriate kinematics equation is Equation 2.4 (v = v0 + at). SOLUTION a. Since the normal force FN balances the weight mg, we know that FN = mg. Using this fact and Equation 4.8, we find that the magnitude of the kinetic frictional force is

(

)

f k = µk FN = µk mg = ( 0.49 )( 81 kg ) 9.8 m/s 2 = 390 N b. Solving Equation 2.4 (v = v0 + at) for v0 gives v0 = v − at. Taking the direction of the player’s slide to be the positive direction, we use Newton’s second law and Equation 4.8 for the kinetic frictional force to write the acceleration a as follows: a=

ΣF − µk mg = = −µk g m m

The acceleration is negative, because it points opposite to the player’s velocity, since the player slows down during the slide. Thus, we find for the initial velocity that

(

)

v0 = v − ( − µk g ) t = 0 m/s − − ( 0.49 ) 9.8 m/s 2 (1.6 s ) = +7.7 m/s

Chapter 4 Problems

191

48. REASONING AND SOLUTION The deceleration produced by the frictional force is a=–

fk m

=

– µ k mg m

= – µk g

The speed of the automobile after 1.30 s have elapsed is given by Equation 2.4 as

v = v0 + at = v0 + ( − µk g ) t = 16.1 m/s − ( 0.720 ) ( 9.80 m/s 2 ) (1.30 s ) = 6.9 m/s ____________________________________________________________________________________________

49.

SSM REASONING Let us assume that the skater is moving horizontally along the +x axis. The time t it takes for the skater to reduce her velocity to vx = +2.8 m/s from v0x = +6.3 m/s can be obtained from one of the equations of kinematics: v x = v0 x + a x t

(3.3a)

The initial and final velocities are known, but the acceleration is not. We can obtain the acceleration from Newton’s second law ( ΣFx = ma x , Equation 4.2a ) in the following manner. The kinetic frictional force is the only horizontal force that acts on the skater, and, since it is a resistive force, it acts opposite to the direction of the motion. Thus, the net force in the x direction is ΣFx = − f k , where fk is the magnitude of the kinetic frictional force. Therefore, the acceleration of the skater is ax = ΣFx /m = − f k / m . The magnitude of the frictional force is f k = µk FN (Equation 4.8), where µ k is the coefficient of kinetic friction between the ice and the skate blades and FN is the magnitude of the normal force. There are two vertical forces acting on the skater: the upward-acting normal force FN and the downward pull of gravity (her weight) mg. Since the skater has no vertical acceleration, Newton's second law in the vertical direction gives (taking upward as the positive direction) ΣFy = FN − mg = 0 . Therefore, the magnitude of the normal force is FN = mg and the magnitude of the acceleration is

ax =

− f k − µk FN − µk m g = = = − µk g m m m

SOLUTION Solving the equation vx = v0 x + a x t for the time and substituting the expression above for the acceleration yields

t=

vx − v0 x ax

=

vx − v0 x − µk g

=

2.8 m/s − 6.3 m/s = 4.4 s − ( 0.081) ( 9.80 m/s 2 )

192 FORCES AND NEWTON'S LAWS OF MOTION

____________________________________________________________________________________________

50. REASONING We assume the car accelerates in the +x direction. The air resistance force fA opposes the car’s motion (see the free-body diagram). The frictional force is static, because the tires do not slip, and points in the direction of the car’s acceleration. The reason for this is that without friction the car’s wheels would simply spin in place, and the car’s acceleration would be severely limited. The frictional force has its maximum value fsMAX because we seek the maximum acceleration before slipping occurs. Applying Newton’s second law (ΣFx = max, Equation 4.2a) to the horizontal motion gives

+y

FN fsMAX

fA

+x

D

W

Free-body diagram of the car

fsMAX − f A = ma x

(1)

where ax is the maximum acceleration we seek. The air resistance force fA is given, and we

(

)

will find the maximum static frictional force from Equation 4.7 fsMAX = µ FN . Because the car’s acceleration has no vertical component, the net vertical force acting on the car must be zero, so the upward normal force FN must balance the two downward forces, the car’s weight W and the downforce D: FN = W + D = mg + D

SOLUTION According to Equation 4.7

(2)

( fsMAX = µs FN ) and Equation (2), the maximum

static frictional force the track can exert on the car is fsMAX = µ FN = µ ( mg + D )

(3)

Now solving Equation (1) for the car’s acceleration ax and then substituting Equation (3) for the static frictional force fsMAX , we obtain

ax = =

fsMAX − f A m

=

µs ( mg + D ) − f A m

( 0.87 ) ( 690 kg ) ( 9.80 m/s2 ) + 4060 N − 1190 N 690 kg

= 12 m/s 2

Chapter 4 Problems

193

51. REASONING The diagram shows the two applied forces that act on the crate. These two forces, plus the kinetic frictional force fk constitute the net force that acts on the crate. Once the net force has been determined, Newtons’ second law, ΣF = ma (Equation 4.1) can be used to find the acceleration of the crate. SOLUTION The sum of the applied forces is F = F1 + F2. The x-component of this sum is Fx = F1 cos 55.0° + F2 = (88.0 N) cos 55.0° + 54.0 N = 104 N. The y-component of F is Fy = F1 sin 55.0° = (88.0 N) sin 55.0° = 72.1 N. The magnitude of F is F = Fx2 + Fy2 =

(104 N )2 + ( 72.1 N )2

+y

F1

55.0°

+x F2

= 127 N

Since the crate starts from rest, it moves along the direction of F. The kinetic frictional force fk opposes the motion, so it points opposite to F. The net force acting on the crate is the

sum of F and fk. The magnitude a of the crate’s acceleration is equal to the magnitude ΣF of the net force divided by the mass m of the crate a=

ΣF − f k + F = m m

(4.1)

According to Equation 4.8, the magnitude fk of the kinetic frictional force is given by

f k = µk FN , where FN is the magnitude of the normal force. In this situation, FN is equal to

the magnitude of the crate’s weight, so FN = mg. Thus, the x-component of the acceleration is

(

)

2 − µk mg + F − ( 0.350 )( 25.0 kg ) 9.80 m/s + 127 N a= = = 1.65 m/s 2 25.0 kg m

The crate moves along the direction of F, whose x and y components have been determined previously. Therefore, the acceleration is also along F. The angle φ that F makes with the xaxis can be found using the inverse tangent function:

Fy F x

φ = tan −1

F1 sin 55.0° −1 = tan F1 cos 55.0° + F2

( 88.0 N ) sin 55.0° = tan −1 = 34.6° above the x axis ( 88.0 N ) cos 55.0° + 54.0 N ____________________________________________________________________________________________

194 FORCES AND NEWTON'S LAWS OF MOTION

52. REASONING AND SOLUTION horizontal directions gives

Newton’s second law applied in the vertical and

L cos 21.0° – W = 0 L sin 21.0° – R = 0

(1) (2)

a. Equation (1) gives 21.0°

L=

W 53 800 N = = 57 600 N cos 21.0° cos 21.0°

L

b. Equation (2) gives

R

R = L sin 21.0° = ( 57 600 N ) sin 21.0° = 20 600 N W

____________________________________________________________________________________________

53.

SSM REASONING In order for the object to move with constant velocity, the net force on the object must be zero. Therefore, the north/south component of the third force must be equal in magnitude and opposite in direction to the 80.0 N force, while the east/west component of the third force must be equal in magnitude and opposite in direction to the 60.0 N force. Therefore, the third force has components: 80.0 N due south and 60.0 N due east. We can use the Pythagorean theorem and trigonometry to find the magnitude and direction of this third force.

SOLUTION The magnitude of the third force is N

F3 = (80.0 N) 2 + (60.0 N) 2 = 1.00 × 10 2 N

The direction of F3 is specified by the angle θ where 80.0 N

θ = tan –1 = 53.1°, south of east 60.0 N

E

θ

80.0 N

θ

F3

60.0 N

____________________________________________________________________________________________

Chapter 4 Problems

195

54. REASONING AND SOLUTION a. In the horizontal direction the thrust F is balanced by the resistive force fr of the water. That is, ΣFx = 0 or fr = F = 7.40 × 105 N b. In the vertical direction, the weight, mg, is balanced by the buoyant force, Fb. So ΣFy = 0 gives Fb = mg = (1.70 × 108 kg)(9.80 m/s2) = 1.67 × 109 N ____________________________________________________________________________________________

55. REASONING The drawing shows the two forces, T and T′ , that act on the tooth. To obtain the net force, we will add the two forces using the method of components (see T′ Section 1.8).

+y +x 16.0°

16.0°

T

SOLUTION The table lists the two vectors and their x and y components: Vector

x component

y component

T

+T cos 16.0°

−T sin 16.0°

T′

−T′ cos 16.0°

−T′ sin 16.0°

T + T′

+T cos 16.0° − T′ cos 16.0°

−T sin 16.0° − T′ sin 16.0°

Since we are given that T = T′ = 21.0 N, the sum of the x components of the forces is ΣFx = +T cos 16.0° − T′ cos 16.0° = +(21 N) cos 16.0° − (21 N) cos 16.0° = 0 N The sum of the y components is ΣFy = −T sin16.0° − T′ sin 16.0° = −(21 N) sin 16.0° − (21 N) sin 16.0° = −11.6 N The magnitude F of the net force exerted on the tooth is F=

( ΣFx ) + ( ΣFy ) 2

2

2 2 = ( 0 N ) + ( −11.6 N ) = 11.6 N

196 FORCES AND NEWTON'S LAWS OF MOTION

______________________________________________________________________________ 56. REASONING At first glance there seems to be very little information given. However, it is enough. In part a of the drawing the bucket is hanging stationary and, therefore, is in equilibrium. The forces acting on it are its weight and the two tension forces from the rope. There are two tension forces from the rope, because the rope is attached to the bucket handle at two places. These three forces must balance, which will allow us to determine the weight of the bucket. In part b of the drawing, the bucket is again in equilibrium, since it is traveling at a constant velocity and, therefore, has no acceleration. The forces acting on the bucket now are its weight and a single tension force from the rope, and they again must balance. In part b, there is only a single tension force, because the rope is attached to the bucket handle only at one place. This will allow us to determine the tension in part b, since the weight is known. SOLUTION Let W be the weight of the bucket, and let T be the tension in the rope as the bucket is being pulled up at a constant velocity. The free-body diagrams for the bucket in parts a and b of the drawing are as follows: T 92.0 N

92.0 N

W

W

Free-body diagram for part a

Free-body diagram for part b

Since the bucket in part a is in equilibrium, the net force acting on it is zero. Taking upward to be the positive direction, we have Σ F = 92.0 N + 92.0 N − W = 0

or

W = 184 N

Similarly, in part b we have ΣF = T − W = 0

or

T = W = 184 N

Chapter 4 Problems

57.

197

SSM REASONING AND SOLUTION The free body diagram for the plane is shown below to the left. The figure at the right shows the forces resolved into components parallel to and perpendicular to the line of motion of the plane.

L

L

T

R

T

R W

θ

W sin θ

W cos θ

If the plane is to continue at constant velocity, the resultant force must still be zero after the fuel is jettisoned. Therefore (using the directions of T and L to define the positive directions), T – R – W(sin θ) = 0 (1) L – W (cos θ) = 0 (2) From Example 13, before the fuel is jettisoned, the weight of the plane is 86 500 N, the thrust is 103 000 N, and the lift is 74 900 N. The force of air resistance is the same before and after the fuel is jettisoned and is given in Example 13 as R = 59 800 N. After the fuel is jettisoned, W = 86 500 N – 2800 N = 83 700 N From Equation (1) above, the thrust after the fuel is jettisoned is T = R + W (sin θ) = [(59 800 N) + (83 700 N)(sin 30.0°)] = 101 600 N From Equation (2), the lift after the fuel is jettisoned is L = W (cos θ) = (83 700 N)(cos 30.0°) = 72 500 N a. The pilot must, therefore, reduce the thrust by 103 000 N – 101 600 N = 1400 N b. The pilot must reduce the lift by 74 900 N – 72 500 N = 2400 N ____________________________________________________________________________________________

198 FORCES AND NEWTON'S LAWS OF MOTION

58. REASONING The worker is standing still. Therefore, he is in equilibrium, and the net force acting on him is zero. The static frictional force fs that prevents him from slipping points upward along the roof, an angle of θ degrees above the horizontal; we choose this as the −x direction (see the free-body diagram). The normal force FN is perpendicular to the roof and thus has no x component. But the gravitational force mg of the earth on the worker points straight down, and thus has a component parallel to the roof. We will use this freebody diagram and find the worker’s mass m by applying Newton’s second law with the acceleration equal to zero.

+y FN

fs

mg sin θ

θ

+x mg

θ Free-body diagram of the worker

SOLUTION The static frictional force fs points in the −x direction, and the x component of the worker’s weight mg points in the +x direction. Because there are no other forces with x components, and the worker’s acceleration is zero, these two forces must balance each other. The x component of the worker’s weight is mg sin θ, therefore fs = mg sin θ. Solving this relation for the worker’s mass, we obtain

m=

fs

g sin θ

=

390 N

(9.80 m/s2 )(sin 36 )

= 68 kg

59. REASONING The sum of the angles the right and left surfaces +y make with the horizontal and the angle between the two surface F1 F2 must be 180.0°. Therefore, the angle that the left surface makes with respect to the horizontal is 180.0° − 90.0° − 45.0° = 45.0°. In 45.0° 45.0° the free-body diagram of the wine bottle, the x axis is the +x horizontal. The force each surface exerts on the bottle is perpendicular to the surfaces, so both forces are directed 45.0° above the horizontal. Letting the surface on the right be surface 1, W and the surface on the left be surface 2, the forces F1 and F2 are as shown in the free-body diagram. The third force acting on the bottle is W, its weight or the gravitational force exerted on it by Free-body diagram of the wine bottle the earth. We will apply Newton’s second law and analyze the vertical forces in this free-body diagram to determine the magnitude F of the forces F1 and F2, using the fact that the bottle is in equilibrium. SOLUTION The vertical components of the forces exerted by the surfaces are F1y = F1 sin 45.0° and F2y = F2 sin 45.0°. But the forces F1 and F2 have the same magnitude F, so the two vertical components become F1y = F2y = F sin 45.0°. Because the bottle is in equilibrium, the upward forces must balance the downward force:

Chapter 4 Problems

F1 y + F2 y = W

or

F sin 45.0 + F sin 45.0 = W

199

2 F sin 45.0 = W

or

Therefore,

(

)

(1.40 kg ) 9.80 m/s2 W mg F= = = = 9.70 N 2 sin 45.0 2sin 45.0 2 sin 45.0

60. REASONING The free-body diagram in the drawing at the right shows the forces that act on the clown (weight = W). In this drawing, note that P denotes the pulling force. Since the rope passes around three pulleys, forces of magnitude P are applied both to the clown’s hands and his feet. The normal force due to the floor is FN , and the maximum static frictional force is fsMAX. At the instant just before the clown’s feet move, the net vertical and net horizontal forces are zero, according to Newton’s second law, since there is no acceleration at this instant.

FN

P fsMAX

P

W

SOLUTION According to Newton’s second law, with upward and to the right chosen as the positive directions, we have FN + P – W = 0 Vertical forces

and

f sMAX – P = 0 Horizontal forces

From the horizontal-force equation we find P = fsMAX. But fsMAX = µsFN . From the vertical-force equation, the normal force is FN = W – P . With these substitutions, it follows that

P = f sMAX = µs FN = µ s (W – P ) Solving for P gives P=

µ sW ( 0.53)(890 N ) = = 310 N 1 + µs 1 + 0.53

____________________________________________________________________________________________

61. REASONING Since the boxes are at rest, they are in equilibrium. According to Equation 4.9b, the net force in the vertical, or y, direction is zero, ΣFy = 0. There are two unknowns in this problem, the normal force that the table exerts on box 1 and the tension in the rope that connects boxes 2 and 3. To determine these unknowns we will apply the relation ΣFy = 0 twice, once to the boxes on the left of the pulley and once to the box on the right.

200 FORCES AND NEWTON'S LAWS OF MOTION

SOLUTION There are four forces acting on the two boxes on the left. The boxes are in equilibrium, so that the net force must be zero. Choosing the +y direction as being the upward direction, we have that (1) − W1 − W2 + FN + T = 0

T

ΣFy

where W1 and W2 are the magnitudes of the weights of the boxes, FN is the magnitude of the normal force that the table exerts on box 1, and T is the magnitude of the tension in the rope. We know the weights. To find the unknown tension, note that the box 3 is also in equilibrium, so that the net force acting on it must be zero. − W3 + T = 0

so that

3 T FN

W3

2 1

W2

T = W3

ΣFy Substituting this expression for T into Equation (1) and solving for the normal force gives

W1

FN = W1 + W2 − W3 = 55 N + 35 N − 28 N = 62 N ____________________________________________________________________________________________

62. REASONING Since the mountain climber is at rest, she is in equilibrium and the net force acting on her must be zero. Three forces comprise the net force, her weight, and the tension forces from the left and right sides of the rope. We will resolve the forces into components and set the sum of the x components and the sum of the y components separately equal to zero. In so doing we will obtain two equations containing the unknown quantities, the tension TL in the left side of the rope and the tension TR in the right side. These two equations will be solved simultaneously to give values for the two unknowns. SOLUTION Using W to denote the weight of the mountain climber and choosing right and upward to be the positive directions, we have the following free-body diagram for the climber: For the x components of the forces we have ΣFx = TR sin 80.0° − TL sin 65.0° = 0

+y

TL

For the y components of the forces we have ΣFy = TR cos 80.0° + TL cos 65.0° − W = 0

Solving the first of these equations for TR, we find that

65.0º 80.0º

W

TR +x

Chapter 4 Problems

TR = TL

201

sin 65.0° sin 80.0°

Substituting this result into the second equation gives TL

sin 65.0° cos 80.0° + TL cos 65.0° − W = 0 sin 80.0°

or

TL = 1.717 W

Using this result in the expression for TR reveals that TR = TL

sin 65.0° sin 65.0° = (1.717W ) = 1.580 W sin 80.0° sin 80.0°

Since the weight of the climber is W = 535 N, we find that TL = 1.717 W = 1.717 ( 535 N ) = 919 N TR = 1.580 W = 1.580 ( 535 N ) = 845 N

63.

SSM REASONING There are four forces that act on the chandelier; they are the forces of tension T in each of the three wires, and the downward force of gravity mg. Under the influence of these forces, the chandelier is at rest and, therefore, in equilibrium. Consequently, the sum of the x components as well as the sum of the y components of the forces must each be zero. The figure below shows a quasi-free-body diagram for the chandelier and the force components for a suitable system of x, y axes. Note that the diagram only shows one of the forces of tension; the second and third tension forces are not shown in the interest of clarity. The triangle at the right shows the geometry of one of the cords, where is the length of the cord, and d is the distance from the ceiling. y

θ

θ

T

θ

d x

mg

We can use the forces in the y direction to find the magnitude T of the tension in any one wire.

SOLUTION that

Remembering that there are three tension forces, we see from the diagram

202 FORCES AND NEWTON'S LAWS OF MOTION

3T sin θ = mg

or

T=

mg mg mg = = 3 sin θ 3(d / ) 3d

Therefore, the magnitude of the tension in any one of the cords is

T=

(44 kg)(9.80 m/s2 )(2.0 m) = 1.9 × 102 N 3(1.5 m)

____________________________________________________________________________________________

64. REASONING The diagram at the right shows the force F that the ground exerts on the end of a crutch. This force, as F mentioned in the statement of the problem, acts along the crutch and, therefore, makes an angle θ with respect to the vertical. The horizontal and vertical components of this force are also shown. The horizontal component, F sin θ, is the static frictional force that prevents the crutch from slipping on the F sin θ floor, so fs = F sin θ . The largest value that the static frictional

F cos θ

θ

force can have before the crutch begins to slip is then given by fsMAX = F sin θ MAX . We also know from Section 4.9 (see Equation 4.7) that the maximum static frictional force is related to the magnitude FN of the normal force by fsMAX = µs FN , where µs is the coefficient of static friction. These two relations will allow us to find θ MAX . SOLUTION The magnitude of the maximum static frictional force is given by fsMAX = µs FN . But, as mentioned in the REASONING section, fsMAX is also the

horizontal component of the force F, so fsMAX = F sin θ MAX . The vertical component of F is F cos θ MAX and is the magnitude FN of the normal force that the ground exerts on the crutch. Thus, we have fsMAX = µs

FN

F cosθ MAX

F sin θ MAX

The force F can be algebraically eliminated from this equation, leaving sin θ MAX = µs cos θ MAX

or

tanθ MAX = µs

The maximum angle that a crutch can have is

θ MAX = tan −1 ( µs ) = tan −1 ( 0.90 ) = 42° ______________________________________________________________________________

Chapter 4 Problems

203

65. REASONING The toboggan has a constant velocity, so it has no acceleration and is in equilibrium. Therefore, the forces acting on the toboggan must balance, that is, the net force acting on the toboggan must be zero. There are three forces present, the kinetic frictional force, the normal force from the inclined surface, and the weight mg of the toboggan. Using Newton’s second law with the acceleration equal to zero, we will obtain the kinetic friction coefficient. SOLUTION In drawing the free-body diagram for the toboggan we choose the +x axis to be parallel to the hill surface and downward, the +y direction being perpendicular to the hill surface. We also use fk to symbolize the frictional force. Since the toboggan is in equilibrium, the zero net force components in the x and y directions are

+y FN

fk 8.00º

+x mg 8.00º

ΣFx = mg sin 8.00° − µ k FN = 0 ΣFy = FN − mg cos8.00° = 0

In the first of these expressions we have used Equation 4.8 for fk to express the kinetic frictional force. Solving the second equation for the normal force FN and substituting into the first equation gives mg sin 8.00° − µ k mg cos 8.00° = 0 F

or

µk =

sin 8.00° = tan 8.00° = 0.141 cos 8.00°

N

66. REASONING The block is in equilibrium in each case. Since the block moves at a constant velocity in each case, it is not accelerating. A zero acceleration is the hallmark of equilibrium. At equilibrium, the net force is zero (i.e., the forces balance to zero), and we will obtain the magnitude of the pushing force by utilizing this fact as it pertains to the vertical or y

(

)

direction. We will use Equation 4.9b ΣFy = 0 for this purpose.

204 FORCES AND NEWTON'S LAWS OF MOTION

+y +y It is important to note, however, that the direction of the kinetic frictional force is +x +x not the same in each case. The frictional force always opposes the fk relative motion between the surface of θ θ fk the block and the wall. Therefore, when P P the block slides upward, the frictional W W force points downward. When the block Free-body diagram for Free-body diagram for slides downward, the frictional force upward motion of the downward motion of points upward. These directions are block the block shown in the free-body diagrams (not to scale) for the two cases. In these drawings W is the weight of the block and fk is the kinetic frictional force.

In each case the magnitude of the frictional force is the same. It is given by Equation 4.8 as fk = kFN, where k is the coefficient of kinetic friction and FN is the magnitude of the normal force. The coefficient of kinetic friction does not depend on the direction of the motion. Furthermore, the magnitude of the normal force in each case is the component of the pushing force that is perpendicular to the wall, or FN = P sin θ. SOLUTION Using Equations 4.9b to describe the balance of forces that act on the block in the y direction and referring to the free-body diagrams, we have Upward motion

ΣFy = P cos θ − W − f k = 0

Downward motion

ΣFy = P cos θ − W + f k = 0

According to Equation 4.8, the magnitude of the kinetic frictional force is fk = kFN, where we have pointed out in the REASONING that the magnitude of the normal force is FN = P sin θ. Substituting into the equations for ΣFy in the two cases, we obtain Upward motion

ΣFy = P cos θ − W − µk P sin θ = 0

Downward motion

ΣFy = P cos θ − W + µ k P sin θ = 0

Solving each case for P, we find that a. Upward motion

P=

W 39.0 N = = 52.6 N cos θ − µk sin θ cos 30.0° − ( 0.250 ) sin 30.0°

Chapter 4 Problems

205

b. Downward motion

P=

67.

W 39.0 N = = 39.4 N cos θ + µk sin θ cos 30.0° + ( 0.250 ) sin 30.0°

SSM REASONING When the bicycle is coasting straight down the hill, the forces that act on it are the normal force FN exerted by the surface of the hill, the force of gravity mg, and the force of air resistance R. When the bicycle climbs the hill, there is one additional force; it is the applied force that is required for the bicyclist to climb the hill at constant speed. We can use our knowledge of the motion of the bicycle down the hill to find R. Once R is known, we can analyze the motion of the bicycle as it climbs the hill.

SOLUTION The figure to the left below shows the free-body diagram for the forces during the downhill motion. The hill is inclined at an angle θ above the horizontal. The figure to the right shows these forces resolved into components parallel to and perpendicular to the line of motion. +y +x FN F N R R mg sin θ mg

mg cos θ

θ

Since the bicyclist is traveling at a constant velocity, his acceleration is zero. Therefore, according to Newton's second law, we have ∑ Fx = 0 and ∑ Fy = 0 . Taking the direction up the hill as positive, we have ∑ Fx = R − mg sin θ = 0 , or R = mg sin θ = (80.0 kg)(9.80 m / s 2 ) sin 15.0° = 203 N When the bicyclist climbs the same hill at constant speed, an applied force P must push the system up the hill. Since the speed is the same, the magnitude of the force of air resistance will remain 203 N. However, the air resistance will oppose the motion by pointing down the hill. The figure at the right shows the resolved forces that act on the system during the uphill motion.

FN

P

mg sinθ mg cosθ R

Using the same sign convention as above, we have ∑ Fx = P − mg sin θ − R = 0 , or

P = R + mg sin θ = 203 N + 203 N =

406 N

____________________________________________________________________________________________

206 FORCES AND NEWTON'S LAWS OF MOTION

68. REASONING Because the kite line is straight, the distance between the kite and the person holding the line is L = 43 m, as shown in the right part of the drawing. In order to find the height h of the kite, we need to know the angle θ that the kite line makes with the horizontal (see the drawing). Once that is known, h = L sin θ will give the kite’s height relative to the person. The key to finding the angle θ is the realization that the tension force T exerted on the kite by the line is parallel to the line itself. Therefore, the tension T is directed at an angle θ below the horizontal (see the free-body diagram below). To find the angle θ, it is sufficient, then, to find one of the components of the tension force, either Tx or Ty, because the magnitude T of the tension is known. With values for Tx or Ty and T, we can use either the sine or cosine function to determine θ. Both the wind’s force f and the tension T have horizontal and vertical components, while the weight force W is purely vertical. Therefore, there are only two horizontal forces acting on the kite (Tx, fx), but three vertical forces (Ty, fy, W), so it will be easier to calculate Tx, the horizontal component of the tension. Because the kite is stationary, the horizontal component of the tension must balance the horizontal component of the force exerted on the kite by the wind: Tx = fx

(1)

f +y fy

L Tx

56° fx

+x

h

θ W T

Ty θ

Free-body diagram of the kite

Height of the kite

SOLUTION From the free-body diagram of the kite, we can see that the x components of the tension and air resistance forces are Tx = T cos θ and fx = f cos 56°. Substituting these expressions into Equation (1), we find that

Tx = f x

T cos θ = f cos 56

or

f cos 56 T

θ = cos −1

or

cos θ =

f cos 56 T

−1 (19 N ) cos 56 = 48 = cos 16 N

Chapter 4 Problems

207

Therefore, the kite’s height relative to the person is h = L sin θ = ( 43 m ) sin 48 = 32 m

69. REASONING The weight of the part of the washcloth off the table is moff g. At the instant just before the washcloth begins to slide, this weight is supported by a force that has magnitude equal to fsMAX, which is the static frictional force that the table surface applies to the part of the washcloth on the table. This force is transmitted “around the bend” in the washcloth hanging over the edge by the tension forces between the molecules of the washcloth, in much the same way that a force applied to one end of a rope is transmitted along the rope as it passes around a pulley. SOLUTION Since the static frictional supports the weight of the washcloth off the table, we have fsMAX = moff g. The static frictional force is fsMAX = µsFN . The normal force FN is applied by the table to the part of the washcloth on the table and has a magnitude equal to the weight of that part of the washcloth. This is so, because the table is assumed to be horizontal and the part of the washcloth on it does not accelerate in the vertical direction. Thus, we have f sMAX = µ s FN = µ s mon g = moff g The magnitude g of the acceleration due to gravity can be eliminated algebraically from this result, giving µsmon = moff . Dividing both sides by mon + moff gives

µs

mon

mon + moff

moff = mon + moff

or

µ s fon = foff

where we have used fon and foff to denote the fractions of the washcloth on and off the table, respectively. Since fon + foff = 1, we can write the above equation on the left as

(

)

µ s 1 – foff = f off

or

foff =

µs 0.40 = = 0.29 1 + µs 1 + 0.40

____________________________________________________________________________________________

208 FORCES AND NEWTON'S LAWS OF MOTION

70. REASONING In addition to the upward buoyant force B and the downward resistive force R, a downward gravitational force mg acts on the submarine (see the free-body diagram), where m denotes the mass of the submarine. Because the submarine is not in contact with any rigid surface, no normal force is exerted on it. We will calculate the acceleration of the submarine from Newton’s second law, using the free-body diagram as a guide.

+y

B

R SOLUTION Choosing up as the positive direction, we sum the forces mg acting on the submarine to find the net force ΣF. According to Free-body diagram Newton’s second law, the acceleration a is of the submarine

(

)

2 ΣF B − R − mg 16 140 N − 1030 N − (1450 kg ) 9.80 m/s a= = = = +0.62 m/s 2 m m 1450 kg

where the positive value indicates that the direction is upward.

71. REASONING According to Newton’s second law, the acceleration has the same direction as the net force and a magnitude given by a = ΣF/m. SOLUTION Since the two forces are perpendicular, the magnitude of the net force is given by the Pythagorean theorem as ΣF =

( 40.0 N )2 + ( 60.0 N )2 . Thus, according to Newton’s

second law, the magnitude of the acceleration is ΣF a= = m

( 40.0 N )2 + ( 60.0 N )2 4.00 kg

= 18.0 m/s 2

The direction of the acceleration vector is given by 60.0 N = 56.3° above the +x axis 40.0 N ______________________________________________________________________________

θ = tan –1

72. REASONING a. Since the fish is being pulled up at a constant speed, it has no acceleration. According to Newton’s second law, the net force acting on the fish must be zero. We will use this fact to determine the weight of the heaviest fish that can be pulled up. b. When the fish has an upward acceleration, we can still use Newton’s second law to find the weight of the heaviest fish. However, because the fish has an acceleration, we will see that the maximum weight is less than that in part (a).

Chapter 4 Problems

209

SOLUTION a. There are two forces acting on the fish (taking the upward vertical direction to be the +y direction): the maximum force of +45 N due to the line, and the weight –W of the fish (negative, because the weight points down). Newton’s second law ( ΣFy = 0, Equation 4.9b )

gives ΣFy = +45 N − W = 0 b. Since

the

fish

has

an

upward

or

W = 45 N

acceleration

( ΣFy = ma y , Equation 4.2b ) becomes

ay,

Newton’s

second

law

ΣFy = +45 N − W ′ = ma y

Where W′ is the weight of the heaviest fish that can be pulled up with an acceleration. Solving this equation for W′ gives W ′ = +45 N − ma y

The mass m of the fish is the magnitude W′ of its weight divided by the magnitude g of the acceleration due to gravity (see Equation 4.5), or m = W′/g. Substituting this relation for m into the previous equation gives

W′ W ′ = +45 N − ay g Solving this equation for W′ yields

45 N 45 N = = 37 N ay 2.0 m/s 2 1+ 1+ g 9.80 m/s 2 ______________________________________________________________________________ W′ =

73.

SSM REASONING If we assume that the acceleration is constant, we can use Equation 2.4 ( v = v0 + at ) to find the acceleration of the car. Once the acceleration is known,

Newton's second law ( ∑ F = ma ) can be used to find the magnitude and direction of the net force that produces the deceleration of the car. SOLUTION The average acceleration of the car is, according to Equation 2.4,

a=

v − v0 t

=

17.0 m/s − 27.0 m/s = − 1.25 m/s 2 8.00 s

210 FORCES AND NEWTON'S LAWS OF MOTION

where the minus sign indicates that the direction of the acceleration is opposite to the direction of motion; therefore, the acceleration points due west. According to Newton's Second law, the net force on the car is ∑ F = ma = (1380 kg)(–1.25 m/s 2 ) = –1730 N The magnitude of the net force is 1730 N . From Newton's second law, we know that the direction of the force is the same as the direction of the acceleration, so the force also points

due west . ____________________________________________________________________________________________

74. REASONING In the absence of air resistance, the two forces acting on the sensor are its weight W and the tension T in the towing cable (see the free-body diagram). We see that Tx is the only horizontal force acting on the sensor, and therefore Newton’s second law ΣFx = max (Equation 4.2a) gives Tx = max. Because the vertical component of the sensor’s acceleration is zero, the vertical component of the cable’s tension T must balance the sensor’s weight: Ty = W = mg. We thus have sufficient information to calculate the horizontal and vertical components of the tension force T, and therefore to calculate its magnitude T from the Pythagorean theorem: T 2 = Tx2 + Ty2 .

+y T Ty +x Tx = max W Free-body diagram of the sensor

SOLUTION Given that Tx = max and that Ty = mg, the Pythagorean theorem yields the magnitude T of the tension in the cable:

T = Tx2 + Ty2 = = (129 kg)

75.

( ma )2 + ( mg )2 =

m2a 2 + m2 g 2 = m a2 + g 2

( 2.84 m/s2 ) + (9.80 m/s2 ) 2

2

= 1320 N

SSM REASONING AND SOLUTION a. Each cart has the same mass and acceleration; therefore, the net force acting on any one of the carts is, according to Newton's second law

∑ F = ma = (26 kg)(0.050 m/s 2 ) = 1.3 N

Chapter 4 Problems

211

b. The fifth cart must essentially push the sixth, seventh, eight, ninth and tenth cart. In other words, it must exert on the sixth cart a total force of

∑ F = ma = 5(26 kg)(0.050 m/s 2 )= 6.5 N ____________________________________________________________________________________________

76. REASONING AND SOLUTION Newton's second law applied to object 1 (422 N) gives

Object 1

Object 2 N1

T = m1a1

T T

Similarly, for object 2 (185 N) W1

T – m2g = m2a2

W

2

If the string is not to break or go slack, both objects must have accelerations of the same magnitude. Then a1 = a and a2 = –a. The above equations become T = m1a T – m2g = – m2a

(1) (2)

a. Substituting Equation (1) into Equation (2) and solving for a yields

a=

m2 g m1 + m2

The masses of objects 1 and 2 are

m1 = W1 / g = ( 422 N ) / ( 9.80 m/s 2 ) = 43.1 kg m2 = W2 / g = (185 N ) / ( 9.80 m/s 2 ) = 18.9 kg The acceleration is a=

m2 g (18.9 kg ) ( 9.80 m/s 2 ) = = 2.99 m/s 2 43.1 kg + 18.9 kg m1 + m2

b. Using this value in Equation (1) gives T = m1a = ( 43.1 kg ) ( 2.99 m/s 2 ) = 129 N ____________________________________________________________________________________________

212 FORCES AND NEWTON'S LAWS OF MOTION

77.

F

N SSM WWW REASONING The speed of the skateboarder at the bottom of the ramp can be found by solving Equation 2.9 ( v 2 = v02 + 2ax , where x is the distance that the skater moves mgsin θ mgcos θ down the ramp) for v. The figure at the right shows the freebody diagram for the skateboarder. The net force ΣF, which θ accelerates the skateboarder down the ramp, is the component of the weight that is parallel to the incline: ∑ F = mg sinθ . Therefore, we know from Newton's second law that the acceleration of the skateboarder down the ramp is

a=

ΣF mg sin θ = = g sin θ m m

SOLUTION Thus, the speed of the skateboarder at the bottom of the ramp is v = v02 + 2ax = v02 + 2 gx sin θ = (2.6 m/s)2 + 2(9.80 m/s 2 )(6.0 m) sin18° = 6.6 m/s

______________________________________________________________________________ 78. REASONING AND SOLUTION From Newton's second law and the equation: v = v0 + at, we have v − v0 F = ma = m t a. When the skier accelerates from rest (v0 = 0 m/s) to a speed of 11 m/s in 8.0 s, the required net force is F =m

v − v0 t

= (73 kg)

(11 m/s) − 0 m/s = 1.0 × 102 N 8.0 s

b. When the skier lets go of the tow rope and glides to a halt (v = 0 m/s) in 21 s, the net force acting on the skier is

F =m

v − v0 t

= (73 kg)

0 m/s − (11 m/s) = −38 N 21 s

The magnitude of the net force is 38 N . ____________________________________________________________________________________________

Chapter 4 Problems

213

79. REASONING The only horizontal force acting on the boat and trailer is the tension in the hitch; therefore, it is the net force. According to Newton’s second law, the tension (or the net force) equals the mass times the acceleration. The mass is known, and the acceleration can be found by applying an appropriate equation of kinematics from Chapter 3. SOLUTION Assume that the boat and trailer are moving in the +x direction. Newton’s second law is ΣFx = ma x (see Equation 4.2a), where the net force is just the tension +T in the hitch, so ΣFx = T . Thus, T = ma x

(1)

Since the initial and final velocities, v0x and vx, and the time t are known, we may use Equation 3.3a from the equations of kinematics to relate these variables to the acceleration: v x = v0 x + a x t

(3.3a)

Solving Equation (3.3a) for ax and substituting the result into Equation (1), we find that

v −v 11 m/s − 0 m/s T = max = m x 0 x = ( 410 kg ) = 160 N 28 s t ____________________________________________________________________________________________

80. REASONING Since we assume that there is no +y frictional force resisting the airplane’s motion, the only horizontal force acting on the airplane arises because of the tension (magnitude = T) in the cable. From Newton’s FN second law ΣF = Ma (Equation 4.1), we conclude that the airplane’s acceleration is given by a = ΣF/M = T/M, fsMAX T where M is the mass of the airplane. The harder the man +x pulls on the cable, the greater the tension T, and the greater the airplane’s acceleration. According to Newton’s third law, however, the cable also exerts an mg opposing horizontal force of magnitude T on the man. Thus, if he is to keep his footing, T cannot exceed the Free-body diagram maximum force of static friction fsMAX the runway of the man exerts on him. Therefore, the airplane’s acceleration is greatest when these two forces have equal magnitudes: T = fsMAX (see the free-body diagram of the man). The maximum static frictional force the runway can exert is determined by the relation fsMAX = µs FN (Equation 4.7). Because the man has no acceleration in the vertical direction, the normal force must balance the downward pull of gravity: FN = mg.

214 FORCES AND NEWTON'S LAWS OF MOTION

SOLUTION Combining FN = mg and fsMAX = µs FN (Equation 4.7), we obtain the maximum tension in the cable: T = fsMAX = µs FN = µs mg

(1)

We can now substitute Equation (1) for the tension into Newton’s second law (a = T/M), and calculate the maximum possible acceleration of the airplane:

(

)

2 T µs mg ( 0.77 )( 85 kg ) 9.80 m/s = = = 5.9 ×10−3 m/s 2 a= M M 109 000 kg

81. REASONING AND SOLUTION If the +x axis is taken to be parallel to and up the ramp, then ΣFx = max gives T – fk – mg sin 30.0° = max where fk = µ kFN . Hence,

Also, ΣFy = may gives

T = max + µ kFN + mg sin 30.0°

(1)

FN – mg cos 30.0° = 0

since no acceleration occurs in this direction. Then FN = mg cos 30.0°

(2)

Substitution of Equation (2) into Equation (1) yields T = max + µ kmg cos 30.0° + mg sin 30.0° T = (205 kg)(0.800 m/s2) + (0.900)(205 kg)(9.80 m/s2)cos 30.0° + (205 kg)(9.80 m/s2)sin 30.0° = 2730 N ____________________________________________________________________________________________

Chapter 4 Problems

215

82. REASONING To determine the man’s upward acceleration by + means of Newton’s second law, we first need to identify all of the forces exerted on him and then construct a free-body diagram. The earth pulls down on the man with a gravitational force W = mg. Once T T he begins accelerating upward, he is no longer in contact with the ground, so there is no normal force acting on him. The pulling force P that he exerts on the rope does not appear in his free-body diagram because it is not a force exerted on him. Each end of the rope exerts a W tension force on him. If we assume that the rope is massless, and ignore friction between the rope and the branch, then the magnitude of the tension T is the same everywhere in the rope. Because the man Free-body diagram of the man pulls down on the free end intentionally and on the other end inadvertently (because it is tied around his waist), Newton’s third law predicts that both ends of the rope pull upward on him. The third law predicts that the free end of the rope pulls up on the man with a force exactly equal in magnitude to that of the 358-N pulling force. Thus, in addition to the downward gravitational force, there are two upward tension forces with magnitudes T = 358 N acting on the man, as illustrated in the free-body diagram. SOLUTION Taking up as the positive direction and applying Newton’s second law to the man’s free-body diagram yields

ΣF = 2T − W = ma Solving for the acceleration a, we find

a=

83.

2 ( 358 N ) 2T − W 2T − mg 2T = = −g = − 9.80 m/s 2 = 0.14 m/s 2 m m m 72.0 kg

SSM REASONING The free-body diagrams for Robin (mass = m) and for the chandelier (mass = M) are given at the right. The tension T in the rope applies an upward force to both. Robin accelerates upward, while the chandelier accelerates downward, each acceleration having the same magnitude. Our solution is based on separate applications of Newton’s second law to Robin and the chandelier.

T Robin

SOLUTION Applying Newton’s second law, we find T – mg = ma Robin Hood

and

T – Mg = – Ma Chandelier

T Chandelier

mg Mg

216 FORCES AND NEWTON'S LAWS OF MOTION

In these applications we have taken upward as the positive direction, so that Robin’s acceleration is a, while the chandelier’s acceleration is –a. Solving the Robin-Hood equation for T gives T = mg + ma Substituting this expression for T into the Chandelier equation gives

mg + ma – Mg = – Ma

M –m a= g M +m

or

a. Robin’s acceleration is (195 kg ) – ( 77.0 kg ) M –m 2 2 a= 9.80 m/s = 4.25 m/s g = + 195 kg + 77.0 kg M m ) ( ) (

(

)

b. Substituting the value of a into the expression for T gives

(

)

T = mg + ma = ( 77.0 kg ) 9.80 m/s 2 + 4.25 m/s2 = 1080 N ____________________________________________________________________________________________

84. REASONING Newton’s second law, Equation 4.2a, can be used to find the tension in the coupling between the cars, since the mass and acceleration are known. The tension in the coupling between the 30th and 31st cars is responsible for providing the acceleration for the 20 cars from the 31st to the 50th car. The tension in the coupling between the 49th and 50th cars is responsible only for pulling one car, the 50th. SOLUTION a. The tension T between the 30th and 31st cars is Tx = (Mass of 20 cars)a x

(

(4.2a)

)(

)

= ( 20 cars ) 6.8 × 103 kg/car 8.0 × 10−2 m / s 2 = 1.1 × 104 N b. The tension T between the 49th and 50th cars is Tx = (Mass of 1 car)a x

(

(4.2a)

)(

)

= (1 car ) 6.8 × 103 kg/car 8.0 × 10−2 m / s 2 = 5.4 × 102 N ____________________________________________________________________________________________

Chapter 4 Problems

85. REASONING The box comes to a halt because the kinetic FN frictional force and the component of its weight parallel to the incline oppose the motion and cause the box to slow down. The distance that the box travels up the incline can be can be found mg sinθ by solving Equation 2.9 ( v 2 = v02 + 2ax ) for x. Before we use this approach, however, we must first determine the acceleration fk of the box as it travels along the incline.

217

mg cos θ

SOLUTION The figure above shows the free-body diagram for the box. It shows the resolved components of the forces that act on the box. If we take the direction up the incline as the positive x direction, then, Newton's second law gives ∑ Fx = – mg sin θ – f k = max

– mg sin θ – µk FN = max

or

where we have used Equation 4.8, f k = µk FN . In the y direction we have ∑ Fy = FN – mg cos θ = 0

or

FN = mg cos θ

since there is no acceleration in the y direction. Therefore, the equation for the motion in the x direction becomes – mg sin θ – µ k mg cos θ = ma x

or

ax = – g (sin θ + µ k cos θ )

According to Equation 2.9, with this value for the acceleration and the fact that v = 0 m/s, the distance that the box slides up the incline is

x=–

v02 2a

=

v02

2 g (sin θ + µk cos θ )

=

(1.50 m/s)2 = 0.265 m 2(9.80 m/s 2 )[sin 15.0° + (0.180)cos 15.0°]

____________________________________________________________________________________________

86. REASONING Since we assume that there is no frictional force resisting the airplane’s motion, the only horizontal force acting on the airplane arises because of the tension (magnitude = T) in the cable. The airplane (mass = M) undergoes a horizontal acceleration caused by the horizontal component Tx = T cos θ of the tension force, where θ is the angle that the cable makes with the horizontal. From Newton’s second law, the acceleration of the airplane is

ax =

ΣFx M

=

T cos θ M

(1)

+y

FN fsMAX

Tx θ

T

+x Ty

mg Free-body diagram of the man

218 FORCES AND NEWTON'S LAWS OF MOTION

The maximum tension in the cable is limited by the condition that the man’s feet must not slip. When the man pulls as hard as possible without slipping, the horizontal component of the tension acting on him matches the maximum static frictional force: Tx = T cos θ = fsMAX (see the free-body diagram of the man). The maximum static frictional force itself is given by fsMAX = µs FN (Equation 4.7). Together, these two relations yield T cos θ = µs FN

(2)

To evaluate the magnitude FN of the normal force that acts on the man, we must consider Newton’s third law. This law indicates that when the man (mass = m) pulls up on the cable, the cable pulls down on him (see the free-body diagram of the man). This additional downward force increases the upward normal force FN the runway exerts on him. Applying Newton’s second law to the vertical direction in this diagram, with zero acceleration, we see that ΣFy = FN − Ty − mg = FN − T sin θ − mg = 0 . Solving for FN yields

FN = T sin θ + mg

(3)

Substituting Equation (3) into Equation (2) yields an expression in which the tension T is the only unknown quantity:

T cos θ = µs (T sin θ + mg )

(4)

We now solve Equation (4) for the tension in the cable: T cos θ = µsT sin θ + µs mg

or

T cos θ − µsT sin θ = µs mg

T=

or

T ( cos θ − µs sin θ ) = µs mg

µs mg cos θ − µs sin θ

(5)

Equation (5) may be substituted into Equation (1) for the airplane’s acceleration: ax =

µs mg cos θ T cos θ = M M ( cos θ − µs sin θ )

(6)

SOLUTION We apply Equation (6) to calculate the acceleration of the airplane:

(

)

( 0.77 )( 85 kg ) 9.80 m/s cos 9.0 µs mg cos θ ax = = M ( cos θ − µs sin θ ) (109 000 kg ) cos 9.0 − 0.77 sin 9.0

(

2

)

= 6.7 × 10−3 m/s 2

Chapter 4 Problems

87. REASONING As the free-body diagram shows, there are two forces acting on the fireman as he slides down the pole: his weight W and the kinetic frictional force fk. The kinetic frictional force opposes the motion of the fireman, so it points upward in the +y direction. In accord with Newton’s second law, the net force, which is the sum of these two forces, is equal to the fireman’s mass times his acceleration. His mass, and therefore his weight, is known, but his acceleration is not. We will turn to one of the equations of kinematics from Chapter 3 to determine the acceleration. SOLUTION Newton’s second law

( ΣFy = ma y , Equation 4.2b )

can

219

+y

fk

W Free-body diagram for the fireman

be applied to this situation: ΣFy = + f k − W = ma y

The magnitude W of the fireman’s weight can be expressed in terms of his mass as W = mg (Equation 4.5), where g is magnitude of the acceleration due to gravity. Solving the equation above for the magnitude of the kinetic frictional force, and using W = mg, gives f k = ma y + W = ma y + mg

(1)

Since the initial and final velocities, v0y and vy, and the displacement y are known, we will use Equation 3.6b from the equations of kinematics to relate these variables to the acceleration: v 2y = v02y + 2a y y . Solving this equation for ay and substituting the result into Equation 1 gives

v 2y − v02y f k = m 2y

+ mg

We note that the fireman slides down the pole, so his displacement is negative, or y = −4.0 m. The magnitude of the kinetic frictional force is, then, v 2y − v02y f k = m 2y

+ mg

(1.4 m/s )2 − ( 0 m/s )2 = ( 86 kg ) + ( 86 kg ) ( 9.80 m/s 2 ) = 820 N ( ) 2 4.0 m − ______________________________________________________________________________

220 FORCES AND NEWTON'S LAWS OF MOTION

88. REASONING Consider the forces that act on each block. Only one + − force contributes to the horizontal net force acting on block 1, as shown in the free-body diagram. This is the −P force −P with which block 2 pushes on block 1. The minus sign in the free-body diagram indicates the Free-body diagram for direction of the force is to the left. This force is part of block 1 the action-reaction pair of forces that is consistent with Newton’s third law. Block 1 pushes forward and to the right against block 2, and block 2 pushes backward and to the left against block 1 with an oppositely directed force of equal magnitude. Two forces contribute to the horizontal net force acting + − on block 2, as shown in the free-body diagram. One is the force P with which block 1 pushes on block 2. P fk According to Newton’s third law, this force has the same magnitude but the opposite direction as the force Free-body diagram for with which block 2 pushes on block 1. The other force block 2 is the kinetic frictional force fk, which points to the left, in opposition to the relative motion between the block and the surface on which it slides. Both blocks decelerate, the magnitude of the deceleration being the same for each block. They have the same deceleration, because they are pressed together. Since the blocks are moving to the right in the drawing, the acceleration vector points to the left, for it reflects the slowing down of the motion. In Case A and in Case B we will apply Newton’s second law separately to each block in order to relate the net force to the acceleration. SOLUTION Referring to the free-body diagram for block 1, we write Newton’s second law as follows: − P = m1 ( −a ) (1) Net force on block 1

where a is the magnitude of the acceleration. The minus sign appears on the right side of this equation because the acceleration, being a deceleration, points to the left, in the negative direction. Referring to the free-body diagram for block 2, we write Newton’s second law as follows: (2) P − f k = m2 ( − a ) Net force on block 2

Solving Equation (1) for a gives a = P / m1 . Substituting this result into Equation (2) gives −P P − f k = m2 m 1

or

P=

m1 f k

m1 + m2

Chapter 4 Problems

221

Substituting this result for P into a = P / m1 gives a=

m1 f k fk P = = m1 m1 ( m1 + m2 ) m1 + m2

We can now use these results to calculate P and a in both cases. a. Case A

P=

Case B

P=

b. Case A

m1 f k

=

( 3.0 kg )( 5.8 N ) =

2.9 N

m1 f k

=

( 6.0 kg )( 5.8 N ) =

3.9 N

m1 + m2 m1 + m2

− fk

a=

m1 + m2

=

3.0 kg + 3.0 kg

6.0 kg + 3.0 kg

−5.8 N = −0.97 m/s 2 3.0 kg + 3.0 kg

The magnitude of the acceleration is 0.97 m/s 2 . Case B

a=

− fk

m1 + m2

=

−5.8 N = −0.64 m/s 2 + 6.0 kg 3.0 kg

The magnitude of the acceleration is 0.64 m/s 2 .

89. SSM REASONING The tension in each coupling bar is responsible for accelerating the objects behind it. The masses of the cars are m1, m2, and m3. We can use Newton’s second law to express the tension in each coupling bar, since friction is negligible:

TA = ( m1 + m2 + m3 ) a

TB = ( m2 + m3 ) a

TC = m3a

Coupling bar A

Coupling bar B

Coupling bar C

In these expressions a = 0.12 m/s2 remains constant. Consequently, the tension in a given bar will change only if the total mass of the objects accelerated by that bar changes as a result of the luggage transfer. Using ∆ (Greek capital delta) to denote a change in the usual fashion, we can express the changes in the above tensions as follows: ∆TA = ∆ ( m1 + m2 + m3 ) a

∆TB = ∆ ( m2 + m3 ) a

∆TC = ( ∆m3 ) a

Coupling bar A

Coupling bar B

Coupling bar C

222 FORCES AND NEWTON'S LAWS OF MOTION

SOLUTION a. Moving luggage from car 2 to car 1 does not change the total mass m1 + m2 + m3, so ∆(m1 + m2 + m3) = 0 kg and ∆TA = 0 N .

The transfer from car 2 to car 1 causes the total mass m2 + m3 to decrease by 39 kg, so ∆(m2 + m3) = –39 kg and

(

)

∆TB = ∆ ( m2 + m3 ) a = ( –39 kg ) 0.12 m/s2 = –4.7 N The transfer from car 2 to car 1 does not change the mass m3, so ∆m3 = 0 kg and ∆TC = 0 N . b. Moving luggage from car 2 to car 3 does not change the total mass m1 + m2 + m3, so ∆(m1 + m2 + m3) = 0 kg and ∆TA = 0 N . The transfer from car 2 to car 3 does not change the total mass m2 + m3, so ∆(m2 + m3) = 0 kg and ∆TB = 0 N . The transfer from car 2 to car 3 causes the mass m3 to increase by 39 kg, so ∆m3 = +39 kg and

(

)

∆TC = ( ∆m3 ) a = ( +39 kg ) 0.12 m/s 2 = +4.7 N

____________________________________________________________________________________________

90. REASONING AND SOLUTION The distance required for the truck to stop is found from x=

v 2 – v02 2a

2 0 m/s ) – v02 ( =

2a

The acceleration of the truck is needed. The frictional force decelerates the crate. The maximum force that friction can supply is fsMAX = µ sFN = µ smg Newton's second law requires that fsMAX = – ma so a = – µ sg Now the stopping distance is

Chapter 4 Problems

x=

v02

2µs g

2 25 m/s ) ( = = 2 ( 0.650 ) ( 9.80 m/s 2 )

223

49.1 m

____________________________________________________________________________________________

91. REASONING AND SOLUTION a. Newton's second law for block 1 (10.0 kg) is T = m1a

(1)

Block 2 (3.00 kg) has two ropes attached each carrying a tension T. Also, block 2 only travels half the distance that block 1 travels in the same amount of time so its acceleration is only half of block 1's acceleration. Newton's second law for block 2 is then

2T − m2 g = − 12 m2 a

(2)

Solving Equation (1) for a, substituting into Equation (2), and rearranging gives T=

1+

1m g 2 2 1 m /m 2 1 4

(

)

= 13.7 N

b. Using this result in Equation (1) yields a=

T 13.7 N = = 1.37 m/s 2 m1 10.0 kg

____________________________________________________________________________________________

92. REASONING AND SOLUTION a. The force acting on the sphere which accelerates it is the horizontal component of the tension in the string. Newton's second law for the horizontal motion of the sphere gives T sin θ = ma The vertical component of the tension in the string supports the weight of the sphere so T cos θ = mg Eliminating T from the above equations results in a = g tan θ .

224 FORCES AND NEWTON'S LAWS OF MOTION

b.

(

)

a = g tan θ = 9.80 m/s 2 tan 10.0° = 1.73 m/s 2

c. Rearranging the result of part a and setting a = 0 m/s2 gives

θ = tan –1 ( a / g ) = 0° ____________________________________________________________________________________________

93.

SSM REASONING AND SOLUTION a. The left mass (mass 1) has a tension T1 pulling it up. Newton's second law gives

T1 – m1g = m1a

(1)

The right mass (mass 3) has a different tension, T3, trying to pull it up. Newton's second for it is T3 – m3g = – m3a (2) The middle mass (mass 2) has both tensions acting on it along with friction. Newton's second law for its horizontal motion is T3 – T1 – µ km2g = m2a

(3)

Solving Equation (1) and Equation (2) for T1 and T3, respectively, and substituting into Equation (3) gives ( m − m1 − µk m2 ) g a= 3 m1 + m2 + m3 Hence,

(

)

25.0 kg − 10.0 kg − ( 0.100 )( 80.0 kg ) 9.80 m/s2 a= = 0.60 m/s 2 10.0 kg + 80.0 kg + 25.0 kg b. From part a:

( ) T3 = m3(g − a) = ( 25.0 kg ) ( 9.80 m/s2 − 0.60 m/s 2 ) = 230 N T1 = m1(g + a) = (10.0 kg ) 9.80 m/s 2 + 0.60 m/s2 = 104 N

____________________________________________________________________________________________

94. REASONING AND SOLUTION a. The static frictional force is responsible for accelerating the top block so that it does not slip against the bottom one. The maximum force that can be supplied by friction is fsMAX = µ sFN = µ sm1g

Chapter 4 Problems

225

Newton's second law requires that fsMAX = m1a, so a = µ sg The force necessary to cause BOTH blocks to have this acceleration is F = (m1 + m2)a = (m1 + m2)µ sg F = (5.00 kg + 12.0 kg)(0.600)(9.80 m/s2) = 1.00 × 102 N b. The maximum acceleration that the two block combination can have before slipping occurs is F a= 17.0 kg Newton's second law applied to the 5.00 kg block is F – µ sm1g = m1a = (5.00 kg)

F 17.0 kg

Hence F = 41.6 N ____________________________________________________________________________________________

95.

SSM REASONING The magnitude of the gravitational force that each part exerts on the other is given by Newton’s law of gravitation as F = Gm1m2 / r 2 . To use this expression, we need the masses m1 and m2 of the parts, whereas the problem statement gives the weights W1 and W2. However, the weight is related to the mass by W = mg, so that for each part we know that m = W/g.

SOLUTION The gravitational force that each part exerts on the other is

F=

Gm1m2 r2

=

G (W1 / g )(W2 / g ) r2

6.67 × 10 –11 N ⋅ m 2 / kg 2 ) (11 000 N )( 3400 N ) ( = = 2 2 2 9.80 m/s 12 m ( ) ( )

1.8 × 10 –7 N

____________________________________________________________________________________________

226 FORCES AND NEWTON'S LAWS OF MOTION

96. REASONING The magnitudes of the initial (v0 = 0 m/s) and final (v = 805 m/s) velocities are known. In addition, data is given for the mass and the thrust, so that Newton’s second law can be used to determine the acceleration of the probe. Therefore, kinematics Equation 2.4 (v = v0 + at) can be used to determine the time t. SOLUTION Solving Equation 2.4 for the time gives t=

v − v0 a

Newton’s second law gives the acceleration as a = (ΣF)/m. Using this expression in Equation 2.4 gives t=

( v − v0 )

( ΣF ) / m

=

m ( v − v0 ) ΣF

=

( 474 kg )(805 m/s − 0 m/s ) = 6.8 ×106 s 56 × 10−3 N

Since one day contains 8.64 × 104 s, the time is

(

t = 6.8 ×106 s

97.

= 79 days ) 8.641 ×day 104 s

SSM REASONING AND SOLUTION According to Equation 3.3b, the acceleration of the astronaut is a y = (v y − v0 y ) / t = v y / t . The apparent weight and the true weight of the astronaut are related according to Equation 4.6. Direct substitution gives

vy FN = mg + ma y = m ( g + a y ) = m g + t True Apparent weight

weight

45 m/s 2 = (57 kg) 9.80 m/s 2 + = 7.3 ×10 N 15 s

____________________________________________________________________________________________

98. REASONING According to Newton's second law ( ∑ F = ma ), the acceleration of the object is given by a = ∑ F / m , where ∑ F is the net force that acts on the object. We must first find the net force that acts on the object, and then determine the acceleration using Newton's second law. SOLUTION The following table gives the x and y components of the two forces that act on the object. The third row of that table gives the components of the net force.

Chapter 4 Problems

Force

x-Component

y-Component

F1

40.0 N

0N

F2

(60.0 N) cos 45.0° = 42.4 N

(60.0 N) sin 45.0° = 42.4 N

∑ F = F1 + F2

82.4 N

42.4 N

227

The magnitude of ∑ F is given by the Pythagorean theorem as

ΣF = (82.4 N) 2 + (42.4) 2 = 92.7 N The angle θ that ∑ F makes with the +x axis is 42.4 N θ = tan −1 = 27.2° 82.4 N

ΣF θ

42.4 N

82.4 N

According to Newton's second law, the magnitude of the acceleration of the object is

a=

∑ F 92.7 N = = 30.9 m/s 2 3.00 kg m

Since Newton's second law is a vector equation, we know that the direction of the right hand side must be equal to the direction of the left hand side. In other words, the direction of the acceleration a is the same as the direction of the net force ∑ F . Therefore, the direction of the acceleration of the object is 27.2° above the +x axis . ____________________________________________________________________________________________

99.

SSM REASONING In order to start the crate moving, an external agent must supply a force that is at least as large as the maximum value fsMAX = µs FN , where µs is the coefficient of static friction (see Equation 4.7). Once the crate is moving, the magnitude of the frictional force is very nearly constant at the value f k = µk FN , where µ k is the coefficient of kinetic friction (see Equation 4.8). In both cases described in the problem statement, there are only two vertical forces that act on the crate; they are the upward normal force FN, and the downward pull of gravity (the weight) mg. Furthermore, the crate has no vertical acceleration in either case. Therefore, if we take upward as the positive direction, Newton's second law in the vertical direction gives FN − mg = 0 , and we see that, in both cases, the magnitude of the normal force is FN = mg .

228 FORCES AND NEWTON'S LAWS OF MOTION

SOLUTION a. Therefore, the applied force needed to start the crate moving is

fsMAX = µs mg = (0.760)(60.0 kg)(9.80 m/s 2 ) = 447 N b. When the crate moves in a straight line at constant speed, its velocity does not change, and it has zero acceleration. Thus, Newton's second law in the horizontal direction becomes P − fk = 0, where P is the required pushing force. Thus, the applied force required to keep the crate sliding across the dock at a constant speed is P = f k = µk mg = (0.410)(60.0 kg)(9.80 m/s 2 ) = 241 N ____________________________________________________________________________________________

100. REASONING Newton’s second law of motion gives the relationship between the net force ΣF and the acceleration a that it causes for an object of mass m. The net force is the vector sum of all the external forces that act on the object. Here the external forces are the drive force, the force due to the wind, and the resistive force of the water. SOLUTION We choose the direction of the drive force (due west) as the positive direction. Solving Newton’s second law ( ΣF = ma ) for the acceleration gives a=

ΣF +4100 N − 800 N − 1200 N = = +0.31 m/s 2 6800 kg m

The positive sign for the acceleration indicates that its direction is due west .

101. REASONING AND SOLUTION The acceleration needed so that the craft touches down with zero velocity is a=

v 2 − v02 2s

2 − (18.0 m/s ) = = 0.982 m/s 2 ( ) 2 −165 m

Newton's second law applied in the vertical direction gives F – mg = ma Then F = m(a + g) = (1.14 × 104 kg)(0.982 m/s2 + 1.60 m/s2) = 29 400 N ____________________________________________________________________________________________

102. REASONING AND SOLUTION The apparent weight is FN = mw(g + a)

Chapter 4 Problems

229

We need to find the acceleration a. Let T represent the force applied by the hoisting cable. Newton's second law applied to the elevator gives T – (mw + me)g = (mw + me)a Solving for a gives

a=

9410 N T −g = − 9.80 m/s2 = 0.954 m/s2 mw + me 60.0 kg + 815 kg

Now the apparent weight is FN = (60.0 kg)(9.80 m/s2 + 0.954 m/s2) = 645 N ____________________________________________________________________________________________

103. SSM REASONING We can use the appropriate equation of kinematics to find the acceleration of the bullet. Then Newton's second law can be used to find the average net force on the bullet.

SOLUTION According to Equation 2.4, the acceleration of the bullet is a=

v − v0 t

=

715 m/s − 0 m/s = 2.86 ×105 m/s 2 2.50 × 10 –3 s

Therefore, the net average force on the bullet is

∑ F = ma = (15 × 10−3 kg)(2.86 ×105 m/s 2 ) = 4290 N ____________________________________________________________________________________________

104. REASONING The magnitude ΣF of the net force acting on the kayak is given by Newton’s second law as ΣF = ma (Equation 4.1), where m is the combined mass of the person and kayak, and a is their acceleration. Since the initial and final velocities, v0 and v, and the displacement x are known, we can employ one of the equations of kinematics from Chapter 2 to find the acceleration.

(

)

SOLUTION Solving Equation 2.9 v 2 = v0 2 + 2ax from the equations of kinematics for

the acceleration, we have

a=

v 2 − v0 2 2x

Substituting this result into Newton’s second law gives

( 0.60 m/s )2 − ( 0 m/s )2 v 2 − v02 ΣF = ma = m = ( 73 kg ) = 32 N 2 ( 0.41 m ) 2x

230 FORCES AND NEWTON'S LAWS OF MOTION

______________________________________________________________________________ 105. REASONING AND SOLUTION a. According to Equation 4.4, the weight of an object of mass m on the surface of Mars would be given by GM M m W= 2 RM where MM is the mass of Mars and RM is the radius of Mars. On the surface of Mars, the weight of the object can be given as W = mg (see Equation 4.5), so mg =

GM M m 2 RM

or

g=

GM M 2 RM

Substituting values, we have g=

(6.67 ×10−11N ⋅ m 2 /kg 2 )(6.46 ×1023 kg) = 3.75 m/s 2 6 2 (3.39 ×10 m)

b. According to Equation 4.5, W = mg = (65 kg)(3.75 m/s2) = 2.4 × 102 N ____________________________________________________________________________________________

106. REASONING Each particle experiences two gravitational forces, one due to each of the remaining particles. To get the net gravitational force, we must add the two contributions, taking into account the directions. The magnitude of the gravitational force that any one particle exerts on another is given by Newton’s law of gravitation as F = Gm1m2 / r 2 . Thus, for particle A, we need to apply this law to its interaction with particle B and with particle C. For particle B, we need to apply the law to its interaction with particle A and with particle C. Lastly, for particle C, we must apply the law to its interaction with particle A and with particle B. In considering the directions, we remember that the gravitational force between two particles is always a force of attraction. SOLUTION We begin by calculating the magnitude of the gravitational force for each pair of particles:

Chapter 4 Problems

FAB =

GmAmB

FBC =

GmB mC

FAC =

Gm AmC

r2

r2

r2

231

6.67 × 10 –11 N ⋅ m 2 / kg 2 ) ( 363 kg )( 517 kg ) ( = = 5.007 × 10 –5 N

( 0.500 m )2

6.67 × 10 –11 N ⋅ m 2 / kg 2 ) ( 517 kg )(154 kg ) ( = = 8.497 × 10 –5 N

( 0.500 m )2

6.67 × 10 –11 N ⋅ m 2 / kg 2 ) ( 363 kg )(154 kg ) ( = = 6.629 × 10 –6 N

( 0.500 m )2

In using these magnitudes we take the direction to the right as positive. a. Both particles B and C attract particle A to the right, the net force being FA = FAB + FAC = 5.007 × 10 –5 N + 6.629 × 10 –6 N = 5.67 × 10 –5 N, right

b. Particle C attracts particle B to the right, while particle A attracts particle B to the left, the net force being FB = FBC – FAB = 8.497 × 10 –5 N – 5.007 × 10 –5 N = 3.49 × 10 –5 N, right

c. Both particles A and B attract particle C to the left, the net force being FC = FAC + FBC = 6.629 × 10 –6 N + 8.497 × 10 –5 N = 9.16 × 10 –5 N, left ____________________________________________________________________________________________

y

107. SSM REASONING AND SOLUTION The system is shown in the drawing. We will let m1 = 21.0 kg , and m2 = 45.0 kg . Then, m1 will move upward, and m2 will move downward. There are two forces that act on each object; they are the tension T in the cord and the weight mg of the object. The forces are shown in the free-body diagrams at the far right.

y

T

x

m1 m2 m1g

We will take up as the positive direction. If the acceleration of m1 is a, then the acceleration of m2 must be –a. From Newton's second law, we have for m1

T

m2g

232 FORCES AND NEWTON'S LAWS OF MOTION

∑ Fy = T − m1g = m1a

(1)

∑ Fy = T − m2 g = – m2 a

(2)

and for m2

a. Eliminating T between these two equations, we obtain a=

m2 – m1 45.0 kg – 21.0 kg 2 2 g = (9.80 m/s ) = 3.56 m/s + m2 + m1 45.0 kg 21.0 kg

b. Eliminating a between Equations (1) and (2), we find T=

2m1m2 2(21.0 kg)(45.0 kg) g= (9.80 m/s 2 ) = 281 N + + m1 m2 21.0 kg 45.0 kg

____________________________________________________________________________________________

108. REASONING Static friction determines the magnitude of the applied force at which either the upper or lower block begins to slide. For the upper block the static frictional force is applied only by the lower block. For the lower block, however, separate static frictional forces are applied by the upper block and by the horizontal surface. The maximum magnitude of any of the individual frictional forces is given by Equation 4.7 as the coefficient of static friction times the magnitude of the normal force. SOLUTION We begin by drawing the free-body diagram for the lower block. MAX

fs, from A FApplied

A MAX

B

fs, from surface Free-body diagram for lower block B

This diagram shows that three horizontal forces act on the lower block, the applied force, and the two maximum static frictional forces, one from the upper block and one from the horizontal surface. At the instant that the lower block just begins to slide, the blocks are in equilibrium and the applied force is balanced by the two frictional forces, with the result that MAX FApplied = fs,MAX from A + fs, from surface

(1)

According to Equation 4.7, the magnitude of the maximum frictional force from the surface is (2) fs,MAX from surface = µs FN = µs 2mg

233

Chapter 4 Problems

Here, we have recognized that the normal force FN from the horizontal surface must balance the weight 2mg of both blocks. It remains now to determine the magnitude of the maximum frictional force fs,MAX from A from the upper block.

MAX

47.0 N

fs, from B

A

To this end, we draw the free-body B diagram for the upper block at the Free-body diagram for instant that it just begins to slip due to upper block A the 47.0-N applied force. At this instant the block is in equilibrium, so that the frictional force from the lower block B balances the 47.0-N force. Thus, fs,MAX from B = 47.0 N , and according to Equation 4.7, we have fs,MAX from B = µs FN = µs mg = 47.0 N

Here, we have recognized that the normal force FN from the lower block must balance the weight mg of only the upper block. This result tells us that smg = 47.0 N. To determine fs,MAX from A we invoke Newton’s third law to conclude that the magnitudes of the frictional

forces at the A-B interface are equal, since they are action-reaction forces. fs,MAX from A = µs mg . Substituting this result and Equation (2) into Equation (1) gives

Thus,

MAX FApplied = fs,MAX from A + fs, from surface = µs mg + µs 2 mg = 3 ( 47.0 N ) = 141 N

______________________________________________________________________________ 109. REASONING AND SOLUTION If the +x axis is taken in the direction of motion, ΣFx = 0 gives F

F – fk – mg sin θ = 0

N

where

fk

fk = µ kFN Then

F

F – µ kFN – mg sin θ = 0

Also, ΣFy = 0 gives so

(1)

FN – mg cos θ = 0 FN = mg cos θ

(2)

Substituting Equation (2) into Equation (1) and solving for F yields

W

25.0°

234 FORCES AND NEWTON'S LAWS OF MOTION

F = mg( sin θ + µ k cos θ ) F = (55.0 kg)(9.80 m/s2)[sin 25.0° + (0.120)cos 25.0°] = 286 N ____________________________________________________________________________________________

110. REASONING Since the wire beneath the limb is at rest, it is in equilibrium and the net force acting on it must be zero. Three forces comprise the net force, the 151-N force from the limb, the 447-N tension force from the left section of the wire, and the tension force T from the right section of the wire. We will resolve the forces into components and set the sum of the x components and the sum of the y components separately equal to zero. In so doing we will obtain two equations containing the unknown quantities, which are the horizontal and vertical components of the tension force T. These two equations will be solved simultaneously to give values for the two unknowns. Knowing the components of the tension force, we can determine its magnitude and direction.

SOLUTION Let Tx and Ty be the horizontal and vertical components of the tension force. The free-body diagram for the wire beneath the limb is as follows: Taking upward and to the right as the positive directions, we find for the x components of the forces that

+y T

447 N

ΣFx = Tx − ( 447 N ) cos 14.0° = 0 Tx = ( 447 N ) cos 14.0° = 434 N For the y components of the forces we have

14.0º

θ Tx

Ty

+x

151 N

ΣFy = Ty + ( 447 N ) sin14.0° − 151 N = 0 Ty = − ( 447 N ) sin14.0° + 151 N = 43 N The magnitude of the tension force is T = Tx2 + Ty2 =

( 434 N )2 + ( 43 N )2

= 436 N

Since the components of the tension force and the angle θ are related by tan θ = Ty / Tx , we find that Ty −1 43 N = tan = 5.7° T 434 N x

θ = tan −1

Chapter 4 Problems

235

111. SSM REASONING The shortest time to pull the person from the cave corresponds to the maximum acceleration, a y , that the rope can withstand. We first determine this acceleration and then use kinematic Equation 3.5b ( y = v0 y t + 12 a y t 2 ) to find the time t. SOLUTION As the person is being pulled from the cave, there are two forces that act on him; they are the tension T in the rope that points vertically upward, and the weight of the person mg that points vertically downward. Thus, if we take upward as the positive direction, Newton's second law gives ∑ Fy = T – mg = ma y . Solving for a y , we have ay =

T T 569 N − 9.80 m/s 2 = 0.92 m/s 2 –g= –g= 2 2 m W /g (5.20 ×10 N)/(9.80 m/s )

Therefore, from Equation 3.5b with v0 y = 0 m/s, we have y = 12 a y t 2 . Solving for t, we find

t=

2y 2(35.1 m) = = 8.7 s ay 0.92 m/s 2

____________________________________________________________________________________________

112. REASONING The free-body diagrams for the large cube (mass = M) and the small cube (mass = m) are shown in the following drawings. In the case of the large cube, we have omitted the weight and the normal force from the surface, since the play no role in the solution (although they do balance). fsMAX

–FN P

FN mg

In these diagrams, note that the two blocks exert a normal force on each other; the large block exerts the force FN on the smaller block, while the smaller block exerts the force –FN on the larger block. In accord with Newton’s third law these forces have opposite directions and equal magnitudes FN . Under the influence of the forces shown, the two blocks have the same acceleration a. We begin our solution by applying Newton’s second law to each one.

SOLUTION According to Newton’s second law, we have

ΣF = P – FN = Ma

FN = ma

Large block

Small block

236 FORCES AND NEWTON'S LAWS OF MOTION

Substituting FN = ma into the large-block expression and solving for P gives P = (M + m) a For the smaller block to remain in place against the larger block, the static frictional force must balance the weight of the smaller block, so that fsMAX = mg. But fsMAX is given by fsMAX = µsFN , where, from the Newton’s second law, we know that FN = ma. Thus, we have µsma = mg or a = g/µs . Using this result in the expression for P gives

P = ( M + m)

( M + m ) g = ( 25 kg + 4.0 kg ) ( 9.80 m/s a= µs

2

0.71

)=

4.0 × 102 N

____________________________________________________________________________________________

113. REASONING According to Newton’s second law, the acceleration of the probe is a = ΣF/m. Using this value for the acceleration in Equation 2.8 and noting that the probe starts from rest (v0 = 0 m/s), we can write the distance traveled by the probe as 1 ΣF x = v0t + 12 at 2 = 2 m

2 t

This equation is the basis for our solution. SOLUTION Since each engine produces the same amount of force or thrust T, the net force is ΣF = 2T when the engines apply their forces in the same direction and

ΣF = T 2 + T 2 = 2T when they apply their forces perpendicularly. Thus, we write the distances traveled in the two situations as follows: 1 2T 2 x= t 2 m

and

Engines fired in same direction

1 2T x = 2 m

2 t⊥

Engines fired perpendicularly

Since the distances are the same, we have 1

2T 2 m

2 1 2T 2 t⊥ t = 2 m

or

2 t 2 = t⊥2

The firing time when the engines apply their forces perpendicularly is, then,

t⊥ =

( 4 2 ) t = ( 4 2 ) ( 28 s ) =

33 s

____________________________________________________________________________________________

Chapter 4 Problems

237

114. REASONING The drawing shows the point between the earth and the moon where the gravitational force exerted on the spacecraft by the earth balances that exerted by the moon. The magnitude of the gravitational force exerted on the spacecraft by the earth is

Fearth = G

mearth mspacecraft r2

while that exerted on the spacecraft by the moon is

Fmoon = G

mmoon mspacecraft

( rearth-moon − r )2

By setting these two expressions equal to each other (since the gravitational forces balance), we will be able to find the distance r. Point where the gravitational forces balance

Earth Moon r rearth-moon

SOLUTION Setting Fearth equal to Fmoon, we have

G

mearth mspacecraft r2

= G

mmoon mspacecraft

( rearth-moon − r )2

Solving this expression for r gives

( rearth-moon ) r=

mearth mmoon

mearth 1+ mmoon

=

( 3.85 ×108 m ) 1 + 81.4

81.4

= 3.47 ×108 m

______________________________________________________________________________

238 FORCES AND NEWTON'S LAWS OF MOTION

115. SSM REASONING AND SOLUTION The free-body diagram is shown at the right. The forces that act on the picture are the pressing force P, the normal force FN exerted on the picture by the wall, the weight mg of the picture, and the force of static friction fsMAX . The maximum magnitude for

MAX

fS FN

P

mg

the frictional force is given by Equation 4.7: = µs FN . The picture is in equilibrium, and, if we take the directions to the right and up as positive, we have in the x direction fsMAX

∑ Fx = P − FN = 0

or

P = FN

∑ Fy = fsMAX − mg = 0

or

fsMAX = mg

and in the y direction

Therefore, fsMAX = µs FN = mg

But since FN = P , we have

µs P = mg

Solving for P, we have

P=

mg

µs

=

(1.10 kg)(9.80 m/s 2 ) = 16.3 N 0.660

____________________________________________________________________________________________

116. REASONING AND SOLUTION a. The rope exerts a tension, T, acting upward on each block. Applying Newton's second law to the lighter block (block 1) gives T – m1g = m1a Similarly, for the heavier block (block 2) T – m2g = – m2a Subtracting the second equation from the first and rearranging yields m – m1 a= 2 g = 3.68 m/s 2 m + m 1 2

b. The tension in the rope is now 908 N since the tension is the reaction to the applied force exerted by the hand. Newton's second law applied to the block is

Chapter 4 Problems

239

T – m1g = m1a Solving for a gives a=

( 908 N ) – 9.80 m/s 2 = 11.8 m/s 2 T –g= 42.0 kg m1

c. In the first case, the inertia of BOTH blocks affects the acceleration whereas, in the second case, only the lighter block's inertia remains. ____________________________________________________________________________________________

117. SSM REASONING AND SOLUTION The penguin comes to a halt on the horizontal surface because the kinetic frictional force opposes the motion and causes it to slow down. The time required for the penguin to slide to a halt (v = 0 m/s) after entering the horizontal patch of ice is, according to Equation 2.4,

t=

v − v0 ax

=

−v0 ax

We must, therefore, determine the acceleration of the penguin as it slides along the horizontal patch (see the following drawing).

θ

fk1

FN1

FN2

mg sin θ

fk2

mg cosθ

mg

Free-body diagram A

Free-body diagram B

For the penguin sliding on the horizontal patch of ice, we find from free-body diagram B and Newton's second law in the x direction (motion to the right is taken as positive) that ∑ Fx = – f k2 = ma x

or

ax =

– f k2 m

=

– µk FN2 m

In the y direction in free-body diagram B, we have ∑ Fy = FN2 – mg = 0 , or FN2 = mg . Therefore, the acceleration of the penguin is

240 FORCES AND NEWTON'S LAWS OF MOTION

ax =

– µk mg m

= – µk g

(1)

Equation (1) indicates that, in order to find the acceleration ax, we must find the coefficient of kinetic friction. We are told in the problem statement that the coefficient of kinetic friction between the penguin and the ice is the same for the incline as for the horizontal patch. Therefore, we can use the motion of the penguin on the incline to determine the coefficient of friction and use it in Equation (1). For the penguin sliding down the incline, we find from free-body diagram A (see the previous drawing) and Newton's second law (taking the direction of motion as positive) that ∑ Fx = mg sin θ – f k1 = max = 0

f k1 = mg sin θ

or

(2)

Here, we have used the fact that the penguin slides down the incline with a constant velocity, so that it has zero acceleration. From Equation 4.8, we know that f k1 = µ k FN1 . Applying Newton's second law in the direction perpendicular to the incline, we have ∑ Fy = FN1 – mg cos θ = 0

or

FN1 = mg cos θ

Therefore, f k1 = µk mg cosθ , so that according to Equation (2), we find f k1 = µ k mg cos θ = mg sin θ Solving for the coefficient of kinetic friction, we have

µk =

sin θ = tan θ cos θ

Finally, the time required for the penguin to slide to a halt after entering the horizontal patch of ice is −v – v0 v0 1.4 m/s t= 0 = = = = 1.2 s ax – µk g g tan θ (9.80 m/s2 ) tan 6.9° ____________________________________________________________________________________________

Chapter 4 Problems

241

118. REASONING The following figure shows the crate on the incline and the free body diagram for the crate. The diagram at the far right shows all the forces resolved into components that are parallel and perpendicular to the surface of the incline. We can analyze the motion of the crate using Newton's second law. The coefficient of friction can be determined from the resulting equations. P sin θ

y

FN

x

MAX

P

MAX

FN

fS

fS

P

θ

θ

P cos θ

θ

mg sin θ

mg

mg cos θ

SOLUTION Since the crate is at rest, it is in equilibrium and its acceleration is zero in all directions. If we take the direction down the incline as positive, Newton's second law indicates that ∑ Fx = P cos θ + mg sin θ − fsMAX = 0 According to Equation 4.7, fsMAX = µ s FN . Therefore, we have P cos θ + mg sin θ − µ s FN = 0

(1)

The expression for the normal force can be found from analyzing the forces that are perpendicular to the incline. Taking up to be positive, we have ∑ Fy = P sin θ + FN – mg cos θ = 0

or

FN = mg cos θ – P sin θ

Equation (1) then becomes P cos θ + mg sin θ − µ s (mg cos θ – P sin θ ) = 0 Solving for the coefficient of static friction, we find that

µs =

P cos θ + mg sin θ (535 N) cos 20.0° + (225 kg)(9.80 m/s 2 ) sin 20.0° = = 0.665 mg cos θ – P sin θ (225 kg)(9.80 m/s 2 ) cos 20.0° – (535 N) sin 20.0°

____________________________________________________________________________________________

242 FORCES AND NEWTON'S LAWS OF MOTION

119. REASONING The free-body diagram for the box is shown in the following drawing on the left. On the right the same drawing is repeated, except that the pushing force P is resolved into its horizontal and vertical components.

P

P sin θ

FN

θ

FN

P cos θ

fk

fk

mg

mg

Since the block is moving at a constant velocity, it has no acceleration, and Newton’s second law indicates that the net vertical and net horizontal forces must separately be zero. SOLUTION Taking upward and to the right as the positive directions, we write the zero net vertical and horizontal forces as follows: FN – mg – P sin θ = 0

P cos θ – f k = 0

Vertical

Horizontal

From the equation for the horizontal forces, we have P cos θ = fk . But the kinetic frictional

force is fk = µkFN . Furthermore, from the equation for the vertical forces, we have FN = mg + P sin θ . With these substitutions, we obtain

P cos θ = f k = µk FN = µk ( mg + P sin θ ) Solving for P gives

P=

µk mg cos θ − µk sin θ

The necessary pushing force becomes infinitely large when the denominator in this expression is zero. Hence, we find that cos θ – µk sin θ = 0 , which can be rearranged to show that sin θ 1 1 = tanθ = or θ = tan –1 = 68° µk cos θ 0.41 ____________________________________________________________________________________________

CHAPTER 5 DYNAMICS OF UNIFORM

CIRCULAR MOTION ANSWERS TO FOCUS ON CONCEPTS QUESTIONS ____________________________________________________________________________________________

1.

(c) The velocity of car A has a constant magnitude (speed) and direction. Since its velocity is constant, car A does not have an acceleration. The velocity of car B is continually changing direction during the turn. Therefore, even though car B has a constant speed, it has an acceleration (known as a centripetal acceleration).

2.

(d) The centripetal (or “center-seeking”) acceleration of the car is perpendicular to its velocity and points toward the center of the circle that the road follows.

3.

(b) The magnitude of the centripetal acceleration is equal to v2/r, where v is the speed of the object and r is the radius of the circular path. Since the radius of the track is smaller at A compared to B, the centripetal acceleration of the car at A has a greater magnitude.

4.

(a) The magnitude ac of the centripetal acceleration is given by ac = v2/r.

5.

(d) The acceleration (known as the centripetal acceleration) and the net force (known as the centripetal force) have the same direction and point toward the center of the circular path.

6.

(a) According to the discussion in Example 7 in Section 5.3, the maximum speed that the cylinder can have is given by vmax = µs gr , where µs is the coefficient of static friction, g is the acceleration due to gravity, and r is the radius of the path.

7.

(d) The radius of path 1 is twice that of path 2. The tension in the cord is the centripetal force. Since the centripetal force is inversely proportional to the radius r of the path, T1 must be one-half of T2.

8.

(a) The centripetal force is given by Fc = mv /r. The centripetal forces for particles 1, 2 and

2

2

2

2

3 are, respectively, 4m0v0 /r0, 3m0v0 /r0, and 2m0v0 /r0. 9.

(d) The centripetal force is directed along the radius and toward the center of the circular path. The component FN sin θ of the normal force is directed along the radius and points toward the center of the path. 2

10. (a) The magnitude of the centripetal force is given by Fc = mv /r. The two cars have the same speed v and the radius r of the turn is the same. The cars also have the same mass m, even though they have different weights due to the different accelerations due to gravity. Therefore, the centripetal accelerations are the same.

244 DYNAMICS OF UNIFORM CIRCULAR MOTION

11. (e) The centripetal force acting on a satellite is provided by the gravitational force. The 2 magnitude of the gravitational force is inversely proportional to the radius squared (1/r ), so if the radius is doubled, the gravitational force is one fourth as great; 1/22 = 1/4. 12. The orbital speed is v = 1.02 × 103 m/s. 13. (b) The magnitude of the centripetal force acting on the astronaut is equal to her apparent 2 weight. The centripetal force is given by Equation 5.3 as Fc = mv /r, which depends on the square (v2) of the astronaut’s speed and inversely (1/r) on the radius of the ring. According to Equation 5.1, r = vT /(2π), the radius is directly proportional to the speed. Thus, the centripetal force is directly proportional to the speed v of the astronaut. As the astronaut walks from the inner ring to the outer ring, her speed doubles and so does her apparent weight. 14. (d) The skier at A is speeding up, so the direction of the acceleration, and hence the net force, must be parallel to the skier’s velocity. At B the skier is momentarily traveling at a constant speed on a circular path of radius r. The direction of the net force, called the centripetal force, must be toward the center of the path. At C the skier is in free-fall, so the net force, which is the gravitational force, is straight downward. 15. (b) According to Newton’s second law, the net force, FN − mg , must equal the mass m 2

times the centripetal acceleration v /r.

Chapter 5 Problems

245

CHAPTER 5 DYNAMICS OF UNIFORM

CIRCULAR MOTION PROBLEMS

1.

SSM REASONING The speed of the plane is given by Equation 5.1: v = 2 π r / T , where T is the period or the time required for the plane to complete one revolution. SOLUTION Solving Equation 5.1 for T we have

T=

2.

2 π r 2 π ( 2850 m ) = = 160 s v 110 m / s 2

REASONING According to ac = v /r (Equation 5.2), the magnitude ac of the centripetal acceleration depends on the speed v of the object and the radius r of its circular path. In Example 2 the object is moving on a path whose radius is infinitely large; in other words, the object is moving along a straight line. SOLUTION Using Equation 5.2, we find the following values for the magnitude of the centripetal acceleration: v 2 (12 m/s ) = = 290 m/s2 r 0.50 m 2

Example 1

ac =

Example 2

v 2 ( 35 m/s ) ac = = = 0 m/s2 r ∞

Example 3

v 2 ( 2.3 m/s ) ac = = = 2.9 m/s2 r 1.8 m

2

2

3.

REASONING AND SOLUTION Since the speed of the object on and off the circle remains constant at the same value, the object always travels the same distance in equal time intervals, both on and off the circle. Furthermore since the object travels the distance OA in the same time it would have moved from O to P on the circle, we know that the distance OA is equal to the distance along the arc of the circle from O to P.

246 DYNAMICS OF UNIFORM CIRCULAR MOTION

The circumference of the circle is 2 π r = 2 π (3.6 m) = 22.6 m . The arc OP subtends an angle of θ = 25° ; therefore, since any circle contains 360°, the arc OP is 25/360 or 6.9 per cent of the circumference of the circle. Thus,

b

gb g

OP = 22.6 m 0.069 = 1.6 m

and, from the argument given above, we conclude that the distance OA is 1.6 m .

4.

REASONING AND SOLUTION Let s represent the length of the path of the pebble after it is released. From Conceptual Example 2, we know that the pebble will fly off tangentially. Therefore, the path s is perperpendicular to the radius r of the circle. Thus, the distances r, s, and d form a right triangle with hypotenuse d as shown in the figure at the right. From the figure we see that cos α =

Target

r 1 r = = d 10 r 10

or

α = cos –1

s Pebble d

α

r

θ

35°

C

1I F G H10 JK= 84°

Furthermore, from the figure, we see that α + θ + 35° = 180° . Therefore,

θ = 145°− α = 145°−84 ° = 61° _____________________________________________________________________________________________

5.

SSM REASONING The magnitude ac of the car’s centripetal acceleration is given by

Equation 5.2 as ac = v2 / r , where v is the speed of the car and r is the radius of the track. 3

The radius is r = 2.6 × 10 m. The speed can be obtained from Equation 5.1 as the circumference (2π r) of the track divided by the period T of the motion. The period is the time for the car to go once around the track (T = 360 s). SOLUTION Since ac = v 2 / r and v = ( 2π r ) / T , the magnitude of the car’s centripetal acceleration is 2π r 2 3 2 4π 2 r 4π ( 2.6 ×10 m ) v T = = 2 = = 0.79 m/s 2 ac = 2 r r T ( 360 s ) 2

______________________________________________________________________________

Chapter 5 Problems

6.

247

REASONING Blood traveling through the aortic arch follows a circular path with a diameter of 5.0 cm and, therefore, a radius of r = 2.5 cm = 0.025 m. We know the speed v of v2 the blood flow, so the relation ac = (Equation 5.2) will give the magnitude of the r blood’s centripetal acceleration. SOLUTION With a blood flow speed of v = 0.32 m/s, the magnitude of the centripetal acceleration in the aortic arch is v 2 ( 0.32 m/s ) = = 4.1 m/s 2 r 0.025 m 2

ac =

7.

REASONING The relationship between the magnitude ac of the centripetal acceleration and the period T of the tip of a moving clock hand can be obtained by using Equations 5.2 and 5.1: v2 2π r ac = v= (5.2) (5.1) r T The period is the time it takes a clock hand to go once around the circle. In these expressions, v is the speed of the tip of the hand and r is the length of the hand. Substituting Equation 5.1 into Equation 5.2 yields 2

2π r 2 v 4π 2 r T ac = = = 2 r r T

(1)

SOLUTION The period of the second hand is Tsecond = 60 s. The period of the minute hand is Tminute = 1 h = 3600 s. Using Equation (1), we find that the ratio of the centripetal acceleration of the tip of the second hand to that of the minute hand is

ac, second ac, minute

=

4π 2 r 2 Tsecond

4π 2 r 2 Tminute

=

2 Tminute 2 Tsecond

2 3600 s ) ( = ( 60 s )2

= 3600

_____________________________________________________________________________________________

8.

2

REASONING The centripetal acceleration is given by Equation 5.2 as ac = v /r. The value of the radius r is given, so to determine ac we need information about the speed v. But the

speed is related to the period T by v = (2π r)/T, according to Equation 5.1. We can substitute this expression for the speed into Equation 5.2 and see that

248 DYNAMICS OF UNIFORM CIRCULAR MOTION

b

2π r / T v2 ac = = r r

g = 4π 2

T

2

r

2

SOLUTION To use the expression obtained in the reasoning, we need a value for the period T. The period is the time for one revolution. Since the container is turning at 2.0 revolutions per second, the period is T = (1 s)/(2.0 revolutions) = 0.50 s. Thus, we find that the centripetal acceleration is

b g b g

2 4 π 2 r 4 π 0.12 m ac = = = 19 m / s 2 2 2 T 0.50 s

_____________________________________________________________________________________________

9.

SSM

Since the magnitude of the centripetal

REASONING AND SOLUTION 2

acceleration is given by Equation 5.2, a C = v / r , we can solve for r and find that r=

( 98.8 m / s ) 2 v2 = = 332 m a C 3.00(9.80 m / s 2 )

_____________________________________________________________________________________________

10. REASONING The magnitude of the centripetal acceleration of any point on the helicopter blade is given by Equation 5.2, a C = v 2 / r , where r is the radius of the circle on which that point moves. From Equation 5.1: v = 2 π r / T . Combining these two expressions, we obtain

aC =

4π 2 r T2

All points on the blade move with the same period T. SOLUTION The ratio of the centripetal acceleration at the end of the blade (point 1) to that which exists at a point located 3.0 m from the center of the circle (point 2) is a C1 a C2

=

4 π 2 r1 / T 2 4 π r2 / T 2

2

=

r1 r2

=

6.7 m = 2.2 3.0 m

_____________________________________________________________________________________________

11. REASONING AND SOLUTION T = 2π r/v. The speed is

The sample makes one revolution in time T as given by

2 –2 3 2 v = rac = (5.00 × 10 m)(6.25 × 10 )(9.80 m/s )

so that

v = 55.3 m/s

Chapter 5 Problems

249

The period is T = 2π (5.00 × 10

–2

m)/(55.3 m/s) = 5.68 × 10

–3

s = 9.47 × 10

–5

min

The number of revolutions per minute = 1/T = 10 600 rev/min . _____________________________________________________________________________________________

12. REASONING AND SOLUTION a. At the equator a person travels in a circle whose radius equals the radius of the earth, 6

r = Re = 6.38 × 10 m, and whose period of rotation is T = 1 day = 86 400 s. We have v = 2πRe/T = 464 m/s The centripetal acceleration is

v 2 ( 464 m/s ) ac = = = 3.37 × 10–2 m/s 2 r 6.38 × 106 m 2

b. At 30.0° latitude a person travels in a circle of radius, 6

r = Re cos 30.0° = 5.53 × 10 m Thus, v = 2π r/T = 402 m/s

and

2

ac = v /r = 2.92 × 10

–2

m/s

2

_____________________________________________________________________________________________

13. REASONING The magnitude Fc of the centripetal force that acts on the skater is given by Equation 5.3 as Fc = mv 2 / r , where m and v are the mass and speed of the skater, and r is the distance of the skater from the pivot. Since all of these variables are known, we can find the magnitude of the centripetal force. SOLUTION The magnitude of the centripetal force is mv 2 ( 80.0 kg )( 6.80 m/s ) = = 606 N Fc = r 6.10 m ______________________________________________________________________________ 2

14. REASONING The centripetal acceleration depends on the speed v and the radius r of the curve, according to ac = v2/r (Equation 5.2). The speeds of the cars are the same, and since they are negotiating the same curve, the radius is the same. Therefore, the cars have the same centripetal acceleration. However, the magnitude Fc of the centripetal force depends on the mass m of the car, as well as the speed and the radius of the curve, according to

250 DYNAMICS OF UNIFORM CIRCULAR MOTION

2

Fc = mv /r (Equation 5.3). Since the speed and the radius are the same for each car, the car with the greater mass, which is car B, experiences the greater centripetal acceleration. SOLUTION We find the following values for the magnitudes of the centripetal accelerations and forces: Car A

ac = Fc =

Car B

b

27 m / s v2 = 120 m r mA v 2 r

r

6.1 m / s 2

(5.2)

2

120 m

b

Fc =

2

b1100 kg gb27 m / sg =

=

27 m / s v2 = ac = 120 m r mB v 2

g= g=

6700 N

(5.3)

2

6.1 m / s 2

b1600 kg gb27 m / sg = =

(5.2)

2

120 m

9700 N

(5.3)

_____________________________________________________________________________________________

15. SSM REASONING AND SOLUTION

The magnitude of the centripetal force on the

ball is given by Equation 5.3: FC = mv 2 / r . Solving for v, we have v=

FC r m

=

(0.028 N)(0.25 m) = 0.68 m / s 0.015 kg

_____________________________________________________________________________________________

16. REASONING At the maximum speed, the maximum centripetal force acts on the tires, and static friction supplies it. The magnitude of the maximum force of static friction is specified by Equation 4.7 as fsMAX = µs FN , where µs is the coefficient of static friction and FN is the magnitude of the normal force. Our strategy, then, is to find the normal force, substitute it into the expression for the maximum frictional force, and then equate the result to the centripetal force, which is Fc = mv 2 / r , according to Equation 5.3. This will lead us to an expression for the maximum speed that we can apply to each car. SOLUTION Since neither car accelerates in the vertical direction, we can conclude that the car’s weight mg is balanced by the normal force, so FN = mg. From Equations 4.7 and 5.3 it follows that mv 2 fsMAX = µs FN = µs mg = Fc = r Thus, we find that

µs mg =

mv 2 r

or

v = µs gr

Chapter 5 Problems

251

Applying this result to car A and car B gives

vA = µs, A gr

vB = µs, B gr

and

In these two equations, the radius r does not have a subscript, since the radius is the same for either car. Dividing the two equations and noting that the terms g and r are eliminated algebraically, we see that

µs, B gr µs, B vB = = vA µs, A g r µs, A

or

vB = vA

µs, B µs, A

= ( 25 m/s )

0.85 = 22 m/s 1.1

17. REASONING AND SOLUTION Initially, the stone executes uniform circular motion in a circle of radius r which is equal to the radius of the tire. At the instant that the stone flies out of the tire, the force of static friction just exceeds its maximum value f sMAX = µ s FN (see Equation 4.7). The force of static friction that acts on the stone from one side of the tread channel is, therefore, fsMAX = 0.90 (1.8 N) = 1.6 N and the magnitude of the total frictional force that acts on the stone just before it flies out is 2 × 1.6 N = 3.2 N . If we assume that only static friction supplies the centripetal force, then, Fc = 3.2 N . Solving Equation 5.3 ( Fc = mv 2 / r ) for the radius r, we have

(

)

6.0 × 10−3 kg (13 m/s)2 mv 2 = = 0.31 m r= 3.2 N Fc _____________________________________________________________________________________________

18. REASONING AND SOLUTION The force P supplied by the man will be largest when the partner is at the lowest point in the swing. The diagram at the right shows the forces acting on the partner in this situation. The centripetal force necessary to keep the partner swinging along the arc of a circle is provided by the resultant of the force supplied by the man and the weight of the partner. From the figure

P − mg =

mv 2 r

Therefore,

P=

mv 2 + mg r

P

mg

252 DYNAMICS OF UNIFORM CIRCULAR MOTION

Since the weight of the partner, W, is equal to mg, it follows that m = (W/g) and P=

(W/g )v 2 [(475 N)/(9.80 m/s 2 )] (4.00 m/s) 2 +W = + (475 N) = 594 N r (6.50 m)

19. REASONING The centripetal force is the name given to the net force pointing toward the center of the circular path. At the lowest point the net force consists of the tension in the arm pointing upward toward the center and the weight pointing downward or away from the 2

center. In either case the centripetal force is given by Equation 5.3 as Fc = mv /r. SOLUTION (a) The centripetal force is

b gb

9 .5 kg 2 .8 m / s mv 2 Fc = = r 0.85 m

g= 2

88 N

(b) Using T to denote the tension in the arm, at the bottom of the circle we have

mv 2 Fc = T − mg = r

( 9.5 kg )( 2.8 m/s ) = 181 N mv 2 T = mg + = ( 9.5kg ) 9.80 m/s 2 + r 0.85 m

(

)

2

20. REASONING When the penny is rotating with the disk (and not sliding relative to it), it is the static frictional force that provides the centripetal force required to keep the penny moving on a circular path. The magnitude fsMAX of the maximum static frictional force is given by f sMAX = µs F N (Equation 4.7), where FN is the magnitude of the normal force and

µs is the coefficient of static friction. Solving this relation for µs gives

µs =

fsMAX FN

(1)

Since the maximum centripetal force that can act on the penny is the maximum static frictional force, we have F c = f sMAX . Since Fc = mv2/r (Equation 5.3), it follows that f sMAX = mv 2 / r . Substituting this expression into Equation (1) yields

Chapter 5 Problems

µs =

fsMAX FN

mv 2 = r FN

253

(2)

The speed of the penny can be determined from the period T of the motion and the radius r according to v = 2π r/T (Equation 5.1). Furthermore, since the penny does not accelerate in the vertical direction, the upward normal force must be balanced by the downward-pointing weight, so that FN = mg, where g is the acceleration due to gravity. Substituting these two expressions for v and FN into Equation (2) gives 2

2π r m 2 mv 4π 2 r T µs = = = r FN r ( mg ) gT2

(3)

SOLUTION Using Equation (3), we find that the coefficient of static friction required to keep the penny rotating on the disk is

µs =

4π 2 ( 0.150 m ) 4π 2 r = = 0.187 2 gT2 9.80 m/s2 (1.80 s )

(

)

______________________________________________________________________________ 21. REASONING AND SOLUTION

The centripetal acceleration of the block is

ac = v2/r = (28 m/s)2/(150 m) = 5.2 m/s2 The angle θ can be obtained from

θ = tan –1

a I F F5.2 m / s IJ= = tan G G J Hg K H9.80 m / s K 2

C

–1

2

28 °

22. REASONING The coefficient µs of static friction is related to the magnitude f sMAX of the maximum static frictional force and the magnitude FN of the normal force acting on the car by fsMAX = µs FN (Equation 4.7), so that:

fsMAX µs = FN The car is going around an unbanked curve, so the centripetal force Fc =

(1) mv 2 r

(Equation 5.3) must be horizontal. The static frictional force is the only horizontal force, so

254 DYNAMICS OF UNIFORM CIRCULAR MOTION

it serves as the centripetal force. The maximum centripetal force occurs when Fc = f sMAX . Therefore, the maximum speed v the car can have without slipping is related to f sMAX by

Fc = fsMAX =

mv 2 r

(2)

Substituting Equation (2) into Equation (1) yields

mv 2 µs = r FN

(3)

In part a the car is subject to two downward-pointing forces, its weight W and the downforce D. The vertical acceleration of the car is zero, so the upward normal force must balance the two downward forces: FN = W + D. Combining this relation with Equation (3), we obtain an expression for the coefficient of static friction:

mv 2 mv 2 mv 2 µs = r = r = FN W + D r ( mg + D )

(4)

SOLUTION a. Since the downforce is D = 11 000 N, Equation (4) gives the coefficient of static friction as

( 830 kg )( 58 m/s ) mv 2 = = 0.91 r ( mg + D ) (160 m ) ( 830 kg ) 9.80 m/s 2 + 11 000 N 2

µs =

(

)

b. The downforce is now absent (D = 0 N). Solving Equation (4) for the speed of the car, we find that

v=

µs r ( mg + D ) m

= µs rg =

=

µs r ( mg + 0 N ) m

=

( 0.91)(160 m ) ( 9.80 m/s2 ) =

µs r ( mg ) m

38 m/s

Chapter 5 Problems

255

23. REASONING a. The free body diagram shows the swing ride and the two forces that act on a chair: the tension T in the cable, and the weight mg of the chair and its occupant. We note that the chair does not accelerate vertically, so the net force ∑ Fy in the vertical direction must be zero, ∑ Fy = 0 . The net force consists of the upward vertical component of the tension and the downward weight of the chair. The fact that the net force is zero will allow us to determine the magnitude of the tension.

60.0°

T +y

15.0 m +x

r mg

b. According to Newton’s second law, the net force ∑ Fx in the horizontal direction is equal to the mass m of the chair and its occupant times the centripetal acceleration ( ac = v 2 / r ) , so that ∑ Fx = mac = mv2 / r . There is only one force in the horizontal direction, the horizontal component of the tension, so it is the net force. We will use Newton’s second law to find the speed v of the chair. SOLUTION a. The vertical component of the tension is +T cos 60.0°, and the weight is −mg, where we have chosen “up” as the + direction. Since the chair and its occupant have no vertical acceleration, we have that ∑ Fy = 0 , so +T cos 60.0° − mg = 0

(1)

∑ Fy

Solving for the magnitude T of the tension gives

(179 kg ) ( 9.80 m/s 2 ) mg T= = = 3510 N cos 60.0° cos 60.0° b. The horizontal component of the tension is +T sin 60.0°, where we have chosen the direction to the left in the diagram as the + direction. Since the chair and its occupant have a centripetal acceleration in this direction, we have

256 DYNAMICS OF UNIFORM CIRCULAR MOTION

v2 T sin 60.0° = mac = m r ∑ Fx

(2)

From the drawing we see that the radius r of the circular path is r = (15.0 m) sin 60.0° = 13.0 m. Solving Equation (2) for the speed v gives

v=

r T sin 60.0° = m

(13.0 m )( 3510 N ) sin 60.0° = 14.9 m/s 179 kg

24. REASONING We will treat this situation as Seawall a circular turn on a banked surface, with the angle θ that the rider leans serving as the banking angle. The banking angle θ is r related to the speed v of the watercraft, the Beginning of turn radius r of the curve and the magnitude g of 2 v the acceleration due to gravity by tan θ = r rg (Equation 5.4). If the rider is closer to the seawall than r, she will hit the wall while making the turn Therefore, the minimum distance at which she must begin the turn is r, the minimum turn radius (see the drawing). SOLUTION Solving Equation 5.4 for r, we obtain the minimum distance:

( 26 m/s ) v2 = = 170 m r= g tan θ 9.80 m/s 2 tan 22 2

(

)

25. REASONING The angle θ at which a friction-free curve is banked depends on the radius r of the curve and the speed v with which the curve is to be negotiated, according to 2 Equation 5.4: tan θ = v /(rg). For known values of θ and r, the safe speed is

v = rg tan θ Before we can use this result, we must determine tan θ for the banking of the track. SOLUTION The drawing at the right shows a cross-section of the track. From the drawing we have 18 m tan θ = = 0. 34 53 m

θ 165 m – 112 m = 53 m

18 m

Chapter 5 Problems

257

a. Therefore, the smallest speed at which cars can move on this track without relying on friction is vmin = (112 m ) ( 9.80 m/s 2 ) ( 0.34 ) = 19 m/s

b. Similarly, the largest speed is vmax = (165 m ) ( 9.80 m/s 2 ) ( 0.34 ) = 23 m/s

v2 (Equation 5.4) determines the banking angle θ that rg a banked curve of radius r must have if a car is to travel around it at a speed v without relying on friction. In this expression g is the magnitude of the acceleration due to gravity. We will solve for v and apply the result to each curve. The fact that the radius of each curve is the same will allow us to determine the unknown speed.

26. REASONING The relation tan θ =

SOLUTION According to Equation 5.4, we have tan θ =

v2 rg

or

v = r g tan θ

Applying this result for the speed to each curve gives vA = r g tan θ A

and

vB = r g tan θ B

Note that the terms r and g are the same for each curve. Therefore, these terms are eliminated algebraically when we divide the two equations. We find, then, that vB vA

=

r g tan θ B r g tan θ A

=

tan θ B

tan θ A

or

vB = vA

tan θ B

tan θ A

= (18 m/s )

tan19° = 22 m/s tan13°

27. REASONING From the discussion on banked curves in Section 5.4, we know that a car can safely round a banked curve without the aid of static friction if the angle θ of the banked curve is given by tan θ = v02 / ( r g ) , where vo is the speed of the car and r is the radius of the curve (see Equation 5.4). The maximum speed that a car can have when rounding an unbanked curve is v0 = µs g r (see Example 7). By combining these two relations, we can find the angle θ.

258 DYNAMICS OF UNIFORM CIRCULAR MOTION

SOLUTION The angle of the banked curve is θ = tan −1 v02 / ( r g ) . Substituting the expression v0 = µs g r into this equation gives v02 µ gr θ = tan = tan −1 s = tan −1 ( µs ) = tan −1 ( 0.81) = 39° rg rg ______________________________________________________________________________ −1

28. REASONING The lifting force L is perpendicular to the jet’s wings. When the jet banks at an angle θ above the horizontal, therefore, the lifting force tilts an angle θ from the vertical (see the free-body diagram). Because the jet has no vertical acceleration during the horizontal turn, the upward vertical component L cos θ of the lifting force balances the jet’s weight: L cos θ = mg, where m is the jet’s mass and g is the acceleration due to gravity. Therefore, the magnitude of the lifting force is L = mg cos θ .

L sin θ

L

θ L cos θ

mg

At this point we know m and g, but not the banking angle θ. Since the jet follows a horizontal circle, the centripetal force must be Free-body diagram horizontal. The only horizontal force acting on the jet is the horizontal component L sin θ of the lifting force, so this must be the centripetal force. The situation is completely analogous to that of a car driving around a banked curve without the assistance of friction. The relation tan θ = v 2 ( rg ) (Equation 5.4), therefore, expresses the relationship between the jet’s unknown banking angle θ, its speed v, the radius r of the turn, and g, all of which are known. SOLUTION The magnitude of the lifting force is

(

)(

2.00 × 105 kg 9.80 m/s2 mg = L= cos θ cos θ

)

Solving the relation tan θ = v 2 ( rg ) (Equation 5.4) for the angle θ, we obtain 2 123 m/s ) ( −1 θ = tan = tan ( 3810 m ) 9.80 m/s2 rg −1 v

2

(

)

= 22.1

Substituting this value for θ into Equation (1) for the lifting force gives

(

)(

)

2.00 × 105 kg 9.80 m/s2 mg L= = = 2.12 × 106 N cos θ cos 22.1°

(1)

Chapter 5 Problems

29. REASONING The distance d is related to the radius r of the circle on which the car travels by d = r/sin 50.0° (see the drawing).

FN 50.0° r

259

+y

Car

+x

We can obtain the radius by noting mg d that the car experiences a centripetal 50.0° force that is directed toward the center of the circular path. This 40.0° force is provided by the component, FN cos 50.0°, of the normal force that is parallel to the radius. Setting this force equal to the mass m of the car times the centripetal acceleration ( ac = v 2 / r ) gives FN cos 50.0° = mac = mv 2 / r . Solving for the radius r and substituting it into the relation d = r/sin 50.0° gives

d=

r = sin 50.0°

mv 2 FN cos 50.0° sin 50.0°

=

mv 2 ( FN cos 50.0°) ( sin 50.0°)

(1)

The magnitude FN of the normal force can be obtained by observing that the car has no

vertical acceleration, so the net force in the vertical direction must be zero, ∑ Fy = 0 . The net force consists of the upward vertical component of the normal force and the downward weight of the car. The vertical component of the normal force is +FN sin 50.0°, and the weight is −mg, where we have chosen the “up” direction as the + direction. Thus, we have that (2) + FN sin 50.0° − mg = 0 ∑ Fy

Solving this equation for FN and substituting it into the equation above will yield the distance d. SOLUTION Solving Equation (2) for FN and substituting the result into Equation (1) gives

d=

mv 2 mv 2 = ( FN cos 50.0°) ( sin 50.0°) mg ( cos 50.0°)( sin 50.0°) sin 50.0°

( 34.0 m/s ) v2 = = = 184 m g cos 50.0° ( 9.80 m/s 2 ) cos 50.0° 2

______________________________________________________________________________

260 DYNAMICS OF UNIFORM CIRCULAR MOTION

30. REASONING The centripetal force Fc required to keep an object of mass m that moves with speed v on a circle of radius r is Fc = mv 2 / r (Equation 5.3). From Equation 5.1, we know that v = 2 π r / T , where T is the period or the time for the suitcase to go around once. Therefore, the centripetal force can be written as m ( 2 π r / T ) 2 4 mπ 2 r Fc = = r T2

(1)

This expression can be solved for T. However, we must first find the centripetal force that acts on the suitcase. SOLUTION Three forces act on the suitcase. They are the weight mg of the suitcase, the force of static friction f sMAX , and the normal force FN exerted on the suitcase by the surface of the carousel. The following figure shows the free body diagram for the suitcase. In this +y diagram, the y axis is along the vertical direction. MAX The force of gravity acts, then, in the –y direction. FN fs The centripetal force that causes the suitcase to θ move on its circular path is provided by the net θ +x force in the +x direction in the diagram. From the diagram, we can see that only the forces FN and f sMAX have horizontal components. Thus, we have

θ

Fc = f sMAX cosθ – FN sin θ , where the minus sign mg indicates that the x component of FN points to the left in the diagram. Using Equation 4.7 for the maximum static frictional force, we can write this result as in equation (2). Fc = µ s FN cosθ – FN sin θ = FN ( µ s cosθ – sin θ )

(2)

If we apply Newton's second law in the y direction, we see from the diagram that FN cosθ + f s MAX sinθ – mg = ma y = 0

or

FN cosθ + µ s FN sinθ – mg = 0

where we again have used Equation 4.7 for the maximum static frictional force. Solving for the normal force, we find mg FN = cosθ + µ s sinθ Using this result in equation (2), we obtain the magnitude of the centripetal force that acts on the suitcase: mg ( µ s cosθ – sin θ ) Fc = FN ( µ s cosθ – sin θ ) = cosθ + µ s sinθ With this expression for the centripetal force, equation (1) becomes

Chapter 5 Problems

mg ( µ s cosθ – sin θ ) cosθ + µ s sinθ

=

261

4 mπ 2 r T2

Solving for the period T, we find T=

4π 2 r ( cos θ + µ s sinθ ) g ( µs cos θ – sin θ )

=

4π 2 (11.0 m) ( cos 36.0° + 0.760 sin 36.0° )

(9.80 m/s2 ) ( 0.760 cos 36.0° – sin 36.0° )

= 45 s

31. REASONING The speed v of a satellite in circular orbit about the earth is given by v = GM E / r (Equation 5.5), where G is the universal gravitational constant, ME is the mass of the earth, and r is the radius of the orbit. The radius is measured from the center of the earth, not the surface of the earth, to the satellite. Therefore, the radius is found by adding the height of the satellite above the surface of the earth to the radius of the earth 6

(6.38 × 10 m). SOLUTION First we add the orbital heights to the radius of the earth to obtain the orbital radii. Then we use Equation 5.5 to calculate the speeds. Satellite A rA = 6.38 × 10 6 m + 360 × 10 3 m = 6.74 × 10 6 m v=

GM E rA

=

c6.67 × 10

−11

N ⋅ m 2 / kg 2

hc5.98 × 10

24

kg

h=

7690 m / s

kg

h=

7500 m / s

6.74 × 10 m 6

Satellite B rA = 6.38 × 10 6 m + 720 × 10 3 m = 7.10 × 10 6 m v=

GM E rA

=

c6.67 × 10

−11

N ⋅ m 2 / kg 2

hc5.98 × 10

24

7 .10 × 10 6 m

_____________________________________________________________________________________________

32. REASONING AND SOLUTION

2

We have for Jupiter v = GMJ/r, where 5

7

7

r = 6.00 × 10 m + 7.14 × 10 m = 7.20 × 10 m Thus, v=

( 6.67 ×10−11 N ⋅ m 2 / kg 2 )(1.90 ×1027 kg ) = 7.20 × 10 m 7

4.20 × 104 m/s

____________________________________________________________________________________________

262 DYNAMICS OF UNIFORM CIRCULAR MOTION

33.

SSM WWW REASONING Equation 5.5 gives the orbital speed for a satellite in a circular orbit around the earth. It can be modified to determine the orbital speed around any planet P by replacing the mass of the earth M E by the mass of the planet M P : v=

GM P / r .

SOLUTION The ratio of the orbital speeds is, therefore, v2 v1

=

GM P / r2 GM P / r1

=

r1 r2

Solving for v2 gives v 2 = v1

r1 r2

= (1.70 × 10 4 m / s)

5.25 × 10 6 m = 1.33 × 10 4 m / s 8.60 × 10 6 m

34. REASONING AND SOLUTION The normal force exerted by the wall on each astronaut is 2

the centripetal force needed to keep him in the circular path, i.e., Fc = mv /r. Rearranging and letting Fc = (1/2)mg yields 2

2

2

r = 2v /g = 2(35.8 m/s) /(9.80 m/s ) = 262 m

35. REASONING In Section 5.5 it is shown that the period T of a satellite in a circular orbit about the earth is given by (see Equation 5.6)

T=

2π r 3/ 2 GM E

where r is the radius of the orbit, G is the universal gravitational constant, and ME is the mass of the earth. The ratio of the periods of satellites A and B is, then,

2π rA3/2 TA TB

=

GM E

2π rB3/2

=

rA3/2 rB3/2

GM E We do not know the radii rA and rB. However we do know that the speed v of a satellite is

equal to the circumference (2π r) of its orbit divided by the period T, so v = 2 π r / T .

Chapter 5 Problems

263

SOLUTION Solving the relation v = 2 π r / T for r gives r = vT / 2π . Substituting this value for r into Equation (1) yields

v T / ( 2π ) TA rA3/2 = 3/2 = A A = 3/ 2 TB rB vB TB / ( 2π ) 3/ 2

( vA TA ) 3/ 2 ( vB TB )

3/ 2

Squaring both sides of this equation, algebraically solving for the ratio TA/TB, and using the fact that vA = 3vB gives

TA vB3 vB3 1 = = 3 = 3 TB vA ( 3vB ) 27 ______________________________________________________________________________ 36. REASONING AND SOLUTION The period of the moon's motion (approximately the length of a month) is given by T=

4π 2 r 3 = GM E

4π 2 ( 3.85 × 108 m ) ( 6.67 ×10−11 N ⋅ m 2 / kg 2 )( 5.98 ×1024 kg ) 3

= 2.38 × 106 s = 27.5 days

37.

SSM REASONING Equation 5.2 for the centripetal acceleration applies to both the plane and the satellite, and the centripetal acceleration is the same for each. Thus, we have

ac =

2 v plane

rplane

=

2 v satellite

rsatellite

or

v plane =

Fr G G Hr

plane

satellite

I JJKv

satellite

The speed of the satellite can be obtained directly from Equation 5.5. SOLUTION Using Equation 5.5, we can express the speed of the satellite as v satellite =

Gm E rsatellite

Substituting this expression into the expression obtained in the reasoning for the speed of the plane gives

264 DYNAMICS OF UNIFORM CIRCULAR MOTION

Fr I Fr v = G J Gr G Hr JK G H b15 mgc6.67 × 10 N ⋅ m =

v plane =

plane

plane

satellite

satellite

satellite

−11

v plane

2

I Gm r JJ r = r K / kg h c5.98 × 10

plane

E

satellite

GmE

satellite

2

24

6.7 × 10 6 m

kg

h=

12 m / s

GM S r −11 2 2 (Equation 5.5), where MS is the mass of the star, G = 6.674 × 10 N ⋅ m / kg is the universal gravitational constant, and r is the orbital radius. This expression can be solved for 2π r MS. However, the orbital radius r is not known, so we will use the relation v = T (Equation 5.1) to eliminate r in favor of the known quantities v and T (the period). Returning to Equation 5.5, we square both sides and solve for the mass of the star:

38. REASONING The speed v of a planet orbiting a star is given by v =

GM S = v2 r

Then, solving v =

or

MS =

rv 2 G

(1)

2π r vT for r yields r = , which we substitute into Equation (1): T 2π

vT 2 v rv v 3T 2π MS = = = G G 2π G 2

(2)

We will use Equation (2) to calculate the mass of the star in part a. In part b, we will solve Equation (2) for the period T of the faster planet, which should be shorter than that of the slower planet.

SOLUTION a. The speed of the slower planet is v = 43.3 km/s = 43.3×103 m/s. Its orbital period in seconds is T = (7.60 yr)[(3.156 × 107 s)/(1 yr)] = 2.40×108 s. Substituting these values into Equation (2) yields the mass of the star:

(

)( 3

) )

43.3 × 103 m/s 2.40 × 108 s v 3T = = 4.65 × 1031 kg MS = 11 2 2 − 2π G 2π 6.674 × 10 N ⋅ m / kg

(

This is roughly 23 times the mass of the sun.

Chapter 5 Problems

265

b. Solving Equation (2) for the orbital period T, we obtain v 3T = MS 2π G

or

T=

2π GM S

(3)

v3

The speed of the faster planet is v = 58.6 km/s = 58.6 × 103 m/s. Equation (3) now gives the orbital period of the faster planet in seconds:

T=

(

)( 3 (58.6 ×103 m/s )

2π 6.674 ×10−11 N ⋅ m2 / kg 2 4.65 ×1031 kg

) = 9.69 ×107 s

Lastly, we convert the period from seconds to years:

(

)

1 yr T = 9.69 ×107 s 7 3.156 ×10 s

= 3.07 yr

39. REASONING The satellite’s true weight W when at rest on the surface of the planet is the gravitational force the planet exerts on it. This force is given by W = GM P m r 2 (Equation 4.4), where G is the universal gravitational constant, MP is the mass of the planet, m is the mass of the satellite, and r is the distance between the satellite and the center of the planet. When the satellite is at rest on the planet’s surface, its distance from the planet’s center is RP, the radius of the planet, so we have W = GM P m RP2 . The satellite’s mass m is given, as is the planet’s radius RP. But we must use the relation T =

2π r 3 2 (Equation 5.6) GM P

to determine the planet’s mass MP in terms of the satellite’s orbital period T and orbital radius r. Squaring both sides of Equation 5.6 and solving for MP, we obtain

T2 =

( ) 2 ( GM P )

22 π 2 r 3 2

Substituting Equation (1) into W =

W=

2

=

4π 2 r 3 GM P

GM P m RP2

or

MP =

4π 2 r 3 GT 2

(1)

(Equation 4.4), we find that

Gm G m 4π 2 r 3 4π 2r 3m M = ( P ) 2 2 = 2 2 RPT RP2 RP G T

(2)

266 DYNAMICS OF UNIFORM CIRCULAR MOTION

SOLUTION All of the quantities in Equation (2), except for the period T, are given in SI base units, so we must convert the period from hours to seconds, the SI base unit for time: T = (2.00 h)[(3600 s)/(1 h)] = 7.20×103 s. The satellite’s orbital radius r in Equation (2) is the distance between the satellite and the center of the orbit, which is the planet’s center. Therefore, the orbital radius is the sum of the planet’s radius RP and the satellite’s height h above the planet’s surface: r = RP + h = 4.15×106 m + 4.1×105 m = 4.56×106 m. We now use Equation (2) to calculate the satellite’s true weight at the planet’s surface:

W=

4π 2 r 3m RP2T 2

=

(

4π 2 4.56 × 106 m

)

3

( 5850 kg )

( 4.15 ×10 m ) ( 7.20 ×10 s ) 2

6

3

2

= 2.45 × 104 N

40. REASONING AND SOLUTION a. The centripetal acceleration of a point on the rim of chamber A is the artificial acceleration due to gravity, 2

aA = vA /rA = 10.0 m/s

2

A point on the rim of chamber A moves with a speed vA = 2π rA/T where T is the period of revolution, 60.0 s. Substituting the second equation into the first and rearranging yields 2

2

rA = aAT /(4π ) = 912 m b. Now rB = rA/4.00 =

228 m 2

c. A point on the rim of chamber B has a centripetal acceleration aB = vB /rB. The point moves with a speed vB = 2π rB/T. Substituting the second equation into the first yields aB =

41.

4π 2 rB T2

=

4π 2 ( 228 m ) = 2.50 m/s 2 2 ( 60.0 s )

SSM REASONING According to Equation 5.3, the magnitude Fc of the centripetal force that acts on each

passenger is Fc = mv 2 / r , where m and v are the mass and speed of a passenger and r is the radius of the turn. From this relation we see that the speed is given by v = Fc r / m . The

Passenger

r 2 mg

centripetal force is the net force required to keep each mg

Chapter 5 Problems

267

passenger moving on the circular path and points toward the center of the circle. With the aid of a free-body diagram, we will evaluate the net force and, hence, determine the speed. SOLUTION The free-body diagram shows a passenger at the bottom of the circular dip. There are two forces acting: her downward-acting weight mg and the upward-acting force 2mg that the seat exerts on her. The net force is +2mg − mg = +mg, where we have taken “up” as the positive direction. Thus, Fc = mg. The speed of the passenger can be found by using this result in the equation above. Substituting Fc = mg into the relation v = Fc r / m yields v=

Fc r m

=

( mg ) r m

= gr =

( 9.80 m/s ) ( 20.0 m ) = 14.0 m/s 2

42. REASONING The rider’s speed v at the top of the loop is related to the centripetal force acting on her mv 2 by Fc = (Equation 5.3). The centripetal force r Fc is the net force, which is the sum of the two vertical forces: W (her weight) and FN (the magnitude of the normal force exerted on her by the electronic sensor). Both forces are illustrated in the “Top of loop” free-body diagram. Because both forces point in the same direction, the magnitude of the centripetal force is Fc = mg + FN . Thus, we

FN = 350 N

FN = 770 N mg

Top of loop free-body diagram

mg = 770 N Stationary free-body diagram

mv 2 . We will solve this relation to find the speed v of the rider. The r reading on the sensor at the top of the loop gives the magnitude FN = 350 N of the downward normal force. Her weight mg is equal to the reading on the sensor when level and stationary (see the “Stationary” free-body diagram). mv 2 SOLUTION Solving mg + FN = for the speed v, we obtain r

have that mg + FN =

v2 =

r ( mg + FN ) m

or

v=

r ( mg + FN ) m

The only quantity not yet known is the rider’s mass m, so we will calculate it from her weight W by using the relation W = mg (Equation 4.5). Thus, m = W/g = (770 N)/(9.80 m/s2) = 79 kg. The speed of the rider at the top of the loop is

268 DYNAMICS OF UNIFORM CIRCULAR MOTION

v=

r ( mg + FN ) m

=

( 21 m )( 770 N + 350 N ) = 17 m/s 79 kg

43. SSM REASONING The centripetal force is the name given to the net force pointing toward the center of the circular path. At point 3 at the top the net force pointing toward the center of the circle consists of the normal force and the weight, both pointing toward the center. At point 1 at the bottom the net force consists of the normal force pointing upward toward the center and the weight pointing downward or away from the center. In either case the centripetal force is given by Equation 5.3 as Fc = mv2/r. SOLUTION At point 3 we have

Fc = F N + mg =

mv 32 r

At point 1 we have

Fc = F N − mg =

mv12 r

Subtracting the second equation from the first gives

2 mg =

mv 32 r

−

mv12 r

Rearranging gives v 32 = 2 gr + v 12

Thus, we find that

c

v 3 = 2 9 .80 m / s 2

+ 15 m / s g = hb3.0 m gb 2

17 m / s

_____________________________________________________________________________________________

44. REASONING The normal force (magnitude FN) that the pilot’s seat exerts on him is part of the centripetal force that keeps him on the vertical circular path. However, there is another contribution to the centripetal force, as the drawing at the right shows. This additional contribution is the pilot’s weight (magnitude W). To obtain the ratio FN/W, we will apply Equation 5.3, which specifies the centripetal force as Fc = mv 2 / r .

FN +

W − SOLUTION Noting that the direction upward (toward the center of the circular path) is positive in the drawing, we see that the centripetal force is Fc = FN − W . Thus, from Equation 5.3 we have

Chapter 5 Problems

Fc = FN − W =

269

mv2 r

The weight is given by W = mg (Equation 4.5), so we can divide the expression for the centripetal force by the expression for the weight and obtain that Fc =

FN − W W

=

mv 2 mg r

FN

or

W

−1 =

v2 gr

Solving for the ratio FN /W, we find that

( 230 m/s ) v2 = 1+ = 1+ = 8.8 W gr 9.80 m/s 2 ( 690 m )

FN

2

(

)

45. REASONING The magnitude Fc of the centripetal force is Fc = mv 2 / r (Equation 5.3). Since the speed v, the mass m, and the radius r are fixed, the magnitude of the centripetal force is the same at each point on the circle. When the ball is at the three o’clock position, the force of gravity, acting downward, is perpendicular to the stick and cannot contribute to the centripetal force. (See Figure 5.21 in the text, point 2, for a similar situation.) At this point, only the tension of T = 16 N contributes to the centripetal force. Considering that the centripetal force is the same everywhere, we can conclude that it has a magnitude of 16 N everywhere. At the twelve o’clock position the tension T and the force of gravity mg both act downward (the negative direction) toward the center of the circle, with the result that the centripetal force at this point is –T – mg. (See Figure 5.21, point 3.) At the six o’clock position the tension points upward toward the center of the circle, while the force of gravity points downward, with the result that the centripetal force at this point is T – mg. (See Figure 5.21, point 1.) SOLUTION Assuming that upward is the positive direction, we find at the twelve and six o’clock positions that

270 DYNAMICS OF UNIFORM CIRCULAR MOTION

− T − mg = −16 N

Twelve o' clock

Centripetal force

b

gc

h

gc

h

T = 16 N − 0.20 kg 9 .80 m / s 2 = 14 N T − mg = 16 N

Six o' clock

Centripetal force

b

T = 16 N + 0.20 kg 9 .80 m / s 2 = 18 N

46. REASONING Because the crest of the hill is a circular arc, the motorcycle’s speed v is related to the centripetal force Fc acting

FN

on the motorcycle: Fc = mv 2 r (Equation 5.3), where m is the mass of the motorcycle and r is the radius of the circular crest. Solving Equation 5.3 for the speed, we obtain v 2 = Fc r m or

mg

v = Fc r m . The free-body diagram shows that two vertical Free-body diagram of forces act on the motorcycle. One is the weight mg of the the motorcycle motorcycle, which points downward. The other is the normal force FN exerted by the road. The normal force points directly opposite the motorcycle’s weight. Note that the motorcycle’s weight must be greater than the normal force. The reason for this is that the centripetal force is the net force produced by mg and FN and must point toward the center of the circle, which lies below the motorcycle. Only if the magnitude mg of the weight exceeds the magnitude FN of the normal force will the centripetal force point downward. Therefore, we can express the magnitude of the centripetal force as Fc = mg − FN. With this identity, the relation v = Fc r m becomes

v=

( mg − FN ) r m

(1)

SOLUTION When the motorcycle rides over the crest sufficiently fast, it loses contact with the road. At that point, the normal force FN is zero. In that case, Equation (1) yields the motorcycle’s maximum speed:

v=

( mg − 0 ) r m

=

m gr = gr = m

(9.80 m/s2 ) ( 45.0 m) = 21.0 m/s

Chapter 5 Problems

271

47. REASONING When the stone is whirled in a horizontal circle, the centripetal force is provided by the tension Th in the string and is given by Equation 5.3 as Th Centripetal force

=

mv 2 r

where m and v are the mass and speed of the stone, and r is the radius of the circle. When the stone is whirled in a vertical circle, the maximum tension occurs when the stone is at the lowest point in its path. The free-body diagram shows the forces that act on the stone in this situation: the tension Tv in the string and the weight mg of the stone. The centripetal force is the net force that points toward the center of the circle. Setting the centripetal force equal to mv 2 / r , as per Equation 5.3, we have +Tv − mg =

mv 2 r

(1)

Tv

Stone mg (2)

Centripetal force

Here, we have assumed upward to be the positive direction. We are given that the maximum tension in the string in the case of vertical motion is 15.0% larger than that in the case of horizontal motion. We can use this fact, along with Equations 1 and 2, to find the speed of the stone.

Solution Since the maximum tension in the string in the case of vertical motion is 15.0% larger than that in the horizontal motion, Tv = (1.000 + 0.150) Th . Substituting the values of Th and Tv from Equations (1) and (2) into this relation gives Tv = (1.000 + 0.150 ) Th

mv 2 mv 2 + mg = (1.000 + 0.150 ) r r Solving this equation for the speed v of the stone yields v=

gr (9.80 m/s 2 ) (1.10 m) = = 8.48 m/s 0.150 0.150

272 DYNAMICS OF UNIFORM CIRCULAR MOTION

Clothes

48. REASONING The drawing at the right shows the two forces that act on a piece of clothing just before it loses contact with the wall of the cylinder. At that instant the centripetal force is provided by the normal force FN and the radial component of the weight. From the drawing, the radial component of the weight is given by

FN

φ

mg

θ

mg cosφ = mg cos (90 ° – θ ) = mg sinθ

Therefore, with inward taken as the positive direction, Equation 5.3 ( Fc = mv 2 / r ) gives

FN + mg sinθ =

mv 2 r

At the instant that a piece of clothing loses contact with the surface of the drum, FN = 0 N, and the above expression becomes mv 2 mg sinθ = r According to Equation 5.1, v = 2π r / T , and with this substitution we obtain g sinθ =

( 2π r / T ) 2 4π 2 r = r T2

This expression can be solved for the period T. Since the period is the required time for one revolution, the number of revolutions per second can be found by calculating 1/T. SOLUTION Solving for the period, we obtain T=

4π 2 r = 2π g sinθ

r = 2π g sinθ

0.32 m

c9.80 m / s hsin 70.0° = 1.17 s 2

Therefore, the number of revolutions per second that the cylinder should make is 1 1 = = 0.85 rev / s T 1.17 s

Chapter 5 Problems

273

49. SSM REASONING In Example 3, it was shown that the magnitudes of the centripetal acceleration for the two cases are Radius = 33 m

a C = 35 m / s 2

Radius = 24 m

a C = 48 m / s 2

According to Newton's second law, the centripetal force is FC = ma C (see Equation 5.3). SOLUTION a. Therefore, when the sled undergoes the turn of radius 33 m, FC = ma C = ( 350 kg)(35 m / s 2 ) = 1.2 × 10 4 N

b. Similarly, when the radius of the turn is 24 m, FC = ma C = ( 350 kg)(48 m / s 2 ) = 1.7 × 10 4 N

50. REASONING Two pieces of information are provided. One is the fact that the magnitude of the centripetal acceleration ac is 9.80 m/s2. The other is that the space station should not rotate faster than two revolutions per minute. This rate of twice per minute corresponds to thirty seconds per revolution, which is the minimum value for the period T of the motion. With these data in mind, we will base our solution on Equation 5.2, which gives the centripetal acceleration as ac = v 2 / r , and on Equation 5.1, which specifies that the speed v on a circular path of radius r is v = 2π r / T .

SOLUTION From Equation 5.2, we have ac =

v2 r

or

r=

v2 ac

Substituting v = 2π r / T into this result and solving for the radius gives

v 2 ( 2π r / T ) = r= ac ac

2

or

r=

acT 2 4π 2

2 9.80 m/s 2 ) ( 30.0 s ) ( = = 223 m

4π 2

51. REASONING Since the tip of the blade moves on a circular path, it experiences a centripetal acceleration whose magnitude ac is given by Equation 5.2 as, ac = v 2 / r , where v is the speed of blade tip and r is the radius of the circular path. The radius is known, and the speed can be obtained by dividing the distance that the tip travels by the time t of travel.

274 DYNAMICS OF UNIFORM CIRCULAR MOTION

Since an angle of 90° corresponds to one fourth of the circumference of a circle, the distance is 14 ( 2π r ) . SOLUTION Since ac = v 2 / r and v = acceleration of the blade tip is

1 4

( 2π r ) / t = π r / ( 2t ) , the magnitude of the centripetal

2

πr 2 π 2 r π ( 0.45 m ) v 2 2t ac = = = 2 = = 6.9 m/s 2 2 r r 4t 4 ( 0.40 s )

52. REASONING AND SOLUTION a. In terms of the period of the motion, the centripetal force is written as Fc = 4π2mr/T2 = 4π2 (0.0120 kg)(0.100 m)/(0.500 s)2 = 0.189 N b. The centripetal force varies as the square of the speed. Thus, doubling the speed would increase the centripetal force by a factor of 2 2 = 4 .

53. REASONING The astronaut in the chamber is subjected to a centripetal acceleration ac that is given by ac = v 2 / r (Equation 5.2). In this expression v is the speed at which the astronaut in the chamber moves on the circular path of radius r. We can solve this relation for the speed. SOLUTION Using Equation 5.2, we have

ac =

54. REASONING

v2 r

or

(

)

v = ac r = 7.5 9.80 m/s 2 (15 m ) = 33 m/s

The person feels the centripetal force acting on his back. This force is

2

Fc = mv /r, according to Equation 5.3. This expression can be solved directly to determine the radius r of the chamber. SOLUTION Solving Equation 5.3 for the radius r gives

b gb

83 kg 3.2 m / s mv 2 r= = FC 560 N

g= 2

1.5 m

_____________________________________________________________________________________________

Chapter 5 Problems

275

55. SSM REASONING As the motorcycle passes over the top of the hill, it will experience a centripetal force, the magnitude of which is given by Equation 5.3: FC = mv 2 / r . The centripetal force is provided by the net force on the cycle + driver system. At that instant, the net force on the system is composed of the normal force, which points upward, and the weight, which points downward. Taking the direction toward the center of the circle (downward) as the positive direction, we have FC = mg − FN . This expression can be solved for FN , the normal force. SOLUTION a. The magnitude of the centripetal force is FC =

mv 2 (342 kg)(25.0 m / s) 2 = = 1.70 × 10 3 N r 126 m

b. The magnitude of the normal force is FN = mg − FC = (342 kg)(9.80 m/s 2 ) − 1.70 × 103 N = 1.66 × 103 N _____________________________________________________________________________________________

56. REASONING The speed of the satellite is given by Equation 5.1 as v = 2π r / T . Since we are given that the period is T = 1.20 × 104 s, it will be possible to determine the speed from Equation 5.1 if we can determine the radius r of the orbit. To find the radius, we will use Equation 5.6, which relates the period to the radius according to T = 2π r 3 / 2 / GM E , where G is the universal gravitational constant and ME is the mass of the earth. SOLUTION According to Equation 5.1, the orbital speed is v=

2π r T

To find a value for the radius, we begin with Equation 5.6:

2π r 3/ 2 T= GM E

r 3/ 2 =

or

T GM E 2π

Next, we square both sides of the result for r3/2:

(

r

3/ 2

)

2

T GM E = 2π

2

or

r3 =

T 2GM E 4π 2

276 DYNAMICS OF UNIFORM CIRCULAR MOTION

We can now take the cube root of both sides of the expression for r3 in order to determine r:

r=

T 2GM E 3 4π 2

1.20 × 10 4 s ) ( 6.67 × 10 −11 N ⋅ m 2 /kg 2 )( 5.98 × 1024 kg ) 3 ( = = 1.13 × 107 m 2

4π 2

With this value for the radius, we can use Equation 5.1 to obtain the speed:

(

)

7 2π r 2π 1.13 × 10 m = = 5.92 × 103 m/s v= 4 T 1.20 × 10 s

57. SSM REASONING AND SOLUTION The centripetal acceleration for any point on the blade a distance r from center of the circle, according to Equation 5.2, is ac = v 2 / r . From Equation 5.1, we know that v = 2 π r / T where T is the period of the motion. Combining these two equations, we obtain ac =

( 2π r / T ) 2 4π 2 r = r T2

a. Since the turbine blades rotate at 617 rev/s, all points on the blades rotate with a period of T = (1/617) s = 1.62 × 10–3 s . Therefore, for a point with r = 0.020 m , the magnitude of the centripetal acceleration is

ac =

4 π 2 ( 0.020 m) = 3.0 × 10 5 m / s 2 2 –3 (1.62 × 10 s)

b. Expressed as a multiple of g, this centripetal acceleration is

c

a c = 3.0 × 10 5 m / s 2

1.00 g I hF G H9.80 m / s JK= 2

3.1 × 10 4 g

_____________________________________________________________________________________________

58. REASONING The centripetal acceleration for any point that is a distance r from the center of the disc is, according to Equation 5.2, ac = v 2 / r . From Equation 5.1, we know that v = 2 π r / T where T is the period of the motion. Combining these two equations, we obtain ac =

( 2π r / T ) 2 4π 2 r = r T2

Chapter 5 Problems

277

SOLUTION Using the above expression for ac , the ratio of the centripetal accelerations of the two points in question is a 2 4 π 2 r2 / T22 r2 / T22 = = a 1 4 π 2 r1 / T12 r1 / T12 Since the disc is rigid, all points on the disc must move with the same period, so T1 = T2 . Making this cancellation and solving for a2 , we obtain a 2 = a1

r2 r1

c

= 120 m / s 2

0.050 m I G hF H0.030 m JK=

2.0 × 10 2 m / s 2

Note that even though T1 = T2 , it is not true that v1 = v 2 . Thus, the simplest way to approach this problem is to express the centripetal acceleration in terms of the period T which cancels in the final step. _____________________________________________________________________________________________

REASONING Let v0 be the initial speed of the ball as it begins its projectile motion. Then, the centripetal force is given by Equation 5.3: FC = mv02 / r . We

59. SSM WWW

are given the values for m and r; however, we must determine the value of v0 from the details of the projectile motion after the ball is released. In the absence of air resistance, the x component of the projectile motion has zero acceleration, while the y component of the motion is subject to the acceleration due to gravity. The horizontal distance traveled by the ball is given by Equation 3.5a (with a x = 0 m/s2): x = v 0 x t = ( v 0 cosθ ) t

with t equal to the flight time of the ball while it exhibits projectile motion. The time t can be found by considering the vertical motion. From Equation 3.3b, v y = v0 y + a y t

After a time t, v y = −v0 y . Assuming that up and to the right are the positive directions, we have −2 v 0 y −2 v 0 sin θ t= = ay ay and x = ( v 0 cosθ )

F−2 v sinθ I G H a JK 0

y

Using the fact that 2sin θ cosθ = sin 2θ , we have

278 DYNAMICS OF UNIFORM CIRCULAR MOTION

x=−

2 v 02 cosθ sin θ ay

=−

v 02 sin 2θ

(1)

ay

Equation (1) (with upward and to the right chosen as the positive directions) can be used to determine the speed v0 with which the ball begins its projectile motion. Then Equation 5.3 can be used to find the centripetal force. SOLUTION Solving equation (1) for v0 , we have v0 =

−x ay sin 2θ

=

− ( 86.75 m)(–9.80 m / s 2 ) = 29 . 3 m / s sin 2(41° )

Then, from Equation 5.3,

mv 02

(7.3 kg)(29.3 m / s) 2 FC = = = 3500 N r 1.8 m _____________________________________________________________________________________________

60. REASONING AND SOLUTION a. The centripetal force is provided by the normal force exerted on the rider by the wall

.

b. Newton's second law applied in the horizontal direction gives 2

2

FN = mv /r = (55.0 kg)(10.0 m/s) /(3.30 m) =

1670 N

c. Newton's second law applied in the vertical direction gives µ sFN – mg = 0 or µ s = (mg)/FN =

0.323

_____________________________________________________________________________________________

61. SSM WWW REASONING If the effects of gravity are not ignored in Example 5, the plane will make an angle θ with the vertical as shown in figure A below. The figure B shows the forces that act on the plane, and figure C shows the horizontal and vertical components of these forces.

Chapter 5 Problems

279

T cos θ T L

θ

L

T sin θ

θ

mg

mg

r

A

B

C

From figure C we see that the resultant force in the horizontal direction is the horizontal component of the tension in the guideline and provides the centripetal force. Therefore,

T sin θ =

mv 2 r

From figure A, the radius r is related to the length L of the guideline by r = L sinθ ; therefore, mv 2 T sin θ = (1) L sin θ

The resultant force in the vertical direction is zero: T cosθ − mg = 0 , so that

T cosθ = mg

(2)

From equation (2) we have T=

mg cos θ

(3)

Equation (3) contains two unknown, T and θ . First we will solve equations (1) and (3) simultaneously to determine the value(s) of the angle θ . Once θ is known, we can calculate the tension using equation (3). SOLUTION Substituting equation (3) into equation (1):

mg I F G Hcos θ JKsin θ = Thus,

mv 2 L sin θ

sin 2 θ v2 = cos θ gL

Using the fact that cos2 θ + sin2 θ = 1, equation (4) can be written

(4)

280 DYNAMICS OF UNIFORM CIRCULAR MOTION

1 – cos 2 θ v2 = cos θ gL or

1 v2 – cos θ = cos θ gL This can be put in the form of an equation that is quadratic in cos θ. Multiplying both sides by cos θ and rearranging yields:

cos 2 θ +

v2 cos θ – 1 = 0 gL

(5)

Equation (5) is of the form ax 2 + bx + c = 0

(6)

with x = cos θ, a = 1, b = v2/(gL), and c = –1. The solution to equation (6) is found from the quadratic formula:

x =

– b ± b 2 − 4 ac 2a

When v = 19.0 m/s, b = 2.17. The positive root from the quadratic formula gives x = cos θ = 0.391. Substitution into equation (3) yields mg (0.900 kg)(9.80 m / s 2 ) T= = = cos θ 0.391

23 N

When v = 38.0 m/s, b = 8.67. The positive root from the quadratic formula gives x = cos θ = 0.114. Substitution into equation (3) yields T=

(0.900 kg)(9.80 m / s 2 ) mg = = cos θ 0.114

77 N

_____________________________________________________________________________________________

CHAPTER 6 WORK AND ENERGY ANSWERS TO FOCUS ON CONCEPTS QUESTIONS

1. (e) When the force is perpendicular to the displacement, as in C, there is no work. When the force points in the same direction as the displacement, as in B, the maximum work is done. When the force points at an angle with respect to the displacement but has a component in the direction of the displacement, as in A, the work has a value between zero and the maximum work. 2. (b) Work is positive when the force has a component in the direction of the displacement. The force shown has a component along the −x and along the +y axis. Therefore, displacements in these two directions involve positive work. 3. (c) The work is given by W = (F cos θ)s, which is zero when F = 0 N, s = 0 m, or θ = 90º. 4. 78 kg·m2/s2 5. (a) The kinetic energy is KE = 12 mv2 . Since the velocity components are perpendicular, the 2 2 + vNorth Pythagorean theorem indicates that v 2 = vEast . Therefore,

(

) ( 3.00 kg ) ( 5.00 m/s ) + (8.00 m/s )

2 2 KE = 12 m vEast + vNorth =

1 2

2

2

6. 115 m/s 7. (e) The work-energy theorem states that W = KEf – KE0. Since work is done, the kinetic energy changes. Since kinetic energy is KE = 12 mv2 , the speed v must also change. Since the instantaneous speed is the magnitude of the instantaneous velocity, the velocity must also change. 8. (d) The work-energy theorem indicates that when the net force acting on the particle does negative work, the kinetic energy decreases. Since each force does negative work, the work done by the net force must be negative, and the kinetic energy must decrease. But when the kinetic energy decreases, the speed must also decrease, since the kinetic energy is proportional to the square of the speed. Since it is stated that the speed increases, this answer is not possible. 9. (b) The work-energy theorem states that the net work done on the particle is equal to the change in the particle’s kinetic energy. However, the speed does not change. Therefore, the kinetic energy does not change, because kinetic energy is proportional to the square of the speed. According to the work-energy theorem, the net work is zero, which will be the case if W1 = − W2.

282 WORK AND ENERGY

10. (d) Since the block starts from rest, the work energy theorem is

W = 12 mvf2 − 12 mv02 =

1 2

( 2.70 kg ) vf2

The final speed can be obtained from this expression, since the work done by the net force is W = (75.0 N) (cos 38.0º) (6.50 m) + (54.0 N) (cos 0.0º) (6.50 m) + (93.0 N) (cos 65.0º) (6.50 m)

11. (c) The gravitational force is a conservative force, and a conservative force does no net work on an object moving around a closed path. 12. (e) Since air resistance always opposes the motion, it is a force that is always directed opposite to the displacement. Therefore, negative work is done. 13. (d) The velocity is constant. Therefore, the speed is also constant, and so is the kinetic energy. However, the total mechanical energy is the kinetic plus the gravitational potential energy, and the car moves up the hill, gaining potential energy as it goes. Thus, in the circumstance described mechanical energy cannot be conserved. 14. (c) The ball comes to a halt at B, when all of its initial kinetic energy is converted into potential energy. 15. (a) The stone’s motion is an example of projectile motion, in which the final vertical height is the same as the initial vertical height. Because friction and air resistance are being ignored, mechanical energy is conserved. Since the initial and final vertical heights are the same, ∆PE = 0 J. Since energy is conserved, it follows that ∆KE = 0 J also. 16. (b) The total mechanical energy is the kinetic energy plus the gravitational potential energy: 2 E = 12 mv 2 + mgh = 12 ( 88.0 kg )(19.0 m/s ) + ( 88.0 kg ) 9.80 m/s 2 ( 55.0 m ) . Since friction

(

)

and air resistance are being ignored, the total mechanical energy is conserved, which means that it has the same value, no matter what the height above sea level is. 17. (d) Since the track is frictionless, the conservation of mechanical energy applies. We take the initial vertical level of the car as the zero level for gravitational potential energy. The highest vertical level that the car can attain occurs when the final speed of the car is zero and all of the initial kinetic energy is converted into gravitational potential energy. According to KE 0 2.2 J the conservation principle, KE0 = mgh, or h = = = 0.45 m . Thus, mg ( 0.50 kg ) 9.80 m/s 2

(

)

the car can pass over hills A, B, C, and D but not hill E. 18. (a) Using the ground as the zero level for measuring height and applying the energy conservation principle gives the following result: 1 2

m ( 7.00 m/s ) + 12 m ( 9.00 m/s ) + mg (11.0 m ) = 12 mvf2 2

2

The mass m of the ball can be eliminated algebraically.

Chapter 6 Answers to Focus on Concepts Questions

283

19. 7.07 m/s 20. (d) The work done by the net nonconservative force is Wnc = KEf + PEf −KE0 − PE0. 21. (b) The principle of conservation of mechanical energy applies only if the work done by the net nonconservative force is zero, Wnc = 0 J. When only a single nonconservative force is present, it is the net nonconservative force, and a force that is perpendicular to the displacement does no work. 22. (a) A single nonconservative force is the net nonconservative force, and the work it does is given by Wnc = ∆KE + ∆PE. Since the velocity is constant, ∆KE = 0 J. Therefore, Wnc = ∆PE = mg(hf − h0). But hf − h0 = −325 m, since hf is smaller than h0. Thus, Wnc = (92.0 kg)(9.80 m/s2)(−325 m). 23. 71.5 J 24. 12 370 J 25. 344 W 26. 130 N 27. 376 J

284 WORK AND ENERGY

CHAPTER 6 WORK AND ENERGY PROBLEMS ______________________________________________________________________________ 1.

SSM

REASONING The work W done by the tension in the tow rope is given by

Equation 6.1 as W = ( F cos θ ) s , where F is the magnitude of the tension, s is the magnitude of the skier’s displacement, and θ is the angle between the tension and the displacement vectors. The magnitude of the displacement (or the distance) is the speed v of the skier multiplied by the time t (see Equation 2.1), or s = v t . SOLUTION Substituting s = v t into the expression for the work, W = ( F cos θ ) s , we have W = ( F cos θ ) v t . Since the skier moves parallel to the boat and since the tow rope is parallel to the water, the angle between the tension and the skier’s displacement is θ = 37.0°. Thus, the work done by the tension is W = ( F cos θ ) vt = [(135 N ) cos 37.0°] ( 9.30 m/s )(12.0 s ) = 1.20 × 104 J ____________________________________________________________________________________________

2.

REASONING a. We will use W = ( F cos θ ) s (Equation 6.1) to determine the work W. The force (magnitude = F) doing the work is the force exerted on you by the elevator floor. This force is the normal force and has a magnitude FN, so F = FN in Equation 6.1. To determine the normal force, we will use the fact that the elevator is moving at a constant velocity and apply Newton’s second law with the acceleration set to zero. Since the force exerted by the elevator and the displacement (magnitude = s) are in the same direction on the upward part of the trip, the angle between them is θ = 0°, with the result that the work done by the force is positive. b. To determine the normal force, we will again use the fact that the elevator is moving at a constant velocity and apply Newton’s second law with the acceleration set to zero. Since the force exerted by the elevator and the displacement are in opposite directions on the downward part of the trip, the angle between them is θ = 180°, and so the work done by the force is negative. SOLUTION a. The free-body diagram at the right shows the three forces that act on you: W is your weight, Wb is the weight of your belongings, and FN is the normal force exerted on you by the floor of the elevator. Since you are moving upward at a constant velocity, your acceleration is zero, you are in equilibrium, and the net force in the y direction must be zero:

FN

W Wb

+y

Chapter 6 Problems

FN − W − Wb = 0 ΣFy

285

(4.9b)

Therefore, the magnitude of the normal force is FN = W + Wb. The work done by the normal force is W = ( FN cos θ ) s = (W + Wb ) ( cos θ ) s

(6.1)

= ( 685 N + 915 N )( cos 0° )(15.2 m ) = 24 300 J b. During the downward trip, you are still in equilibrium since the elevator is moving with a constant velocity. The magnitude of the normal force is now FN = W. The work done by the normal force is W = ( FN cos θ ) s = (W cos θ ) s = ( 685 N )( cos 180° )(15.2 m ) = −10 400 J

(6.1)

______________________________________________________________________________ 3.

REASONING The work done by the tension in the cable is given by Equation 6.1 as W = (T cos θ ) s . Since the elevator is moving upward at a constant velocity, it is in equilibrium, and the magnitude T of the tension must be equal to the magnitude mg of the elevator’s weight; T = mg. SOLUTION a. The tension and the displacement vectors point in the same direction (upward), so the angle between them is θ = 0°. The work done by the tension is W = (T cos θ ) s = ( mg cos θ ) s

(6.1)

= (1200 kg ) ( 9.80 m/s 2 ) cos 0° ( 35 m ) = 4.1 × 105 J

b. The weight and the displacement vectors point in opposite directions, so the angle between them is θ = 180°. The work done by the weight is

W = ( mg cos θ ) s = (1200 kg ) ( 9.80 m/s 2 ) cos 180° ( 35 m ) = −4.1 × 105 J

(6.1)

______________________________________________________________________________ 4.

REASONING a. The work done by the gravitational force is given by Equation 6.1 as W = (F cos θ) s. The gravitational force points downward, opposite to the upward vertical displacement of 4.60 m. Therefore, the angle θ is 180º.

286 WORK AND ENERGY

b. The work done by the escalator is done by the upward normal force that the escalator exerts on the man. Since the man is moving at a constant velocity, he is in equilibrium, and the net force acting on him must be zero. This means that the normal force must balance the man’s weight. Thus, the magnitude of the normal force is FN = mg, and the work that the escalator does is also given by Equation 6.1. However, since the normal force and the upward vertical displacement point in the same direction, the angle θ is 0º. SOLUTION a. According to Equation 6.1, the work done by the gravitational force is W = ( F cos θ ) s = ( mg cos θ ) s = ( 75.0 kg ) ( 9.80 m/s 2 ) cos180° ( 4.60 m ) = −3.38 × 103 J b. The work done by the escalator is W = ( F cos θ ) s = ( FN cos θ ) s = ( 75.0 kg ) ( 9.80 m/s 2 ) cos 0° ( 4.60 m ) = 3.38 × 103 J

5.

SSM REASONING AND SOLUTION Solving Equation 6.1 for the angle θ , we obtain

W 1.10 × 103 J −1 = cos = 42.8° Fs (30.0 N)(50.0 m) ______________________________________________________________________________

θ = cos −1

6.

REASONING AND SOLUTION According to Equation 6.1, W = Fs cos θ, the work is

a.

W = (94.0 N)(35.0 m) cos 25.0° = 2980 J

b.

W = (94.0 N)(35.0 m) cos 0° = 3290 J

______________________________________________________________________________ 7.

REASONING AND SOLUTION a. In both cases, the lift force L is perpendicular to the displacement of the plane, and, therefore, does no work. As shown in the drawing in the text, when the plane is in the dive, there is a component of the weight W that points in the direction of the displacement of the plane. When the plane is climbing, there is a component of the weight that points opposite to the displacement of the plane. Thus, since the thrust T is the same for both cases, the net force in the direction of the displacement is greater for the case where the plane is diving. Since the displacement s is the same in both cases, more net work is done during the dive .

Chapter 6 Problems

287

b. The work done during the dive is Wdive = (T + W cos 75°) s , while the work done during the climb is Wclimb = (T + W cos 115°) s . Therefore, the difference between the net work done during the dive and the climb is

Wdive – Wclimb = (T + W cos 75°) s – (T + W cos 115°) s = Ws (cos 75° – cos 115°)

= (5.9 × 10 4 N)(1.7× 103 m)(cos 75° – cos 115°) = 6.8 × 10 7 J ______________________________________________________________________________ 8.

REASONING The drawing shows three of the +y 29.0° forces that act on the cart: F is the pushing force that F the shopper exerts, f is the frictional force that f +x opposes the motion of the cart, and mg is its weight. The displacement s of the cart is also shown. Since the cart moves at a constant velocity along the +x mg s direction, it is in equilibrium. The net force acting on it in this direction is zero, ΣFx = 0. This relation can be used to find the magnitude of the pushing force. The work done by a constant force is given by Equation 6.1 as W = ( F cos θ ) s , where F is the magnitude of the force, s is the

magnitude of the displacement, and θ is the angle between the force and the displacement. SOLUTION a. The x-component of the net force is zero, ΣFx = 0, so that

F cos 29.0° − f = 0

(4.9a)

ΣFx

The magnitude of the force that the shopper exerts is F =

48.0 N f = = 54.9 N . cos 29.0° cos 29.0°

b. The work done by the pushing force F is W = ( F cos θ ) s = ( 54.9 N )( cos 29.0° )( 22.0 m ) = 1060 J

(6.1)

c. The angle between the frictional force and the displacement is 180°, so the work done by the frictional force f is W = ( f cos θ ) s = ( 48.0 N )( cos 180.0° )( 22.0 m ) = −1060 J

d. The angle between the weight of the cart and the displacement is 90°, so the work done by the weight mg is

288 WORK AND ENERGY

(

W = ( mg cos θ ) s = (16.0 kg ) 9.80 m/s 2

) ( cos 90°)( 22.0 m ) = 0 J

______________________________________________________________________________ 9.

SSM REASONING AND SOLUTION a. According to Equation 6.1, the work done by the applied force is 3

W = Fs cos θ = (2.40 × 102 N)(8.00 m) cos 20.0° = 1.80 × 10 J b. According to Equation 6.1, the work done by the frictional force is Wf = fks cos θ, where

f k = µs (mg − F sin θ ) = (0.200) (85.0 kg)(9.80 m/s2 ) − (2.40 × 102 N)(sin 20.0°) = 1.50 × 102 N Therefore, 3

2

Wf = (1.50 × 10 N)(8.00 m) cos 180° = −1.20 × 10 J

______________________________________________________________________________ 10. REASONING The amount of work W done by a wind force F is given by W = ( F cos θ ) s (Equation 6.1), where F is the magnitude of the force, θ is the angle between the direction of the force and the direction of the boat’s motion (which is due north), and s is the magnitude of the boat’s displacement. In the two scenarios described, the magnitudes of both wind forces F1 and F2 are the same and equal F, but the displacements and angles are different. Therefore, we can express the work done in the first case as W1 = ( F cos θ1 ) s1 and that done

in the second case as W2 = ( F cos θ 2 ) s2 . We will use these two expressions, along with the fact that the wind does the same amount of work in both cases ( W1 = W2 ) to find the angle θ1 between due north and the direction of the wind’s force. SOLUTION Setting the two expressions for work done by the wind equal, and solving for the angle θ1, we find

( F cosθ1 ) s1 = ( F cos θ2 ) s2 W1

W2

or

cos θ1 =

s2 cos θ 2 s1

or

s2 cos θ 2 s1

θ1 = cos −1

In the second case, the force of the wind points due north, the same direction that the boat sails. Therefore, the second angle is zero (θ2 = 0°), and angle that the force of the wind makes with due north is

Chapter 6 Problems

θ1 = cos

(

289

)

( 47 m ) cos 0 = 25 52 m

−1

11. REASONING The work done by a constant force of magnitude F that makes an angle θ with a displacement of magnitude s is given by W = ( F cos θ ) s (Equation 6.1). The magnitudes and directions of the pulling forces exerted by the husband and wife differ, but the displacement of the wagon is the same in both cases. Therefore, we can express the work WH done by the husband’s pulling force as WH = ( FH cos θ H ) s . Similarly, the work done by

the wife can be written as WW = ( FW cos θ W ) s . We know the directional angles of both

forces, the magnitude FH of the husband’s pulling force, and that both forces do the same amount of work (WW = WH).

SOLUTION Setting the two expressions for work equal to one another, and solving for the magnitude FW of the wife’s pulling force, we find

( FW cos θ W ) s = ( FH cos θ H ) s WW

or

FW =

WH

FH cos θ H cos θ W

=

( 67 N ) ( cos 58 cos 38

) = 45 N

12. REASONING AND SOLUTION The net work done on the car is WT = WF + Wf + Wg + WN WT = Fs cos 0.0° + f s cos 180° – mgs sin 5.0° + FNs cos 90° Rearranging this result gives WT F= + f + mg sin 5.0° s

150 × 103 J 2 3 + 524 N + (1200 kg ) 9.80 m/s sin 5.0° = 2.07 × 10 N 290 m ______________________________________________________________________________ =

(

)

290 WORK AND ENERGY

13. REASONING The work W done by the net external force acting on the skier can be found from the work-energy theorem in the form of W = 12 mvf2 − 12 mv02 (Equation 6.3) because the final and initial kinetic energies of the skier can be determined. The kinetic energy

( mv ) 1 2

2

is increasing, because the skier’s speed is increasing. Thus, the work will be

positive, reflecting the fact that the net external force must be in the same direction as the displacement of the skier to make the skier pick up speed. SOLUTION The work done by the net external force acting on the skier is

W = 12 mvf2 − =

1 2

1 mv 2 0 2

( 70.3 kg )(11.3 m /s )2 − 12 ( 70.3 kg )( 6.10 m /s )2 =

3

3.2 × 10 J

______________________________________________________________________________ 14. REASONING The car’s kinetic energy depends upon its mass and speed via KE = 12 mv 2 (Equation 6.2). The total amount of work done on the car is equal to the difference between its final and initial kinetic energies: W = KE f − KE 0 (Equation 6.3). We will use these two relationships to determine the car’s mass. SOLUTION Combining KE = 12 mv 2 (Equation 6.2) and W = KE f − KE 0 (Equation 6.3), we obtain 1 m v2 − v2 = W or W = 12 mvf2 − 12 mv02 f 0 2

(

)

Solving this expression for the car’s mass m, and noting that 185 kJ = 185 000 J, we find that 2 (185 000 J ) 2W m= 2 2 = = 1450 kg vf − v0 ( 28.0 m/s )2 − ( 23.0 m/s )2

15.

SSM WWW REASONING AND SOLUTION The work done on the arrow by the bow is given by W = Fs cos 0° = Fs This work is converted into kinetic energy according to the work energy theorem. W = 12 mvf2 − 12 mv02 Solving for vf, we find that

Chapter 6 Problems

2W + v02 = m

vf =

2 Fs + v02 = m

2 ( 65 N )( 0.90 m ) –3

291

+ ( 0 m/s ) = 39 m/s 2

75 × 10 kg ______________________________________________________________________________

16. REASONING a. Because we know the flea’s mass m, we can find its speed vf once we know its kinetic energy KE f = 12 mvf2 (Equation 6.2) at the instant it leaves the ground. The work-energy

theorem W = KE f − KE 0 (Equation 6.3) relates KEf to the total work W done on the flea during push-off. Because it starts from rest, the flea has no initial kinetic energy, and we see that the total work done on the flea is equal to its final kinetic energy: W = KE f − ( 0 J ) = KE f = 12 mvf2 . Since air resistance and the flea’s weight are being ignored, the only force doing work on the flea is the upward force Fground exerted by the ground, so Wground = W = 12 mvf2 .

(

)

b. We will make use of the definition of work Wground = Fground cos θ s (Equation 6.1) to calculate the flea’s upward displacement s while the force Fground is doing work on the flea. SOLUTION a. Solving Wground = 12 mvf2 for the final speed vf of the flea, we find 1 2

mvf2 = Wground

vf2 =

or

2Wground

or

m

vf =

2Wground m

Therefore,

vf =

(

2 +2.4 ×10−4 J 1.9 ×10

−4

kg

) = 1.6 m/s

b. The force Fground of the ground on the flea is upward, the same as the direction of the flea’s displacement s while it is pushing off. Therefore, the angle θ between Fground and s is

(

)

zero. Solving Wground = Fground cos θ s for the displacement magnitude s, we obtain s=

Wground Fground cos θ

=

+2.4 × 10−4 J

( 0.38 N ) ( cos 0

)

= 6.3 × 10−4 m

292 WORK AND ENERGY

17. REASONING The work done by the catapult Wcatapult is one contribution to the work done by the net external force that changes the kinetic energy of the plane. The other contribution is the work done by the thrust force of the plane’s engines Wthrust. According to the workenergy theorem (Equation 6.3), the work done by the net external force Wcatapult + Wthrust is equal to the change in the kinetic energy. The change in the kinetic energy is the given kinetic energy of 4.5 × 107 J at lift-off minus the initial kinetic energy, which is zero since the plane starts at rest. The work done by the thrust force can be determined from Equation 6.1 [W = (F cos θ) s], since the magnitude F of the thrust is 2.3 × 105 N and the magnitude s of the displacement is 87 m. We note that the angle θ between the thrust and the displacement is 0º, because they have the same direction. In summary, we will calculate Wcatapult from Wcatapult + Wthrust = KEf − KE0. SOLUTION According to the work-energy theorem, we have Wcatapult + Wthrust = KEf − KE0 Using Equation 6.1 and noting that KE0 = 0 J, we can write the work energy theorem as follows: Wcatapult + ( F cos θ ) s = KE f Work done by thrust

Solving for Wcatapult gives

Wcatapult = KE f − ( F cosθ ) s Work done by thrust

= 4.5 ×107 J − ( 2.3 ×105 N ) cos 0° ( 87 m ) = 2.5 ×107 J

18. REASONING AND SOLUTION From the work-energy theorem, Equation 6.3, 1 1 1 W = mvf2 − mv02 = m vf2 − v02 2 2 2 1 2 2 a. W = (7420 kg) (8450 m/s) – (2820 m/s) = 2

2.35 × 1011 J

b. W = (7420 kg) (2820 m/s) − (8450 m/s) = –2.35 × 10 J 2 ______________________________________________________________________________ 1

2

2

11

Chapter 6 Problems

19.

293

SSM REASONING The work done to launch either object can be found from 1

1

2

2

Equation 6.3, the work-energy theorem, W = KEf − KE 0 = mvf − mv0 . 2 2 SOLUTION a. The work required to launch the hammer is

(

)

W = 12 mvf2 − 12 mv02 = 12 m vf2 − v02 = 12 (7.3 kg) (29 m/s)2 − ( 0 m/s ) = 3.1×103 J

2

b. Similarly, the work required to launch the bullet is

(

)

W = 12 m vf2 − v02 = 12 (0.0026 kg) (410 m/s)2 − ( 0 m/s ) = 2.2 × 102 J

2

______________________________________________________________________________ 20. REASONING a. The work W done by the force is related to the change in the asteroid’s kinetic energy by the work-energy theorem W = 12 mvf2 − 12 mv02 (Equation 6.3). Since the final speed of the asteroid is vf = 5500 m/s and is less than the initial speed of v0 = 7100 m/s, the work obtained from Equation 6.3 will be negative. This is consistent with the fact that, since the asteroid is slowing down, it is decelerating, so that the force must point in a direction that is opposite to the direction of the displacement. b. Since the work W is known from part (a), we can use W = ( F cos θ ) s (Equation 6.1) to determine the magnitude F of the force. The direction of the force is opposite to the displacement of the asteroid, so the angle between them is θ = 180°. SOLUTION a. The work-energy theorem states that the work done by the force acting on the asteroid is equal to its final kinetic energy minus its initial kinetic energy: 2

W = 12 mvf − =

1 2

1 mv 2 0 2

( 4.5 × 10 kg ) (5500 m /s ) 4

(6.3) 2

−

1 2

( 4.5 × 10 kg ) ( 7100 m /s ) 4

2

11

= −4.5 × 10

J

b. The work done by the force acting on the asteroid is given by W = ( F cos θ ) s (Equation 6.1). Since the force acts to slow down the asteroid, the force and the displacement vectors point in opposite directions, so the angle between them is θ = 180°. The magnitude of the force is

294 WORK AND ENERGY

−4.5 × 1011 J W 5 F= = = 2.5 × 10 N 6 s cos θ 1.8 × 10 m cos180°

(

)

(6.1)

______________________________________________________________________________ 21. REASONING AND SOLUTION The work energy theorem, W = 12 mvf2 − 12 mv02 , gives

vf =

2W 2 + v0 m

where

W = Fs cos 180° = − (4.0 × 105 N)(2500 × 103 m) = − 1.0 × 1012 J Now vf =

(

2 −1.0 × 1012 J 4

) + (11 000 m/s )2 = 9 × 103 m/s

5.0 × 10 kg ______________________________________________________________________________

22. REASONING Since the person has an upward acceleration, there must be a net force acting in the upward direction. The net force ΣFy is related to the acceleration ay by Newton’s second law, ΣFy = ma y , where m is the mass of the person. This relation will allow us to determine the tension in the cable. The work done by the tension and the person’s weight can be found directly from the definition of work, Equation 6.1. SOLUTION a. The free-body diagram at the right shows the two forces that act on the person. Applying Newton’s second law, we have

T

+y

s

T − mg = ma y ΣFy

Solving for the magnitude of the tension in the cable yields

mg

T = m(ay + g) = (79 kg)(0.70 m/s2 + 9.80 m/s2) = 8.3 × 102 N

b. The work done by the tension in the cable is WT = (T cos θ ) s = (8.3 × 102 N) (cos 0°) (11 m) = 9.1 × 103 J c. The work done by the person’s weight is

(6.1)

295

Chapter 6 Problems

WW = ( mg cos θ ) s = (79 kg) ( 9.8 m/s 2 ) (cos 180°) (11 m) = −8.5 × 103 J

(6.1)

d. The work-energy theorem relates the work done by the two forces to the change in the kinetic energy of the person. The work done by the two forces is W = WT + WW: WT + WW = 12 mvf2 − 12 mv02 W

(6.3)

Solving this equation for the final speed of the person gives

vf = v02 +

2 (WT + WW ) m

2 ( 9.1 × 103 J − 8.5 × 103 J ) = 4 m / s 79 kg ______________________________________________________________________________ =

23.

( 0 m/s )

2

+

SSM REASONING According to the work-energy theorem, the kinetic energy of the sled increases in each case because work is done on the sled. The work-energy theorem is 1 2

1 2

given by Equation 6.3: W = KE f − KE 0 = mvf2 − mv02 . The work done on the sled is given by Equation 6.1: W = ( F cos θ )s . The work done in each case can, therefore, be expressed as

W1 = (F cos 0°)s =

1 mv2f 2

1 2

− mv02 = ∆KE1

and 1 2

1 2

W2 = (F cos 62°)s = mvf2 − mv02 = ∆KE2 The fractional increase in the kinetic energy of the sled when θ = 0° is ∆KE1 (F cos 0°)s = = 0.38 KE 0 KE0

Therefore,

Fs = (0.38) KE0

(1)

The fractional increase in the kinetic energy of the sled when θ = 62° is ∆KE2 (F cos 62°)s Fs = = (cos 62°) KE 0 KE 0 KE 0

Equation (1) can be used to substitute for Fs in Equation (2).

(2)

296 WORK AND ENERGY

SOLUTION Combining Equations (1) and (2), we have ∆KE2 Fs (0.38) KE 0 = (cos 62°) = (cos 62°) = (0.38)(cos 62°) = 0.18 KE 0 KE 0 KE 0

Thus, the sled's kinetic energy would increase by 18 % . ______________________________________________________________________________ 24. REASONING The initial speed v0 of the skier can be obtained by applying the workenergy theorem: W = 12 mvf2 − 12 mv02 (Equation 6.3). This theorem indicates that the initial kinetic energy 12 mv02 of the skier is related to the skier’s final kinetic energy work W done on the skier by the kinetic frictional force according to 1 2

1 mv 2 f 2

and the

mv02 = 12 mvf2 − W

Solving for the skier’s initial speed gives

v0 = vf2 −

2W m

(1)

The work done by the kinetic frictional force is given by W = ( f k cosθ )s (Equation 6.1), where fk is the magnitude of the kinetic frictional force and s is the magnitude of the skier’s displacement. Because the kinetic frictional force points opposite to the displacement of the skier, θ = 180°. According to Equation 4.8, the kinetic frictional force has a magnitude of f k = µk FN , where µ k is the coefficient of kinetic friction and FN is the magnitude of the normal force. Thus, the work can be expressed as W = ( f k cosθ )s = ( µ k FN cos180°)s Substituting this expression for W into Equation (1), we have that v0 = vf2 −

2 µ F ( cos180° ) s 2W = vf2 − k N m m

(2)

Since the skier is sliding on level ground, the magnitude of the normal force is equal to the weight mg of the skier (see Example 10 in Chapter 4), so FN = mg. Substituting this relation into Equation (2) gives v0 = vf2 −

2 µk m g ( cos180° ) s m

= vf2 − 2 µ k g ( cos180° ) s

Chapter 6 Problems

297

SOLUTION Since the skier comes to a halt, vf = 0 m/s. Therefore, the initial speed is 2 v0 = vf2 − 2 µk g ( cos180° ) s = ( 0 m/s ) − 2 ( 0.050 ) ( 9.80 m/s 2 ) ( cos180° ) ( 21 m ) = 4.5 m/s

____________________________________________________________________________________________

25.

SSM WWW REASONING When the satellite goes from the first to the second orbit, its kinetic energy changes. The net work that the external force must do to change the orbit can be found from the work-energy theorem:

1 2

1 2

W = KE f − KE 0 = mvf2 − mv02 .

The

speeds vf and v0 can be obtained from Equation 5.5 for the speed of a satellite in a circular orbit of radius r. Given the speeds, the work energy theorem can be used to obtain the work. SOLUTION According to Equation 5.5, v = GME r . Substituting into the work-energy theorem, we have 2 2 GM E GM E GM E m 1 1 1 2 2 2 2 1 1 1 W = 2 mvf − mv0 = 2 m vf − v0 = 2 m − = − 2 r r 2 f 0 rf r0 Therefore,

(

W=

(6.67 ×10

)

–11

2

2

N ⋅ m /kg )(5.98 ×10 2

24

kg)(6200 kg)

1 1 11 × − = 1.4 × 10 J 6 7 7.0 × 10 m 3.3 × 10 m ______________________________________________________________________________ 26. REASONING a. The magnitude s of the displacement that occurs while the snowmobile coasts to a halt is the distance that we seek. The work W done by the net external force F acting on the snowmobile during the coasting phase is given by W = (F cos θ) s (Equation 6.1), and we can use this expression to obtain s. To do so, we will need to evaluate W. This we will do with the aid of the work-energy theorem as given by W = 12 mvf2 − 12 mv02 (Equation 6.3), since we know the mass m and the final and initial velocities vf and v0, respectively. The net external force F during the coasting phase is just the horizontal force of kinetic friction (magnitude = fk), since the drive force is shut off and the vertical forces balance (there is no vertical acceleration). This frictional force has a magnitude of 205 N. This follows because, while the snowmobile is moving at a constant 5.50 m/s, it is not accelerating, and the drive force must be balancing the frictional force, which exists both before and after the drive force is shut off. Finally, to use W = (F cos θ) s (Equation 6.1) to determine s, we will

298 WORK AND ENERGY

need a value for the angle θ between the frictional force and the displacement. The frictional force points opposite to the displacement, so θ = 180°. b. By using only the work-energy theorem it is not possible to determine the time t during which the snowmobile coasts to a halt. To determine t it is necessary to use the equations of kinematics. We can use these equations, if we assume that the acceleration during the coasting phase is constant. The acceleration is determined by the force of friction (assumed constant) and the mass of the snowmobile, according to Newton’s second law. For example, we can use s = 12 ( v0 + vf ) t (Equation 2.7) to determine t once we know the magnitude of the displacement s, which is the distance in which the snowmobile coasts to a halt. SOLUTION a. According to Equation 6.3, the work energy theorem, is

W = 12 mvf2 − 12 mv02 As discussed in the REASONING, we know that W = (fk cos θ) s. Substituting this result into the work-energy theorem gives

( f k cosθ ) s = 12 mvf2 − 12 mv02

(1)

In Equation (1), fk = 205 N (see the REASONING), the angle θ between the frictional force and the displacement is θ = 180º (see the REASONING), the final speed is vf = 0 m/s (the snowmobile coasts to a halt), and the initial speed is given as v0 = 5.50 m/s. Solving Equation (1) for s gives s=

b.

m ( vf2 − v02 )

(136 kg ) ( 0 m/s ) − ( 5.50 m/s ) = 2 f k cos θ 2 ( 205 N ) cos180° 2

As explained in the REASONING, we can use s =

2

= 10.0 m

1 2

(v

0

+ vf ) t (Equation 2.7) to

determine the time t during which the snowmobile coasts to a halt. Solving this equation for t gives 2 (10.0 m ) 2s t= = = 3.64 s v0 + vf 5.50 m/s + 0 m/s

27. REASONING AND SOLUTION The net work done on the plane can be found from the work-energy theorem:

(

W = 12 mvf2 − 12 mv02 = 12 m vf2 − v02

)

Chapter 6 Problems

299

The tension in the guideline provides the centripetal force required to keep the plane moving in a circular path. Since the tension in the guideline becomes four times greater, Tf = 4 T0 or mvf2 rf

= 4

mv02 r0

Solving for vf2 gives 2 vf

=

4v02 rf r0

Substitution of this expression into the work-energy theorem gives

W=

4v 2 r

1 m 2

0 f

r0

− v02

=

1 mv 2 0 2

rf 4 r 0

− 1

Therefore, 14 m 1 2 2 W = (0.90 kg)(22 m/s) 4 − 1 = 5.4 ×10 J 2 16 m ______________________________________________________________________________

28. REASONING It is useful to divide this problem into two parts. The first part involves the skier moving on the snow. We can use the work-energy theorem to find her speed when she comes to the edge of the cliff. In the second part she leaves the snow and falls freely toward the ground. We can again employ the work-energy theorem to find her speed just before she lands. SOLUTION The drawing at the right shows the three forces that act on the skier as she glides on the snow. The forces are: her weight mg, the normal force FN, and the kinetic frictional force fk. Her displacement is labeled as s. The work-energy theorem, Equation 6.3, is W = 12 mvf2 − 12 mv02

FN

+y

fk s 65.0° 25.0°

mg

where W is the work done by the net external force that acts on the skier. The work done by each force is given by Equation 6.1, W = ( F cos θ ) s , so the work-energy theorem becomes

300 WORK AND ENERGY

( mg cos 65.0 ) s + ( f k cos 180 ) s + ( FN cos 90 ) s

= 12 mvf2 − 12 mv02

W

Since cos 90° = 0, the third term on the left side can be eliminated. The magnitude fk of the kinetic frictional force is given by Equation 4.8 as f k = µ k FN . The magnitude FN of the normal force can be determined by noting that the skier does not leave the surface of the slope, so ay = 0 m/s2. Thus, we have that ΣFy = 0, so FN − mg cos 25.0° = 0

or

FN = mg cos 25.0°

ΣFy

The magnitude of the kinetic frictional force becomes f k = µ k FN = µ k mg cos 25.0 . Substituting this result into the work-energy theorem, we find that

( mg cos 65.0 ) s + ( µk mg cos 25.0 ) ( cos 180 ) s

= 12 mvf2 − 12 mv02

W

Algebraically eliminating the mass m of the skier from every term, setting cos 180° = –1 and v0 = 0 m/s, and solving for the final speed vf gives vf = 2 gs ( cos 65.0 − µ k cos 25.0

)

= 2 ( 9.80 m/s 2 ) (10.4 m ) cos 65.0 − ( 0.200 ) cos 25.0 = 7.01 m/s The drawing at the right shows her displacement s during free fall. Note that the displacement is a vector that starts where she leaves the slope and ends where she touches the ground. The only force acting on her during the free fall is her weight mg. The work-energy theorem, Equation 6.3, is

3.50 m θ s mg

W = 12 mvf2 − 12 mv02 The work W is that done by her weight, so the work-energy theorem becomes

( mg cosθ ) s W

= 12 mvf2 − 12 mv02

In this expression θ is the angle between her weight (which points vertically downward) and her displacement. Note from the drawing that s cos θ = 3.50 m. Algebraically eliminating the mass m of the skier from every term in the equation above and solving for the final speed vf gives

Chapter 6 Problems

301

vf = v02 + 2 g ( s cos θ ) =

( 7.01 m/s )

2

+ 2 ( 9.80 m/s 2 ) ( 3.50 m ) = 10.9 m/s

______________________________________________________________________________ 29. REASONING The change in gravitational potential energy for both the adult and the child is ∆PE = mghf − mgh0, where we have used Equation 6.5. Therefore, ∆PE = mg(hf − h0). In this expression hf − h0 is the vertical height of the second floor above the first floor, and its value is not given. However, we know that it is the same for both staircases, a fact that will play the central role in our solution. SOLUTION Solving ∆PE = mg(hf − h0) for hf − h0, we obtain

hf − h0 =

( ∆PE )Adult mAdult g

and

hf − h0 =

( ∆PE )Child mChild g

Since hf − h0 is the same for the adult and the child, we have

( ∆PE )Adult ( ∆PE )Child =

mAdult g

mChild g

Solving this result for (∆PE)Child gives

( ∆PE )Child =

( ∆PE )Adult mChild ( 2.00 ×103 J ) (18.0 kg ) mAdult

=

81.0 kg

= 444 J

30. REASONING At a height h above the ground, the gravitational potential energy of each clown is given by PE = mgh (Equation 6.5). At a height hJ, Juggle’s potential energy is, therefore, PEJ = mJghJ, where mJ is Juggles’ mass. Similarly, Bangles’ potential energy can be expressed in terms of his mass mB and height hB: PEB = mBghB. We will use the fact that these two amounts of potential energy are equal (PEJ = PEB) to find Juggles’ mass mJ.

SOLUTION Setting the gravitational potential energies of both clowns equal, we solve for Juggles’ mass and obtain

mJ g hJ = mB g hB PE

J

PE

B

or

mJ =

mB hB hJ

=

( 86 kg )( 2.5 m ) = ( 3.3 m )

65 kg

302 WORK AND ENERGY

31.

SSM REASONING During each portion of the trip, the work done by the resistive force is given by Equation 6.1, W = ( F cos θ )s . Since the resistive force always points opposite to the displacement of the bicyclist, θ = 180° ; hence, on each part of the trip, W = ( F cos 180 °)s = −Fs . The work done by the resistive force during the round trip is the algebraic sum of the work done during each portion of the trip. SOLUTION a. The work done during the round trip is, therefore, Wtotal = W1 + W2 = −F1s1 − F2 s2

= −(3.0 N)(5.0 × 103 m) − (3.0 N)(5.0× 103 m) = –3.0 × 10 4 J b. Since the work done by the resistive force over the closed path is not zero, we can conclude that the resistive force is not a conservative force . ______________________________________________________________________________ 32. REASONING AND SOLUTION a. From the definition of work, W = Fs cos θ. For upward motion θ = 180° so W = – mgs = – (71.1 N)(2.13 m – 1.52 m) = –43 J

b. The change in potential energy is ∆PE = mghf – mgh0 = (71.1 N)(2.13 m – 1.52 m) = +43 J ______________________________________________________________________________

33. REASONING The work done by the weight of the basketball is given by Equation 6.1 as W = ( F cos θ ) s , where F = mg is the magnitude of the weight, θ is the angle between the weight and the displacement, and s is the magnitude of the displacement. The drawing shows that the weight and displacement are parallel, so that θ = 0°. The potential energy of the mg basketball is given by Equation 6.5 as PE = mgh, where h is the height of the ball above the ground. SOLUTION a. The work done by the weight of the basketball is

W = ( F cos θ ) s = mg (cos 0°)(h0 − hf) = (0.60 kg)(9.80 m/s )(6.1 m − 1.5 m) = 27 J 2

b. The potential energy of the ball, relative to the ground, when it is released is

s

303

Chapter 6 Problems

2

PE0 = mgh0 = (0.60 kg)(9.80 m/s )(6.1 m) = 36 J

(6.5)

c. The potential energy of the ball, relative to the ground, when it is caught is PEf = mghf = (0.60 kg)(9.80 m/s2)(1.5 m) = 8.8 J

(6.5)

d. The change in the ball’s gravitational potential energy is ∆PE = PEf − PE0 = 8.8 J – 36 J = −27 J

We see that the change in the gravitational potential energy is equal to –27 J = −W , where W is the work done by the weight of the ball (see part a). ______________________________________________________________________________ 34. REASONING The gravitational force on Rocket Man is conservative, but the force generated by the propulsion unit is nonconservative. Therefore, the work WP done by the propulsive force is equal to the net work done by all external nonconservative forces: WP = Wnc. We will use Wnc = Ef − E0 (Equation 6.8) to calculate Wnc in terms of Rocket Man’s initial mechanical energy and final mechanical energy. Because Rocket Man’s speed and height increase after he leaves the ground, both his kinetic energy KE = 12 mv 2 (Equation 6.2) and potential energy PE = mgh (Equation 6.5) increase. Therefore, his final mechanical energy Ef = KEf + PEf is greater than his initial mechanical energy E0 = KE0 + PE0, and the work WP = Wnc = Ef − E0 done by the propulsive force is positive.

SOLUTION We make the simplifying assumption that Rocket Man’s initial height is h0 = 0 m. Thus, his initial potential energy PE 0 = mgh0 is zero, and his initial kinetic energy KE 0 = 12 mv02 is also zero (because he starts from rest). This means he has no initial

mechanical energy, and we see that the work done by the propulsive force is equal to his final mechanical energy: WP = Ef − 0 = Ef . Expressing Rocket Man’s final mechanical energy Ef in terms of his final kinetic energy and his final potential energy, we obtain the work done by the propulsive force:

WP = 12 mvf2 + mghf

(

)

= 12 (136 kg )( 5.0 m/s ) + (136 kg ) 9.80 m/s2 (16 m ) = +2.3 ×104 J 2

304 WORK AND ENERGY

35.

SSM REASONING AND SOLUTION a. The work done by non-conservative forces is given by Equation 6.7b as Wnc = ∆KE + ∆PE

∆PE = Wnc – ∆KE

so

Now ∆KE =

1 2

mvf2 −

1 2

mv02 =

1 2

(55.0 kg)[(6.00 m/s) 2 − (1.80 m/s)2 ] = 901 J

and ∆PE = 80.0 J – 265 J – 901 J = −1086 J

b. ∆PE = mg (h – h0) so h – h0 =

∆ PE –1086 J = = –2.01 m mg (55.0 kg ) 9.80 m/s 2

(

)

This answer is negative, so that the skater’s vertical position has changed by 2.01 m and the skater is below the starting point . ______________________________________________________________________________ 36. REASONING The girl’s gravitational potential energy PE = mgh (Equation 6.5) increases as she rises. This increase comes at the expense of her kinetic energy KE = 12 mv 2 (Equation 6.2), which decreases as she rises. Because air resistance is negligible, all of the kinetic energy she loses is transformed into potential energy, so that her total mechanical energy E = KE + PE remains constant. We will use this principle to determine the distance she rises during this interval. SOLUTION The conservation principle gives KE f + PE f = KE 0 + PE 0 Ef

or

PE f − PE 0 = KE 0 − KE f

or

mg ( hf − h0 ) = KE 0 − KE f

E0

Solving this expression for hf − h0 , which is the distance she rises, we obtain

hf − h0 =

37.

KE0 − KEf mg

=

440 J − 210 J

( 35 kg ) ( 9.80 m/s2 )

= 0.67 m

SSM REASONING No forces other than gravity act on the rock since air resistance is being ignored. Thus, the net work done by nonconservative forces is zero, Wnc = 0 J.

Chapter 6 Problems

305

Consequently, the principle of conservation of mechanical energy holds, so the total mechanical energy remains constant as the rock falls. If we take h = 0 m at ground level, the gravitational potential energy at any height h is, according to Equation 6.5, PE = mgh . The kinetic energy of the rock is given by 2

1

Equation 6.2: KE = mv 2 . In order to use Equation 6.2, we must have a value for v at 2 2

each desired height h. The quantity v can be found from Equation 2.9 with v0 = 0 m/s, since the rock is released from rest. At a height h, the rock has fallen through a distance 2 (20.0 m) – h , and according to Equation 2.9, v = 2ay = 2a[(20.0 m) – h]. Therefore, the kinetic energy at any height h is given by KE = ma[(20.0 m)– h ]. The total energy at any height h is the sum of the kinetic energy and potential energy at the particular height. SOLUTION The calculations are performed below for h = 10.0 m . The table that follows also shows the results for h = 20.0 m and h = 0 m. 2

PE = mgh = (2.00 kg)(9.80 m/s )(10.0 m) = 196 J KE = ma[ (20.0 m) – h ] = (2.00 kg)(9.80 m/s 2 )(20.0 m – 10.0 m) = 196 J E = KE + PE = 196 J + 196 J =

h (m) 20.0 10.0 0

KE (J) 0 196 392

PE (J) 392 196 0

392 J

E (J) 392 392 392

Note that in each case, the value of E is the same, because mechanical energy is conserved. ______________________________________________________________________________ 38. REASONING a. and b. Since there is no friction, there are only two forces acting on each box, the gravitational force (its weight) and the normal force. The normal force, being perpendicular to the displacement of a box, does no work. The gravitational force does work. However, it is a conservative force, so the principle of conservation of mechanical energy applies to the motion of each box, and we can use it to determine the speed in each case. This principle reveals that the final speed depends only on the initial speed, and the initial and final heights. The final speed does not depend on the mass of the box or on the steepness of the slope. Thus, both boxes reach B with the same speed. c. The kinetic energy is given by the expression KE = 12 mv 2 (Equation 6.2). Both boxes have the same speed v when they reach B. However, the heavier box has the greater mass. Therefore, the heavier box has the greater kinetic energy.

306 WORK AND ENERGY

SOLUTION a. The conservation of mechanical energy states that 1 2

2

mvB + mghB =

1 mv 2 A 2

+ mghA

(6.9b)

Solving for the speed vB at B gives

vB =

vA + 2 g ( hA − hB ) = 2

( 0 m /s )2 + 2 ( 9.80 m / s 2 ) ( 4.5 m − 1.5 m )

=

7.7 m /s

b. The speed of the heavier box is the same as that of the lighter box: 7.7 m /s . c. The ratio of the kinetic energies is

( KEB )heavier box ( KEB )lighter box

( = (

) )

1 mv 2 B 2 heavier box 2 1 mv B 2 lighter box

=

mheavier box mlighter box

=

44 kg = 11 kg

4.0

______________________________________________________________________________ 39. REASONING We can find the landing speed vf from the final kinetic energy KEf, since KEf = 12 mvf2 . Furthermore, we are ignoring air resistance, so the conservation of mechanical energy applies. This principle states that the mechanical energy with which the pebble is initially launched is also the mechanical energy with which it finally strikes the ground. Mechanical energy is kinetic energy plus gravitational potential energy. With respect to potential energy we will use ground level as the zero level for measuring heights. The pebble initially has kinetic and potential energies. When it strikes the ground, however, the pebble has only kinetic energy, its potential energy having been converted into kinetic energy. It does not matter how the pebble is launched, because only the vertical height determines the gravitational potential energy. Thus, in each of the three parts of the problem the same amount of potential energy is converted into kinetic energy. As a result, the speed that we will calculate from the final kinetic energy will be the same in parts (a), (b), and (c). SOLUTION a. Applying the conservation of mechanical energy in the form of Equation 6.9b, we have 1 2

mvf2 + mghf =

Final mechanical energy at ground level

1 2

mv02 + mgh0

Initial mechanical energy at top of building

Solving for the final speed gives vf = v02 + 2 g ( h0 − hf ) =

(14.0 m/s )

2

+ 2 ( 9.80 m/s 2 ) ( 31.0 m ) − ( 0 m ) = 28.3 m/s

Chapter 6 Problems

307

b. The calculation and answer are the same as in part (a). c. The calculation and answer are the same as in part (a).

40. REASONING The distance h in the drawing in the text is the difference between the skateboarder’s final and initial heights (measured, for example, with respect to the ground), or h = hf − h0. The difference in the heights can be determined by using the conservation of mechanical energy. This conservation law is applicable because nonconservative forces are negligible, so the work done by them is zero (Wnc = 0 J). Thus, the skateboarder’s final total mechanical energy Ef is equal to his initial total mechanical energy E0: 1 mv 2 f 2

+ mghf =

1 2

mv02 + mgh0

Ef

(6.9b)

E0

Solving Equation 6.9b for hf − h0, we find that hf − h0 =

1 v2 2 0

− 12 vf2 g

h

SOLUTION Using the fact that v0 = 5.4 m/s and vf = 0 m/s (since the skateboarder comes to a momentary rest), the distance h is

h=

1 v2 2 0

− 12 vf2

1 ( 5.4 2

2

m/s ) − 12 ( 0 m/s )

2

= = 1.5 m g 9.80 m/s 2 ______________________________________________________________________________ 41. REASONING Since air resistance is being neglected, the only force that acts on the golf ball is the conservative gravitational force (its weight). Since the maximum height of the trajectory and the initial speed of the ball are known, the conservation of mechanical energy can be used to find the kinetic energy of the ball at the top of the highest point. The conservation of mechanical energy can also be used to find the speed of the ball when it is 8.0 m below its highest point. SOLUTION a. The conservation of mechanical energy, Equation 6.9b, states that 1 mv 2 f 2

2

+ mghf = 12 mv0 + mgh0

KEf Solving this equation for the final kinetic energy, KEf, yields

308 WORK AND ENERGY

KE f = 12 mv0 + mg ( h0 − hf 2

=

1 2

)

( 0.0470 kg )( 52.0 m / s )2 + ( 0.0470 kg ) ( 9.80 m / s 2 ) ( 0 m − 24.6 m ) =

52.2 J

b. The conservation of mechanical energy, Equation 6.9b, states that 1 mv 2 f 2

+ mghf =

Ef

1 mv 2 0 2

+ mgh0

Ef

The mass m can be eliminated algebraically from this equation, since it appears as a factor in every term. Solving for vf and noting that the final height is hf = 24.6 m – 8.0 m = 16.6 m, we have that

vf =

v0 + 2 g ( h0 − hf ) = 2

( 52.0 m / s )2 + 2 ( 9.80 m / s 2 ) ( 0 m − 16.6 m ) =

48.8 m / s

______________________________________________________________________________ 42. REASONING If air resistance is ignored, the only nonconservative force that acts on the person is the tension in the rope. However, since the tension always points perpendicular to the circular path of the motion, it does no work, and the principle of conservation of mechanical energy applies. SOLUTION Let the initial position of the person be at the top of the cliff, a distance h0 above the water. Then, applying the conservation principle to path 2, we have 1 mvf2 2

Ef

+ mghf =

( at water )

1 mv02 2

+ mgh0

(6.9b)

E0 ( at release point )

Solving this relation for the speed at the release point, we have

v0 = vf2 + 2 g ( hf − h0 ) But vf = 13.0 m/s, since the speed of entry is the same for either path. Therefore, 2 v0 = vf2 + 2 g ( hf − h0 ) = (13.0 m/s ) + 2 ( 9.80 m/s 2 ) ( −5.20 m ) = 8.2 m/s

_____________________________________________________________________________________________

Chapter 6 Problems

43. REASONING AND SOLUTION Since friction and air resistance are negligible, mechanical energy is conserved. Thus, the ball will have the same speed at the bottom of its swing whether it is moving toward or away from the crane. As the drawing shows, when the cable swings to an angle of 20.0°, the ball will rise a distance of

309

20.0º L L − L cos 20.0º

L cos 20.0º h0 = 0 m

h = L (1 – cos 20.0°) where L is the length of the cable. Applying the conservation of energy with h0 = 0 m gives 1 2

mv02 = 12 mvf2 + mgh

Solving for vf gives vf = v02 − 2 gL (1 − cos 20.0° ) =

( 5.00 m/s )

2

− 2 ( 9.80 m/s 2 ) (12.0 m )(1 − cos 20.0° ) = 3.29 m/s

______________________________________________________________________________ 44. REASONING To find the maximum height H above the end of the track we will analyze the projectile motion of the skateboarder after she leaves the track. For this analysis we will use the principle of conservation of mechanical energy, which applies because friction and air resistance are being ignored. In applying this principle to the projectile motion, however, we will need to know the speed of the skateboarder when she leaves the track. Therefore, we will begin by determining this speed, also using the conservation principle in the process. Our approach, then, uses the conservation principle twice.

SOLUTION Applying the conservation of mechanical energy in the form of Equation 6.9b, we have 1 mvf2 + mghf = 12 mv02 + mgh0 2 Final mechanical energy at end of track

Initial mechanical energy on flat part of track

We designate the flat portion of the track as having a height h0 = 0 m and note from the drawing that its end is at a height of hf = 0.40 m above the ground. Solving for the final speed at the end of the track gives vf = v02 + 2 g ( h0 − hf ) =

( 5.4 m/s )

2

+ 2 ( 9.80 m/s 2 ) ( 0 m ) − ( 0.40 m ) = 4.6 m/s

310 WORK AND ENERGY

This speed now becomes the initial speed v0 = 4.6 m/s for the next application of the conservation principle. At the maximum height of her trajectory she is traveling horizontally with a speed vf that equals the horizontal component of her launch velocity. Thus, for the next application of the conservation principle vf = (4.6 m/s) cos 48º. Applying the conservation of mechanical energy again, we have 1 2

mvf2 + mghf

=

Final mechanical energy at maximum height of trajectory

1 2

mv02 + mgh0

Initial mechanical energy upon leaving the track

Recognizing that h0 = 0.40 m and hf = 0.40 m + H and solving for H give 1 2

H=

45.

mvf2 + mg ( 0.40 m ) + H = 12 mv02 + mg ( 0.40 m )

v02 − vf2 2g

=

( 4.6 m/s )

2

− ( 4.6 m/s ) cos 48°

2 ( 9.80 m/s 2 )

2

= 0.60 m

SSM REASONING Friction and air resistance are being ignored. The normal force from the slide is perpendicular to the motion, so it does no work. Thus, no net work is done by nonconservative forces, and the principle of conservation of mechanical energy applies. SOLUTION Applying the principle of conservation of mechanical energy to the swimmer at the top and the bottom of the slide, we have 1 1 2 2 mvf + mghf = mv0 + mgh0 2 2 Ef E0

If we let h be the height of the bottom of the slide above the water, hf = h , and h0 = H . Since the swimmer starts from rest, v0 = 0 m/s, and the above expression becomes 1 2 v 2 f

+ gh = gH

Solving for H, we obtain v2 H = h+ f 2g Before we can calculate H, we must find vf and h. Since the velocity in the horizontal direction is constant, ∆x 5.00 m vf = = = 10.0 m/s ∆t 0.500 s

Chapter 6 Problems

311

The vertical displacement of the swimmer after leaving the slide is, from Equation 3.5b (with down being negative), 1

2

2

1

2

y = 2 ay t = 2 (–9.80 m/s )(0.500 s) = −1.23 m

Therefore, h = 1.23 m. Using these values of vf and h in the above expression for H, we find v2 (10.0 m/s)2 H = h + f = 1.23 m + = 6.33 m 2g 2(9.80 m/s2 ) ______________________________________________________________________________

46. REASONING We are neglecting air resistance and friction, which eliminates two nonconservative forces from consideration. Another nonconservative force acting on the semitrailer is the normal force exerted by the ramp. But this force does no work because it is perpendicular to the ramp and the semitrailer’s velocity. Hence, the semi-trailer’s total mechanical energy is conserved, and we will apply the energy conservation principle to determine the initial velocity v0 of the truck. SOLUTION The truck coasts to a stop, so we know that vf = 0 m/s. Therefore, the energy conservation principle gives

( 12 mvf2 + mghf ) = ( 12 mv02 + mgh0 ) Ef

or m g ( hf − h0 ) =

1 2

m v02 or g ( hf − h0 ) = 12 v02 (1)

E0

Note that hf − h0 is the vertical height up which the truck climbs on the ramp, as the drawing indicates. The drawing also indicates that hf − h0 is related to the distance d the truck coasts along the ramp and the ramp’s angle θ of inclination. The side hf − h0 is opposite the angle θ, and d is the hypotenuse, so we solve sin θ =

d hf−h0

θ

hf − h0

(Equation 1.1) to find that hf − h0 = d sin θ . d Substituting d sin θ for hf − h0 in Equation (1), we obtain gd sin θ = 12 v02

or

v02 = 2 gd sin θ

or

v0 = 2 gd sin θ

When the semitrailer enters the ramp, therefore, its speed is

(

)

v0 = 2 9.80 m/s 2 (154 m ) sin14.0 = 27.0 m/s

312 WORK AND ENERGY

47. REASONING AND SOLUTION If air resistance is ignored, the only nonconservative force that acts on the skier is the normal force exerted on the skier by the snow. Since this force is always perpendicular to the direction of the displacement, the work done by the normal force is zero. We can conclude, therefore, that mechanical energy is conserved. 1 mv02 2

1 2

+ mgh0 = mvf2 + mghf

Since the skier starts from rest v0 = 0 m/s. Let hf define the zero level for heights, then the final gravitational potential energy is zero. This gives 1 2

mgh0 = mvf2

(1)

At the crest of the second hill, the two forces that act on the skier are the normal force and the weight of the skier. The resultant of these two forces provides the necessary centripetal force to keep the skier moving along the circular arc of the hill. When the skier just loses contact with the snow, the normal force is zero and the weight of the skier must provide the necessary centripetal force.

mg =

mvf2

so that

r

vf2 = gr

F

N

mg

(2)

Substituting this expression for vf2 into Equation (1) gives 1 2

mgh0 = mgr

Solving for h0 gives

r 36 m = = 18 m 2 2 ______________________________________________________________________________ h0 =

48. REASONING The principle of conservation of mechanical energy provides the solution to both parts of this problem. In applies in both cases because the tracks are frictionless (and we assume that air resistance is negligible). When the block on the longer track reaches its maximum height, its final total mechanical energy consists only of gravitational potential energy, because it comes to a momentary halt at this point. Thus, its final speed is zero, and, as a result, its kinetic energy is also zero. In contrast, when the block on the shorter track reaches the top of its trajectory after leaving the track, its final total mechanical energy consists of part kinetic and part potential energy. At the top of the trajectory the block is still moving horizontally, with a velocity that equals the horizontal velocity component that it had when it left the track (assuming that air

Chapter 6 Problems

313

resistance is negligible). Thus, at the top of the trajectory the block has some kinetic energy. We can expect that the height H is greater than the height H1 + H2. To see why, note that both blocks have the same initial kinetic energy, since each has the same initial speed. Moreover, the total mechanical energy is conserved. Therefore, the initial kinetic energy is converted to potential energy as each block moves upward. On the longer track, the final total mechanical energy is all potential energy, whereas on the shorter track it is only part potential energy, as discussed previously. Since gravitational potential energy is proportional to height, the final height is greater for the block with the greater potential energy, or the one on the longer track. SOLUTION The two inclines in the problem are as follows:

H2 H

v0

v0

H1 θ

θ

Shorter track

Longer track

a. Longer track: Applying the principle of conservation of mechanical energy for the longer track gives 2 2 1 1 = 2 mvf + mghf 2 mv0 + mgh0 Final total mechanical energy

Initial total mechanical energy

Recognizing that vf = 0 m/s, hf = H, and h0 = 0 m, we find that 1 2

m ( 0 m/s ) + mgH = 12 mv02 + mg ( 0 m ) 2

H=

v02 2g

=

( 7.00 m/s )

2

2 ( 9.80 m/s 2 )

= 2.50 m

b. Shorter track: We apply the conservation principle to the motion of the block after it leaves the track, in order to calculate the final height at the top of the trajectory, where hf = H1 + H2 and the final speed is vf. For the initial position at the end of the track we have h0 = H1 and an initial speed of v0T. Thus, it follows that 1 2

mvf2 + mg ( H1 + H 2 ) =

Final total mechanical energy

1 2

2 mv0T + mgH1

Initial total mechanical energy

314 WORK AND ENERGY

Solving for H1 + H2 gives

H1 + H 2 =

2 v0T + 2 gH1 − vf2

(1)

2g

At the top of its trajectory the block is moving horizontally with a velocity that equals the horizontal component of the velocity with which it left the track, that is, vf = v0T cos θ. Thus, a value for v0T is needed and can be obtained by applying the conservation principle to the motion of the block that takes place on the track. For the motion on the track the final speed is the speed v0T at which the block leaves the track, and the final height is hf = H1 = 1.25 m. The initial speed is the given value of v0 = 7.00 m/s, and the initial height is h0 = 0 m. The conservation principle reveals that 1 2

2 mv0T + mgH1

=

Final total mechanical energy

v0T = v02 − 2 gH1 =

( 7.00 m/s )

2

1 2

mv02 + mg ( 0 m )

Initial total mechanical energy

− 2 ( 9.80 m/s 2 ) (1.25 m ) = 4.95 m/s

For use in Equation (1) we find, then, that the speed with which the block leaves the track is v0T = 4.95 m/s and the final speed at the top of the trajectory is vf = v0T cos θ = (4.95 m/s) cos θ. In addition, hf = H1 + H2 at the top of the trajectory, and h0 = H1 = 1.25 m for the initial position at the end of the track. Thus, Equation (1) reveals that 2 v0T + 2 gH1 − vf2 H1 + H 2 = 2g =

( 4.95 m/s )

2

+ 2 ( 9.80 m/s 2 ) (1.25 m ) − ( 4.95 m/s ) cos 50.0° 2 ( 9.80 m/s 2 )

2

= 1.98 m

As expected, H1 + H2 is less than H, and the block rises to a greater height on the longer track. ______________________________________________________________________________ 49. REASONING AND SOLUTION When the car is at the top of the track the centripetal force consists of the full weight of the car. 2

mv /r = mg

315

Chapter 6 Problems

Applying the conservation of energy between the bottom and the top of the track gives 2

(1/2)mv + mg(2r) = (1/2)mv0

2

Using both of the above equations 2

v0 = 5gr so 2

2

2

r = v0 /(5g) = (4.0 m/s) /[5(9.8 m/s )] = 0.33 m ______________________________________________________________________________ 50. REASONING AND SOLUTION If air resistance is ignored, the only nonconservative force that acts on the person is the normal force exerted on the person by the surface. Since this force is always perpendicular to the direction of the displacement, the work done by the normal force is zero. We can conclude, therefore, that mechanical energy is conserved. 1 mv02 2

1 2

+ mgh0 = mvf2 + mghf

(1)

where the final state pertains to the position where the person leaves the surface. Since the person starts from rest v0 = 0 m/s. Since the radius of the surface is r,

r

θ

h0 = r, and hf = r cos θf where θf is the angle at which the person leaves the surface. Equation (1) becomes mgr = mvf2 + mg ( r cos θ f ) 1 2

r cos θ

r

(2) F

N

In general, as the person slides down the surface, the two forces that act on him are the normal force FN and the weight mg. The centripetal force required to keep the person moving in the circular path is the resultant of FN and the radial component of the weight, mg cos θ.

mg cos θ

When the person leaves the surface, the normal force is zero, and the radial component of the weight provides the centripetal force.

mg cos θ f =

mvf2 r

⇒

vf2 = gr cos θf

(3)

Substituting this expression for vf2 into Equation (2) gives mgr = 12 mg ( r cos θf ) + mg ( r cos θf )

θ

θ θ

mg

316 WORK AND ENERGY

Solving for θf gives 2 ______________________________________________________________________________

θf = cos −1 = 48° 3

51. REASONING The work Wnc done by the nonconservative force exerted on the surfer by the wave is given by Equation 6.6, which states that this work is equal to the surfer’s change in kinetic energy, 12 mvf2 − 12 mv02 , plus the change in her potential energy, mghf − mgh0 :

Wnc =

( 12 mvf2 − 12 mv02 ) + ( mghf − mgh0 )

(6.6)

All the terms in this equation are known, so we can evaluate the work.

SOLUTION We note that the difference between her final and initial heights is hf − h0 = −2.7 m, negative because her initial height h0 is greater than her final height hf. Thus, the work done by the nonconservative force is:

(

)

Wnc = 12 m vf2 − v02 + mg ( hf − h0 ) =

1 2

( 59 kg ) ( 9.5 m/s )2 − (1.4 m/s )2 + ( 59 kg ) ( 9.80 m/s2 ) ( −2.7 m ) = 1.0 ×103 J

______________________________________________________________________________ 52. REASONING As the firefighter slides down the pole from a height h0 to the ground (hf = 0 m), his potential energy decreases to zero. At the same time, his kinetic energy increases as he speeds up from rest (v0 = 0 m/s) to a final speed vf,. However, the upward nonconservative force of kinetic friction fk, acting over a downward displacement h0, does a

(

)

negative amount of work on him: Wnc = f k cos180 h0 = − f k h0 (Equation 6.1). This work decreases his total mechanical energy E. Applying the work-energy theorem (Equation 6.8), with hf = 0 m and v0 = 0 m/s, we obtain

Wnc = − f k h0 = Wnc

( 12 mvf2 + mghf ) − ( 12 mv02 + mgh0 )

( 12 mvf2 + 0 J ) − ( 0 J + mgh0 ) = 12 mvf2 − mgh0 Ef

E0

(6.8)

(1)

Chapter 6 Problems

317

SOLUTION Solving Equation (1) for the height h0 gives mgh0 − f k h0 = 12 mvf2

h0 ( mg − f k ) = 12 mvf2

or

or

h0 =

mvf2

2 ( mg − f k )

The distance h0 that the firefighter slides down the pole is, therefore, 2 93 kg )( 3.4 m/s ) ( h0 = = 2 ( 93 kg ) ( 9.80 m/s 2 ) − 810 N

53.

5.3 m

SSM REASONING AND SOLUTION The work-energy theorem can be used to determine the change in kinetic energy of the car according to Equation 6.8:

Wnc =

(

1 2

mvf2 + mghf

)−(

1 2

mv02 + mgh0

Ef

)

E0

The nonconservative forces are the forces of friction and the force due to the chain mechanism. Thus, we can rewrite Equation 6.8 as

Wfriction + Wchain = ( KE f + mghf ) − ( KE 0 + mgh0 ) We will measure the heights from ground level. Solving for the change in kinetic energy of the car, we have KE f − KE 0 = Wfriction + Wchain − mghf + mgh0

= − 2.00×104 J + 3.00 × 104 J − (375 kg)(9.80 m/s 2 )(20.0 m) + (375 kg)(9.80 m/s 2 )(5.00 m) = –4.51×104 J ______________________________________________________________________________ 54. REASONING We will use the work-energy theorem Wnc = Ef − E0 (Equation 6.8) to find the speed of the student. Wnc is the work done by the kinetic frictional force and is negative because the force is directed opposite to the displacement of the student. SOLUTION The work-energy theorem states that 2

Wnc = 12 mvf + mghf − Ef

(

1 mv 2 0 2

+ mgh0 E0

)

(6.8)

318 WORK AND ENERGY

Solving for the final speed gives vf =

2Wnc m

+ v02 − 2 g ( hf − h0 )

(

3

2 −6.50 × 10 J

) + ( 0 m /s )

(

)

( −11.8 m ) = 8.6 m / s 83.0 kg ______________________________________________________________________________ =

55.

2

− 2 9.80 m /s

2

SSM REASONING The work-energy theorem can be used to determine the net work done on the car by the nonconservative forces of friction and air resistance. The workenergy theorem is, according to Equation 6.8,

Wnc =

(

1 mv 2 f 2

) (

+ mghf −

1 mv 2 0 2

+ mgh0

)

The nonconservative forces are the force of friction, the force of air resistance, and the force provided by the engine. Thus, we can rewrite Equation 6.8 as

Wfriction + Wair + Wengine =

(

1 mv 2 f 2

) (

+ mghf −

1 mv 2 0 2

+ mgh0

)

This expression can be solved for Wfriction + Wair .

SOLUTION We will measure the heights from sea level, where h0 = 0 m. Since the car starts from rest, v0 = 0 m/s. Thus, we have

Wfriction + Wair = m

(

1 v2 2 f

)

+ ghf − Wengine

3 2 2 2 6 = (1.50 × 10 kg) 12 (27.0 m/s) + (9.80 m/s )(2.00 × 10 m) – (4.70 × 10 J) 6

= –1.21 × 10 J ______________________________________________________________________________ 56. REASONING According to Equation 6.6, the work Wnc done by nonconservative forces, such as kinetic friction and air resistance, is equal to the object’s change in kinetic energy, 1 mv 2 − 1 mv 2 plus the change in its potential energy, mgh − mgh : f 0 f 0 2 2

Wnc =

( 12 mvf2 − 12 mv02 ) + ( mghf − mgh0 )

This relation will be used in both parts (a) and (b) of the problem.

(6.6)

Chapter 6 Problems

319

SOLUTION a. Since nonconservative forces are absent, the work done by them is zero, so Wnc = 0 J. In this case, we can algebraically rearrange Equation 6.6 to show that the initial total mechanical energy (kinetic energy plus potential energy) E0 at the top is equal to the final total mechanical energy Ef at the bottom: 1 mv 2 f 2

+ mghf =

Ef

1 2

mv02 + mgh0 E0

Solving for the final speed vf of the skeleton and rider gives

vf = v02 + 2 g ( h0 − hf ) The initial speed of the rider is v0 = 0 m/s, and the drop in height is h0 − hf = 104 m , so the final speed at the bottom of the run is 2 vf = v02 + 2 g ( h0 − hf ) = ( 0 m/s ) + 2 ( 9.80 m/s 2 ) (104 m ) = 45.1 m/s

b. The work Wnc done by the nonconservative forces follows directly from Equation 6.6:

(

)

Wnc = 12 m vf2 − v02 + mg ( hf − h0 ) =

1 2

( 86.0 kg ) ( 35.8 m/s )2 − ( 0 m/s )2 + (86.0 kg ) ( 9.80 m/s 2 ) ( −104 m ) =

−3.25 ×104 J

Note that the difference in heights, hf − h0 = −104 m , is a negative number because the final height hf is less than the initial height h0. ______________________________________________________________________________ 57. REASONING a. Because the kinetic frictional force, a nonconservative force, is present, it does negative work on the skier. The work-energy theorem, in the form of Equation 6.6, may be used to find the work done by this force. b. Once the work done by the kinetic frictional force is known, the magnitude of the kinetic frictional force can be determined by using the definition of work, Equation 6.1, since the magnitude of the skier’s displacement is known.

320 WORK AND ENERGY

SOLUTION a. The work Wnc done by the kinetic frictional force is related to the object’s kinetic and potential energies by Equation 6.6: Wnc =

(

1 2

) + ( mgh

mvf2 − 12 mv02

− mgh0 )

f

s

The initial height of the skier at the bottom of the hill is h0 = 0 m, and the final height is hf = s sin 25° (see the drawing). Thus, the work is Wnc =

(

=

1 2

1 mv 2 f 2

2

− 12 mv0

25°

) + mg ( s sin 25° − h ) 0

( 63 kg )( 4.4 m / s )2 − 12 ( 63 kg )( 6.6 m / s )2

(

+ ( 63 kg ) 9.80 m / s

2

) (1.9 m ) sin 25° − 0 m =

−270 J

b. The work done by the kinetic frictional force is, according to Equation 6.1, Wnc = ( f k cos 180° ) s , where fk is the magnitude of the kinetic frictional force, and s is the magnitude of the skier’s displacement. The displacement of the skier is up the hill and the kinetic frictional force is directed down the hill, so the angle between the two vectors is θ = 180° and cos θ = –1. Solving the equation above for fk, we have Wnc −270 J = = 140 N − s −1.9 m ______________________________________________________________________________ fk =

58. REASONING AND SOLUTION Equation 6.8, we have

Wnc =

(

According to the work-energy theorem as given in

1 mvf2 2

+ mghf

)−(

1 mv02 2

+ mgh0

)

The metal piece starts at rest and is at rest just as it barely strikes the bell, so that

(

2

)

vf = v0 = 0 m/s. In addition, hf = h and h0 = 0 m, while Wnc = 0.25 12 M v , where M and v are the mass and speed of the hammer. Thus, the work-energy theorem becomes

0.25

(

1 M v2 2

Solving for the speed of the hammer, we find

) = mgh

hf

Chapter 6 Problems

321

2mgh 2(0.400 kg)(9.80 m/s2 )(5.00 m) = = 4.17 m/s 0.25M 0.25 (9.00 kg) ______________________________________________________________________________ v =

59. REASONING We can determine the force exerted on the diver by the water if we first find the work done by the force. This is because the work can be expressed using Equation 6.1 as Wnc = (F cos θ)s, where F is the magnitude of the force from the water and s is the magnitude of the displacement in the water. Since the force that the water exerts is nonconservative, we have included the subscript “nc” in labeling the work. To calculate the work we will use the work-energy theorem in the form of Equation 6.8: Wnc = ( 12 mvf2 + mghf ) − ( 12 mv02 + mgh0 ) . In this theorem Wnc is the work done by the net nonconservative force, which, in this case, is just the force exerted by the water.

SOLUTION We write Equation 6.8 as follows:

Wnc =

(

1 2

mvf2 + mghf ) −

Final mechanical energy

(

1 2

mv02 + mgh0 ) = mghf − mgh0

(1)

Initial mechanical energy

where we have used the fact that the diver is at rest initially and finally, so v0 = vf = 0 m/s. According to Equation 6.1 the work done on the diver by the force of the water is

Wnc = (F cos θ)s = (F cos 180º)s = −Fs The angle θ between the force of the water and the displacement is 180º, because the force opposes the motion. Substituting this result into Equation (1) gives − Fs = mghf − mgh0

(2)

Identifying the final position under the water as hf = 0 m und using upward as the positive direction, we know that the initial position on the tower must be h0 = 3.00 m + 1.10 m = 4.10 m. Thus, solving Equation (2) for F, we find that

F=

mg ( h0 − hf )

=

( 67.0 kg ) ( 9.80 m/s 2 ) ( 4.10 m ) − ( 0 m )

= 2450 N s 1.10 m ______________________________________________________________________________ 60. REASONING AND SOLUTION The force exerted by the bat on the ball is the only nonconservative force acting. The work due to this force is Wnc = mvf – mv0 + mg ( hf – h0 ) 1

2

2

1

2

2

322 WORK AND ENERGY

Taking h0 = 0 m at the level of the bat, v0 = 40.0 m/s just before the bat strikes the ball and vf to be the speed of the ball at hf = 25.0 m, we have

vf =

2Wnc m

2

+ v0 – 2 ghf

2(70.0 J) 2 2 + ( 40.0 m/s ) –2(9.80 m/s )(25.0 m) = 45.9 m/s 0.140 kg ______________________________________________________________________________ =

61.

SSM WWW REASONING After the wheels lock, the only nonconservative force acting on the truck is friction. The work done by this conservative force can be determined from the work-energy theorem. According to Equation 6.8,

Wnc = Ef − E0 =

(

1 mv 2 f 2

) (

+ mghf −

1 mv 2 0 2

+ mgh0

)

(1)

where Wnc is the work done by friction. According to Equation 6.1, W = ( F cos θ )s ; since the force of kinetic friction points opposite to the displacement, θ = 180° . According to Equation 4.8, the kinetic frictional force has a magnitude of f k = µk FN , where µ k is the coefficient of kinetic friction and FN is the magnitude of the normal force. Thus, Wnc = ( f k cosθ ) s = µ k FN (cos 180°) s = − µ k FN s Since the truck is sliding down an incline, we refer to the free-body diagram in order to determine the magnitude of the normal force FN . The free-body diagram at the right shows the three forces that act on the truck. They are the normal force FN , the force of kinetic friction fk , and the weight mg . The weight has been resolved into its components, and these vectors are shown as dashed arrows. From the free body diagram and the fact that the truck does not accelerate perpendicular to the incline, we can see that the magnitude of the normal force is given by

(2)

FN

fk

mg 15° mg cos 15°

mg sin 15°

FN = mg cos15.0°

Therefore, Equation (2) becomes Wnc = − µk ( mg cos15.0° ) s

(3)

Chapter 6 Problems

323

Combining Equations (1), (2), and (3), we have

− µk ( mg cos 15.0° ) s =

(

1 mv 2 f 2

+ mghf

)−(

1 mv 2 0 2

+ mgh0

)

(4)

This expression may be solved for s. SOLUTION When the car comes to a stop, vf = 0 m/s. If we take hf = 0 m at ground level, then, from the drawing at the right, we see that h0 = s sin 15.0° . Equation (4) then gives

s 15°

h0

(11.1 m/s )2 = = 13.5 m s= 2 g ( µ k cos15.0° − sin15.0° ) 2 ( 9.80 m/s 2 ) ( 0.750 ) cos15.0° − sin15.0° v02

______________________________________________________________________________ 62. REASONING The work W done is equal to the average power P multiplied by the time t, or (6.10a) W = Pt The average power is the average power generated per kilogram of body mass multiplied by Armstrong’s mass. The time of the race is the distance s traveled divided by the average speed v , or t = s / v (see Equation 2.1).

SOLUTION a. Substituting t = s / v into Equation 6.10a gives

135 ×103 m W s 6 75.0 kg W = Pt = P = 6.50 = 5.48 ×10 J 12.0 m/s v kg

(

)

P b. Since 1 joule = 2.389 × 10−4 nutritional calories, the work done is 2.389 ×10−4 nutritional calories W = 5.48 ×106 joules = 5.48 ×106 joules 1 joule

(

)

= 1.31× 103 nutritional calories ______________________________________________________________________________

324 WORK AND ENERGY

63.

SSM REASONING AND SOLUTION One kilowatt⋅ hour is the amount of work or energy generated when one kilowatt of power is supplied for a time of one hour. From 3 Equation 6.10a, we know that W = P t . Using the fact that 1 kW = 1.0 × 10 J/s and that 1h = 3600 s, we have

1.0 kWh = (1.0 × 10 3 J/s)(1 h)= (1.0 × 103 J/s)(3600 s)= 3.6 × 10 6 J ______________________________________________________________________________ 64. REASONING The average power is defined as the work divided by the time, Equation 6.10a, so both the work and time must be known. The time is given. The work can be obtained with the aid of the work-energy theorem as formulated in Wnc =

(

1 2

)

mvf2 − 12 mv02 + ( mghf − mgh0 ) (Equation 6.6). Wnc is the work done by the lifting

force acting on the helicopter. In using this equation, we note that two types of energy are changing: the kinetic energy

( mv ) and the gravitational potential energy (mgh). 1 2

2

The

kinetic energy is increasing, because the speed of the helicopter is increasing. The gravitational potential energy is increasing, because the height of the helicopter is increasing. SOLUTION The average power is P=

Wnc t

(6.10a)

where Wnc is the work done by the nonconservative lifting force and t is the time. The work is related to the helicopter’s kinetic and potential energies by Equation 6.6:

Wnc =

(

1 2

)

mvf2 − 12 mv02 + ( mghf − mgh0 )

Thus, the average power is W P = nc = t

P=

1 2

(

1 mv 2 f 2

2

− 12 mv0

) + ( mgh

f

t

− mgh0 )

=

1m 2

(v

2 f

2

− v0

) + mg ( h

f

− h0 )

t

( 810 kg ) ( 7.0 m /s )2 − ( 0 m /s )2 + (810 kg ) ( 9.80 m /s2 ) [8.2 m − 0 m]

= 2.4 × 104 W

3.5 s ______________________________________________________________________________

Chapter 6 Problems

325

in energy 65. REASONING The average power is given by Equation 6.10b ( P = ChangeTime ). The time is given. Since the road is level, there is no change in the gravitational potential energy, and the change in energy refers only to the kinetic energy. According to Equation 6.2, the kinetic energy is 12 mv 2 . The speed v is given, but the mass m is not. However, we can obtain the mass from the given weights, since the weight is mg.

SOLUTION a. Using Equations 6.10b and 6.2, we find that the average power is 2 2 Change in energy KE f − KE 0 12 mvf − 12 mv0 P= = = Time Time Time

Since the car starts from rest, v0 = 0 m/s, and since the weight is W = mg, the mass is m = W/g. Therefore, the average power is 1 2

P=

mvf2 − 12 mv02 Wvf2 = Time 2 g ( Time )

(1)

Using the given values for the weight, final speed, and the time, we find that

9.0 ×103 N ) ( 20.0 m/s ) Wvf2 ( P= = = 3.3 ×104 W (44 hp) 2 2 g ( Time ) 2 ( 9.80 m/s ) ( 5.6 s ) 2

b. From Equation (1) it follows that

1.4 ×104 N ) ( 20.0 m/s ) Wvf2 ( P= = = 5.1×104 W (68 hp) 2 g ( Time ) 2 ( 9.80 m/s 2 ) ( 5.6 s ) 2

66. REASONING The work W done is given by Equation 6.1 as W = ( F cos θ ) s , where F is the magnitude of the force that you exert on the rowing bar, s is the magnitude of the bar’s displacement, and θ is the angle between the force and the displacement. Solving this equation for F gives F=

W ( cos θ ) s

(6.1)

The work is also equal to the average power P multiplied by the time t, or W = Pt

(6.10a)

326 WORK AND ENERGY

Substituting Equation (6.10a) into (6.1) gives F=

W Pt = ( cos θ ) s ( cos θ ) s

SOLUTION The angle θ is 0°, since the direction of the pulling force is the same as the displacement of the rowing bar. Thus, the magnitude of the force exerted on the handle is

( 82 W )(1.5 s ) Pt = = 1.0 × 102 N ( cos θ ) s ( cos 0° )(1.2 m ) ______________________________________________________________________________ F=

67. REASONING The average power developed by the cheetah is equal to the work done by the cheetah divided by the elapsed time (Equation 6.10a). The work, on the other hand, can be related to the change in the cheetah’s kinetic energy by the work-energy theorem, Equation 6.3.

SOLUTION a. The average power is P=

W t

(6.10a)

where W is the work done by the cheetah. This work is related to the change in the cheetah’s kinetic energy by Equation 6.3, W = 12 mvf2 − 12 mv02 , so the average power is

W P= = t =

1 mv 2 f 2 1 2

2

− 12 mv0

t (110 kg )( 27 m / s )2 −

1 2

4.0 s

(110 kg )( 0 m / s )2

= 1.0 × 104 W

b. The power, in units of horsepower (hp), is 1 hp 4 P = 1.0 × 10 W = 13 hp 745.7 W ______________________________________________________________________________

(

)

Work Time (Equation 6.10a). The net work done on the glider is equal to the change in its kinetic energy, according to the work-energy theorem W = KE f − KE 0 (Equation 6.3). The glider

68. REASONING

The power required to accelerate the glider is found from P =

accelerates from rest, so its initial kinetic energy is zero. Therefore, the average power 1 mv 2 KE f mvf2 f 2 required is P = = = . The constant tension in the cable produces a constant Time t 2t

Chapter 6 Problems

327

acceleration in the glider, permitting us to determine the time from the expression x = 12 ( v0 + vf ) t (Equation 2.7), where x is the glider’s displacement. SOLUTION Since the glider starts from rest, we have v0 = 0 m/s. Therefore, solving 2x 2x 2x = = . x = 12 ( v0 + vf ) t (Equation 2.7) for the time t, we obtain t = v0 + vf 0 m/s + vf vf

Substituting

mv 2 2x for t in P = f , we find that the average power required of the winch is vf 2t mvf3 (184 kg )( 26.0 m/s ) P= = = = 1.68 ×104 W 4 ( 48.0 m ) 2x 4x 2 vf 3

mvf2

69.

SSM REASONING AND SOLUTION In the drawings below, the positive direction is to the right. When the boat is not pulling a skier, the engine must provide a force F1 to overcome the resistive force of the water FR. Since the boat travels with constant speed, these forces must be equal in magnitude and opposite in direction. – FR + F1 = ma = 0

FR

or FR = F1

F1

(1)

When the boat is pulling a skier, the engine must provide a force F2 to balance the resistive force of the water FR and the tension in the tow rope T. – FR + F2 – T = ma = 0

T

F

or F2 = FR + T

R

F

2

(2)

The average power is given by P = Fv , according to Equation 6.11 in the text. Since the boat moves with the same speed in both cases, v1 = v2, and we have P1 P 2 = F1 F2

or

F2 = F1

Using Equations (1) and (2), this becomes F1 + T = F1

P2 P1

P2 P1

328 WORK AND ENERGY

Solving for T gives P T = F1 2 − 1 P1

The force F1 can be determined from P1 = F1v1 , thereby giving

7.50 × 104 W 8.30 × 104 W P1 P 2 2 1 − − 1 = 6.7 × 10 N = 7.50 × 104 W 12 m/s v1 P1 ______________________________________________________________________________ T=

70. REASONING AND SOLUTION The following drawings show the free-body diagrams for the car in going both up and down the hill. The force F R is the combined force of air resistance and friction, and the forces FU and F D are the forces supplied by the engine in going uphill and downhill respectively. Going up the hill

FN

Going down the hill

FN

FU

FR FD

FR mg sin θ

mg cos θ

mg sin θ

mg cos θ

Writing Newton's second law in the direction of motion for the car as it goes uphill, taking uphill as the positive direction, we have

FU − FR − mg sin θ = ma = 0 Solving for FU , we have

FU = FR + mg sin θ

Similarly, when the car is going downhill, Newton's second law in the direction of motion gives FR − FD − mg sinθ = ma = 0 so that FD = FR − mg sinθ Since the car needs 47 hp more to sustain the constant uphill velocity than the constant downhill velocity, we can write P U = P D + ∆P

Chapter 6 Problems

329

where P U is the power needed to sustain the constant uphill velocity, PD is the power needed to sustain the constant downhill velocity, and ∆ P = 47 hp . In terms of SI units, 746 W 4 ∆P = ( 47 hp ) = 3.51 × 10 W 1 hp Using Equation 6.11 ( P = Fv ), the equation P U = P D + ∆P can be written as

FU v = FD v + ∆P Using the expressions for FU and FD , we have Solving for θ, we find

( FR + mg sin θ )v = ( FR − mg sin θ )v + ∆P

3.51 × 104 W –1 = sin = 2.0° 2(1900 kg)(9.80 m/s 2 )(27 m/s) ______________________________________________________________________________ ∆P 2mgv

θ = sin –1

71.

SSM REASONING The area under the force-versus-displacement graph over any displacement interval is equal to the work done over that displacement interval. SOLUTION a. Since the area under the force-versus-displacement graph from s = 0 to 0.50 m is greater for bow 1, it follows that bow 1 requires more work to draw the bow fully .

b. The work required to draw bow 1 is equal to the area of the triangular region under the force-versus-displacement curve for bow 1. Since the area of a triangle is equal one-half times the base of the triangle times the height of the triangle, we have 1 2

W1 = (0.50m)(350 N) = 88 J For bow 2, we note that each small square under the force-versus-displacement graph has an area of (0.050 m)(40.0 N) = 2.0 J

We estimate that there are approximately 31.3 squares under the force-versus-displacement graph for bow 2; therefore, the total work done is 2.0 J = 63 J (31.3 squares) square

Therefore, the additional work required to stretch bow 1 as compared to bow 2 is

330 WORK AND ENERGY

88 J – 63 J =

25 J

______________________________________________________________________________ 72. REASONING When the force varies with the displacement, the work is the area beneath the graph of the force component F cos θ along the displacement as a function of the magnitude s of the displacement. Here, the shape of this area is a triangle. The area of a triangle is one-half times the base times the height of the triangle. SOLUTION The base of the triangle is 1.60 m, and the “height” is 62.0 N. Therefore, the area, which is the work, is

F cos θ 62.0 N

W = 12 Base × Height =

1 2

(1.60 m )( 62.0 N )

0

1.60 m

s

= 49.6 J

73. REASONING AND SOLUTION a. The work done is equal to the colored area under the graph. Therefore, the work done by the net force on the skater from 0 to 3.0 m is

Area under graph = ( 31 N − 0 N )( 3.0 m − 0 m ) = 93 J from 0 to 3.0 m b. From 3.0 m to 6.0 m, the net force component F cos θ is zero, and there is no area under the curve. Therefore, no work is done . c. The speed of the skater will increase from 1.5 m/s when s = 0 m to a final value vf at s = 3.0 m . Since the force drops to zero at s = 3.0 m , the speed remains constant at the value vf from s = 3.0 m to s = 6.0 m ; therefore, the speed at s = 6.0 m is vf . We can find vf from the work-energy theorem (Equation 6.3). According to the work-energy theorem, the work done on the skater is given by 1 2

1 2

W = mvf2 − mv02 Solving for vf , we find 2W 2(93 J) 2 2 + v0 = + (1.5 m/s) = 2.3 m/s m 65 kg ______________________________________________________________________________ vf =

74. REASONING The area under the force-versus-displacement graph over any displacement interval is equal to the work done over that interval. From Example 14, we know that the

Chapter 6 Problems

331

total work done in drawing back the string of the bow is 60.5 J, corresponding to a total area under the curve of 242 squares (each square represents 0.250 J of work). The percentage of the total work done over any displacement interval is, therefore, Work done during interval Total work done during all intervals

× 100 =

Number of squares under curve during interval Total number of squares under entire curve

× 100

SOLUTION a. We estimate that there are 130 squares under the curve from s = 0 to 0.306 m ; therefore, the percentage of work done during the interval in question is 130 squares × 100 = 54% 242 squares

b. Similarly, we estimate that there are 112 squares under the curve in the interval from s = 0.306 m to 0.500 m; therefore, the percentage of work done during the interval in question is 112 squares × 100 = 46% 242 squares ______________________________________________________________________________ 75. REASONING The final speed vf of the object is related to its final kinetic energy by KE f = 12 mvf2 (Equation 6.2). The object’s mass m is given, and we will use the work-energy

theorem W = KE f − KE 0 (Equation 6.3) to determine its final kinetic energy. We are helped by the fact that the object starts from rest, and therefore has no initial kinetic energy KE0. Thus, the net work done on the object equals its final kinetic energy: W = KE f = 12 mvf2 . In a graph of F cos θ versus s, such as that given, the net work W is the total area under the graph. For the purpose of calculation, it is convenient to divide this area into two pieces, a triangle (from s = 0 m to s = 10.0 m) and a rectangle (from s = 10.0 m to s = 20.0 m). Both pieces have the same width (10.0 m) and height (10.0 N), so the triangle has half the area of the rectangle.

2W . m done on the object from s = 0 m to s = 10.0 m

SOLUTION Solving W = 12 mvf2 for the final speed of the object, we obtain vf = The net work W is the sum of the work W0,10

(the triangular area) and the work W10,20 done on the object from s = 10.0 m to s = 20.0 m (the rectangular area). We calculate the work W10,20 by multiplying the width (10.0 m) and

height (10.0 N) of the rectangle: W10,20 = ( Width )( Height ) . The triangle’s area, which is the work W0,10, is half this amount: W0,10 = 12 W10,20 . Therefore, the net work

W = W0,10 + W10,20 done during the entire interval from s = 0 m to s = 20.0 m is

332 WORK AND ENERGY

W = 12 W10,20 + W10,20 = 23 W10,20 =

3 2

( Width )( Height )

We can now calculate the final speed of the object from vf =

vf =

2

2W : m

( 32 ) ( Width )( Height ) = 3 (10.0 m )(10.0 N ) = 7.07 m/s m

6.00 kg

76. REASONING AND SOLUTION The work done by the retarding force is given by Equation 6.1: W = ( F cos θ )s . Since the force is a retarding force, it must point opposite to the direction of the displacement, so that θ = 180° . Thus, we have

W = (F cosθ )s = (3.0 × 103 N)(cos 180°)(850 m) = –2.6 × 106 J The work done by this force is negative , because the retarding force is directed opposite to the direction of the displacement of the truck. ______________________________________________________________________________ 77.

SSM REASONING The only two forces that act on the gymnast are his weight and the force exerted on his hands by the high bar. The latter is the (non-conservative) reaction force to the force exerted on the bar by the gymnast, as predicted by Newton's third law. This force, however, does no work because it points perpendicular to the circular path of motion. Thus, Wnc = 0 J, and we can apply the principle of conservation of mechanical energy. SOLUTION The conservation principle gives 1 mv f2 2

1 2

+ mghf = mv 02 + mgh0 Ef

E0

Since the gymnast's speed is momentarily zero at the top of the swing, v0 = 0 m/s. If we take hf = 0 m at the bottom of the swing, then h0 = 2 r , where r is the radius of the circular path followed by the gymnast's waist. Making these substitutions in the above expression and solving for vf , we obtain

vf = 2gh0 = 2g(2r) = 2(9.80 m/s2 )(2 × 1.1 m) = 6.6 m/s ______________________________________________________________________________

Chapter 6 Problems

333

78. REASONING a. Since there is no air friction, the only force that acts on the projectile is the conservative gravitational force (its weight). The initial and final speeds of the ball are known, so the conservation of mechanical energy can be used to find the maximum height that the projectile attains. b. When air resistance, a nonconservative force, is present, it does negative work on the projectile and slows it down. Consequently, the projectile does not rise as high as when there is no air resistance. The work-energy theorem, in the form of Equation 6.6, may be used to find the work done by air friction. Then, using the definition of work, Equation 6.1, the average force due to air resistance can be found. SOLUTION a. The conservation of mechanical energy, as expressed by Equation 6.9b, states that 1 mv 2 f 2

1 mv 2 0 2

+ mghf =

Ef

+ mgh0

E0

The mass m can be eliminated algebraically from this equation since it appears as a factor in every term. Solving for the final height hf gives 1 2

hf =

( v02 − vf2 ) + h

0

g

Setting h0 = 0 m and vf = 0 m/s, the final height, in the absence of air resistance, is 2

hf =

2

vo − vf 2g

=

(18.0 m / s )2 − ( 0 m/s )2

(

2 9.80 m / s 2

)

= 16.5 m

b. The work-energy theorem is

Wnc =

(

1 mv 2 f 2

− 12 mv02

) + ( mgh

f

− mgh0 )

(6.6)

where Wnc is the nonconservative work done by air resistance. According to Equation 6.1,

the work can be written as Wnc = ( FR cos 180° ) s , where FR is the average force of air

resistance. As the projectile moves upward, the force of air resistance is directed downward, so the angle between the two vectors is θ = 180° and cos θ = –1. The magnitude s of the displacement is the difference between the final and initial heights, s = hf – h0 = 11.8 m. With these substitutions, the work-energy theorem becomes

(

)

− FR s = 12 m vf2 − vo2 + mg ( hf − h0 )

334 WORK AND ENERGY

Solving for FR gives

FR =

=

1m 2

1 2

( vf2 − vo2 ) + mg ( hf − h0 ) −s

( 0.750 kg ) ( 0 m/s )2 − (18.0 m/s )2 + ( 0.750 kg ) ( 9.80 m/s2 ) (11.8 m ) = − (11.8 m )

2.9 N

______________________________________________________________________________ 79.

SSM REASONING AND SOLUTION We will assume that the tug-of-war rope remains parallel to the ground, so that the force that moves team B is in the same direction as the displacement. According to Equation 6.1, the work done by team A is

W = (F cosθ )s = (1100 N)(cos 0°)(2.0 m) = 2.2 × 10 3 J ______________________________________________________________________________ 80. REASONING AND SOLUTION The vertical height of the skier is h = s sin 14.6° so s

∆PE = mgs sin 14.6°

h 14.6°

= (75.0 kg)(9.80 m/s2)(2830 m) sin 14.6° = 5.24 × 105 J ______________________________________________________________________________ 81. REASONING The net work done by the pushing force and the frictional force is zero, and our solution is focused on this fact. Thus, we express this net work as WP + Wf = 0, where WP is the work done by the pushing force and Wf is the work done by the frictional force. We will substitute for each individual work using Equation 6.1 [W = (F cos θ) s] and solve the resulting equation for the magnitude P of the pushing force. SOLUTION According to Equation 6.1, the work done by the pushing force is WP = (P cos 30.0°) s = 0.866 P s

The frictional force opposes the motion, so the angle between the force and the displacement is 180°. Thus, the work done by the frictional force is Wf = (fk cos 180°) s = – fk s

Chapter 6 Problems

Equation 4.8 indicates that the magnitude of the kinetic frictional force is fk = µ kFN, where FN is the magnitude of the normal force acting on the crate. The free-body diagram shows the forces acting on the crate. Since there is no acceleration in the vertical direction, the y component of the net force must be zero:

+y FN P fk

30.0º

+x

mg

FN − mg − P sin 30.0° = 0 Therefore,

335

FN = mg + P sin 30.0°

It follows, then, that the magnitude of the frictional force is fk = µ k FN = µ k (mg + P sin 30.0°) Suppressing the units, we find that the work done by the frictional force is Wf = – fk s = − (0.200)[(1.00 × 102 kg)(9.80 m/s2) + 0.500P]s = −(0.100P + 196)s Since the net work is zero, we have WP + Wf = 0.866 Ps − (0.100P + 196)s = 0 Eliminating s algebraically and solving for P gives P = 256 N .

82. REASONING The average power generated by the tension in the cable is equal to the work done by the tension divided by the elapsed time (Equation 6.10a). The work can be related to the change in the lift’s kinetic and potential energies via the work-energy theorem.

SOLUTION a. The average power is P=

Wnc t

(6.10a)

where Wnc is the work done by the nonconservative tension force. This work is related to the lift’s kinetic and potential energies by Equation 6.6, Wnc = so the average power is

P=

Wnc t

( =

1 mv 2 f 2

− 12 mv02

) + ( mgh

f

t

(

1 2

)

mvf2 − 12 mv02 + ( mghf − mgh0 ) ,

− mgh0 )

336 WORK AND ENERGY

Since the lift moves at a constant speed, v0 = vf, the average power becomes

83.

( 4 skiers ) 65

(

)

kg 9.80 m / s 2 (140 m ) 3 skier P= = = 3.0 × 10 W t 60 s ( 2 min ) 1 min ______________________________________________________________________________ mg ( hf − h0 )

SSM REASONING AND SOLUTION a. The lost mechanical energy is Elost = E0 – Ef The ball is dropped from rest, so its initial energy is purely potential. The ball is momentarily at rest at the highest point in its rebound, so its final energy is also purely potential. Then

Elost = mgh0 – mghf = ( 0.60 kg ) ( 9.80 m/s 2 ) (1.05 m ) – ( 0.57 m ) = 2.8 J b. The work done by the player must compensate for this loss of energy. Elost = ( F cos θ ) s work done by player

⇒

F=

Elost

( cos θ ) s

=

2.8 J = 35 N ( cos 0° )( 0.080 m )

______________________________________________________________________________ 84. REASONING When launched with the minimum initial speed v0, the puck has just enough initial kinetic energy to reach the teammate with a final speed vf of zero. All of the initial kinetic energy has served the purpose of compensating for the work done by friction in opposing the motion. Friction is a nonconservative force, so to determine v0 we will use the work-energy theorem in the form of Equation 6.8: Wnc = ( 12 mvf2 + mghf ) − ( 12 mv02 + mgh0 ) .

Wnc is the work done by the net nonconservative force, which, in this case, is just the kinetic frictional force. This work can be expressed using Equation 6.1 as Wnc = (fk cos θ)s, where fk is the magnitude of the kinetic frictional force and s is the magnitude of the displacement, or the distance between the players. We are given neither the frictional force nor the distance between the players, but we are given the initial speed that enables the puck to travel half way. This information will be used to evaluate Wnc. SOLUTION Noting that a hockey rink is flat (hf = h0), we write Equation 6.8 as follows

Chapter 6 Problems

Wnc =

(

1 2

mvf2 + mghf ) −

Final mechanical energy

(

1 2

mv02 + mgh0 )

= 12 mvf2 − 12 mv02 = − 12 mv02

337

(1)

Initial mechanical energy

where we have used the fact that the final speed is vf = 0 m/s when the initial speed v0 has its minimum value. According to Equation 6.1 the work done on the puck by the kinetic frictional force is Wnc = (fk cos θ)s = (fk cos 180º)s = − fks In this expression the angle θ is 180º, because the frictional force points opposite to the displacement. With this substitution Equation (1) becomes

− f k s = − 12 mv02

or

f k s = 12 mv02

(2)

To evaluate the term fks we note that Equation (2) also applies to the failed attempt at the pass, so that f k ( 12 s ) = 12 m (1.7 m/s )

2

or

f k s = m (1.7 m/s )

2

Substituting this result into Equation (2) gives

m (1.7 m/s ) = 12 mv02 2

85.

or

v0 = 2 (1.7 m/s ) = 2.4 m/s 2

SSM WWW REASONING Gravity is the only force acting on the vaulters, since friction and air resistance are being ignored. Therefore, the net work done by the nonconservative forces is zero, and the principle of conservation of mechanical energy holds. SOLUTION Let E2f represent the total mechanical energy of the second vaulter at the ground, and E20 represent the total mechanical energy of the second vaulter at the bar. Then, the principle of mechanical energy is written for the second vaulter as 1 1 2 2 mv2f + mgh2f = mv20 + mgh20 2 2 E2f E20

Since the mass m of the vaulter appears in every term of the equation, m can be eliminated algebraically. The quantity h20 is h20 = h, where h is the height of the bar. Furthermore, when the vaulter is at ground level, h2f = 0 m. Solving for v20 we have v 20 = v 22 f − 2 gh

(1)

338 WORK AND ENERGY

In order to use Equation (1), we must first determine the height h of the bar. The height h can be determined by applying the principle of conservation of mechanical energy to the first vaulter on the ground and at the bar. Using notation similar to that above, we have 1 1 2 2 mv1f + mgh1f = mv10 + mgh10 2 2 E1f E10

Where E10 corresponds to the total mechanical energy of the first vaulter at the bar. The height of the bar is, therefore,

v2 − v2 (8.90 m/s)2 − (1.00 m/s)2 h = h10 = 1f 10 = = 3.99 m 2g 2(9.80 m/s2 ) The speed at which the second vaulter clears the bar is, from Equation (1), 2

2

2

v20 = v2f − 2 gh = (9.00 m/s) − 2(9.80 m/s )(3.99 m) = 1.7 m/s ______________________________________________________________________________ 86. REASONING AND SOLUTION At the bottom of the circular path of the swing, the centripetal force is provided by the tension in the rope less the weight of the swing and rider. That is,

mv r

2

= T − mg

FC

Solving for the mass yields m=

T 2

v +g r

The energy of the swing is conserved if friction is ignored. The initial energy, E0, when the swing is released is completely potential energy and is E0 = mgh0, where h0 = r (1 – cos 60.0°) =

1r 2

is the vertical height of the swing. At the bottom of the path the swing's energy is entirely kinetic and is Ef = 12 mv

Setting Eo = Ef and solving for v gives

2

Chapter 6 Problems

339

v = 2gh0 = gr

The expression for the mass now becomes

8.00 × 102 N T = = 40.8 kg 2 g 2 9.80 m/s 2 ______________________________________________________________________________ m=

(

)

87. REASONING The magnitude fk of the average kinetic friction force exerted on the skateboarder is related to the amount of work Wk the done by that force and the distance s he

slides along the ramp according to Wk = ( f k cos θ ) s (Equation 6.1). Solving this equation for the kinetic friction force, we find

fk =

Wk

(1)

s cos θ

The ramp is semicircular, so sliding from the top to the bottom means sliding one-fourth of the circumference of a circle with a radius of r = 2.70 m. Therefore, s = 14 ( 2π r ) = 12 π r . Because the kinetic friction force is always directed opposite the skateboarder’s velocity, the angle θ in Equation (1) is 180°. Thus, Equation (1) becomes

fk = To

complete

Wnc =

the

Wk

1 π r cos180 2

solution,

we

=−

2Wk

(2)

πr

will

use

the

work-energy

theorem

( 12 mvf2 − 12 mv02 ) + ( mghf − mgh0 ) (Equation 6.6) to find the work Wk done by the

nonconservative kinetic friction force. We can do this because the normal force exerted by the ramp, which is also a nonconservative force, is always perpendicular to the skateboarder’s velocity. Consequently, the normal force does no work, so the net work Wnc done by all nonconservative forces is just the work Wk done by the kinetic friction force:

Wnc = Wk =

( 12 mvf2 − 12 mv02 ) + ( mghf − mgh0 ) = 12 m ( vf2 − v02 ) + mg ( hf − h0 )

(3)

SOLUTION Substituting Equation (3) into Equation (2), we obtain

(

)

2 12 m vf2 − v02 + mg ( hf − h0 ) fk = − =− πr πr 2Wk

(4)

340 WORK AND ENERGY

The skateboarder falls from rest, so we have v0 = 0 m/s. He starts a distance h above the top of the ramp, which itself is a distance r above the bottom of the ramp. His vertical displacement, therefore, is hf − h0 = − ( h + r ) . The algebraic sign of this displacement is negative because he moves downward. Suppressing units, Equation (4) then yields

(

)

2 2 2 12 m vf2 − 0 − mg ( h + r ) = − mvf − 2mg ( h + r ) = 2mg ( h + r ) − mvf fk = − πr πr πr

(5)

Thus, the average force of kinetic friction acting on the skateboarder as he slides down the ramp is fk =

(

)

2 ( 61.0 kg ) 9.80 m/s 2 (1.80 m + 2.70 m ) − ( 61.0 kg )( 6.40 m/s )

π ( 2.70 m )

2

= 3.40 × 102 N

CHAPTER 7 IMPULSE AND MOMENTUM ANSWERS TO FOCUS ON CONCEPTS QUESTIONS _____________________________________________________________________________________________

1.

(b) Kinetic energy, 12 mv 2 , is a scalar quantity and is the same for both cars. Momentum, mv, is a vector quantity that has a magnitude and a direction. The two cars have different directions, so they have different momenta.

2.

(d) Momentum is a vector quantity that has a magnitude and a direction. The magnitudes (m0v0) and directions (due north) are the same for both runners.

3.

The magnitude of the total momentum is 47 700 kg ⋅ m/s .

4.

(d) According to the impulse-momentum theorem, the impulse is equal to the final momentum minus the initial momentum. Therefore, the impulses are the same in both cases, since the final momenta are the same and the initial momenta are the same. The impulse is also the product of the net average force and the time of impact. Since the impulses are equal and the time of impact is larger for the air mattress than for the ground, the air mattress exerts a smaller net average force on the high jumper.

5.

The impulse is +324 kg ⋅ m/s .

6.

(c) According to the impulse-momentum theorem, the impulse is equal to the change in the particle’s momentum. The magnitude of the momentum change is the same in regions A and C; therefore, in these regions the particle experiences impulses of the same magnitude. The momentum in region B is constant. Therefore, the change in momentum is zero, and so is the impulse.

7.

(b) According to the impulse-momentum theorem, the net average force is equal to the change in the particle’s momentum divided by the time interval. This ratio is greatest in region C. The ratio is equal to zero in region B, since the change in the particle’s momentum is zero there.

8.

Magnitude of net average force in Region A = 1.0 N, Magnitude of net average force in Region C = 4.0 N

9.

(d) Since there are no external forces acting on the rocket, momentum is conserved before, during, and after the separation. This means that the momentum of the rocket before the separation (+150 000 kg⋅m/s) is equal to the sum of the momenta of the two stages after the separation [+250 000 kg⋅m/s + (− 100 000 kg⋅m/s) = +150 000 kg⋅m/s].

342 IMPULSE AND MOMENTUM

10. (a) The net external force acting on the ball/earth system is zero. The gravitational forces

that the ball and earth exert on each other are internal forces, or forces that the objects within the system exert on each other. The space probe is also an isolated system, since there are no external forces acting on it. 11. (c) The net external force that acts on the objects is zero, so the total linear momentum is conserved. Therefore, the total momentum before the collision (+12 kg⋅m/s − 2 kg⋅m/s = +10 kg⋅m/s) equals the total momentum after the collision (−4 kg⋅m/s +14 kg⋅m/s = +10 kg⋅m/s). 12. (b) The collision between the objects is an elastic collision, so the total kinetic energy is conserved. Therefore, the total kinetic energy before the collision (8 J + 6 J = 14 J) is equal to the total kinetic energy after the collision (10 J + 4 J = 14 J). 13. (c) Since the net external force acting on the two objects during the collision is zero, the total linear momentum of the system is conserved. In other words, the total linear momentum before the collision (−3 kg⋅m/s + 4 kg⋅m/s = +1 kg⋅m/s) equals the total linear momentum after the collision (+1 kg⋅m/s). Furthermore, some kinetic energy is lost during a completely inelastic collision, which is the case here. The final kinetic energy (4 J) is less that the total initial kinetic energy (1 J + 6 J = 7 J). 14. (d) The net external force acting on the two objects during the collision is zero, so the total momentum of the system is conserved. In two dimensions this means that the x-component of the initial total momentum (+16 kg⋅m/s) is equal to the x-component of the final total momentum. Since the x-component of the final momentum of object 1 is +6 kg⋅m/s, the x-component of the final momentum of object 2 must be +10 kg⋅m/s (+16 kg⋅m/s − 6 kg⋅m/s = +10 kg⋅m/s). 15. (c) The net external force acting on the two objects during the collision is zero, so the total momentum of the system is conserved. In two dimensions this means that the y-component of the initial total momentum (0 kg⋅m/s) is equal to the y-component of the final total momentum. Since the y-component of the final momentum of object 1 is −5 kg⋅m/s, then the y-component of the final momentum of object 2 must be +5 kg⋅m/s [0 kg⋅m/s − (−5 kg⋅m/s) = +5 kg⋅m/s] . 16. x coordinate of the center of mass = −1.5 m, y coordinate of the center of mass = 0 m 17. Velocity of center of mass = 2.0 m/s, along the +x axis

Chapter 7 Problems

CHAPTER 7

343

IMPULSE AND MOMENTUM

PROBLEMS 1.

SSM REASONING According to Equation 7.1, the impulse J produced by an average force F is J = F∆t , where ∆t is the time interval during which the force acts. We will apply this definition for each of the forces and then set the two impulses equal to one another. The fact that one average force has a magnitude that is three times as large as that of the other average force will then be used to obtain the desired time interval. SOLUTION Applying Equation 7.1, we write the impulse of each average force as follows: J1 = F1∆t1

and

J 2 = F 2 ∆t 2

But the impulses J1 and J2 are the same, so we have that F1∆t1 = F 2 ∆t2 . Writing this result in terms of the magnitudes of the forces gives F 1∆t1 = F 2 ∆t2

or

F ∆t2 = 1 ∆t1 F2

The ratio of the force magnitudes is given as F 1 / F 2 = 3 , so we find that F ∆t2 = 1 ∆t1 = 3 ( 3.2 ms ) = 9.6 ms F2

2.

REASONING The impulse-momentum theorem, as expressed in Equation 7.4, states that the impulse acting on each car is equal to the final momentum of the car minus its initial momentum:

( ΣF) ∆t = Impulse

m vf Final momentum

− m v0 Initial momentum

or

ΣF =

m vf − m v0 ∆t

where ΣF is the net average force that acts on the car, and ∆t is the time interval during which the force acts.

SOLUTION We assume that the velocity of each car points in the +x direction. The net average force acting on each car is:

344 IMPULSE AND MOMENTUM

3.

Car A

ΣF =

m v f − m v 0 (1400 kg )( +27 m/s ) − (1400 kg )( 0 m/s ) = = +4200 N 9.0 s ∆t

Car B

ΣF =

m v f − m v 0 (1900 kg )( +27 m/s ) − (1900 kg )( 0 m/s ) = = +5700 N ∆t 9.0 s

REASONING The impulse that the roof of the car applies to the hailstones can be found from the impulse-momentum theorem, Equation 7.4. Two forces act on the hailstones, the average force F exerted by the roof, and the weight of the hailstones. Since it is assumed that F is much greater than the weight of the hailstones, the net average force ( ΣF ) is equal to F . SOLUTION From Equation 7.4, the impulse that the roof applies to the hailstones is:

F ∆t = Impulse

mvf Final momentum

−

mv 0

= m ( vf − v 0 )

Initial momentum

Solving for F (with up taken to be the positive direction) gives m F = ( vf − v 0 ) = (0.060 kg/s) [ (+15 m/s) − (−15 m/s)] = +1.8 N ∆t This is the average force exerted on the hailstones by the roof of the car. The positive sign indicates that this force points upward. From Newton's third law, the average force exerted by the hailstones on the roof is equal in magnitude and opposite in direction to this force. Therefore, Force on roof = −1.8 N The negative sign indicates that this force points downward .

4.

REASONING According to the impulse-momentum theorem, the rocket’s final momentum m v f differs from its initial momentum by an amount equal to the impulse ( ΣF ) ∆t of the net force exerted on it: ( ΣF ) ∆t = mvf − mv 0 (Equation 7.4). We are ignoring gravitational

and frictional forces, so this impulse is due entirely to the force generated by the motor. The magnitude of the motor’s impulse is given as ( ΣF ) ∆t = 29.0 N ⋅ s , so we will obtain the

rocket’s final speed by using Equation 7.4.

Chapter 7 Problems

345

The rocket starts from rest, so v0 = 0 m/s, and the impulse-momentum

SOLUTION

theorem becomes ( ΣF ) ∆t = mvf . Therefore, the rocket’s final speed vf is the magnitude of the motor impulse divided by the rocket’s mass: vf =

5.

( ΣF ) ∆t = 29.0 N ⋅ s = 166 m/s m

0.175 kg

SSM REASONING The impulse that the volleyball player applies to the ball can be found from the impulse-momentum theorem, Equation 7.4. Two forces act on the volleyball while it’s being spiked: an average force F exerted by the player, and the weight of the ball. As in Example 1, we will assume that F is much greater than the weight of the ball, so the weight can be neglected. Thus, the net average force ( ΣF ) is equal to F . SOLUTION From Equation 7.4, the impulse that the player applies to the volleyball is

F ∆t = Impulse

mv f Final momentum

−

mv 0 Initial momentum

= m( v f − v 0 ) = (0.35 kg) [ (–21 m/s) – (+4.0 m/s)] = –8.7 kg ⋅ m/s The minus sign indicates that the direction of the impulse is the same as that of the final velocity of the ball.

6.

REASONING During the collision, the bat exerts an impulse on the ball. The impulse is the product of the average force that the bat exerts and the time of contact. According to the impulse-momentum theorem, the impulse is also equal to the change in the momentum of the ball. We will use these two relations to determine the average force exerted by the bat on the ball. SOLUTION The impulse J is given by Equation 7.1 as J = F∆t , where F is the average force that the bat exerts on the ball and ∆t is the time of contact. According to the impulsemomentum theorem, Equation 7.4, the net average impulse ( ΣF ) ∆t is equal to the change in the ball’s momentum; ( ΣF ) ∆t = mv f − mv 0 . Since we are ignoring the weight of the ball, the bat’s force is the net force, so ΣF = F . Substituting this value for the net average force into the impulse-momentum equation and solving for the average force gives

F=

mv f − mv 0 ∆t

=

( 0.149 kg )( −45.6 m/s ) − ( 0.149 kg )( +40.2 m/s ) = 1.10 ×10−3 s

−11 600 N

346 IMPULSE AND MOMENTUM

where the positive direction for the velocity has been chosen as the direction of the incoming ball.

7.

REASONING The impulse-momentum theorem (Equation 7.4) states that the impulse of an applied force is equal to the change in the momentum of the object to which the force is applied. We will use this theorem to determine the final momentum from the given value of the initial momentum. The impulse is the average force times the time interval during which the force acts, according to Equation 7.1. The force and the time interval during which it acts are given, so we can calculate the impulse. SOLUTION According to the impulse-momentum theorem, the impulse applied by the retrorocket is J = mvf − mv0 (7.4) The impulse is J = F ∆t (Equation 7.1), which can be substituted into Equation 7.4 to give

F ∆t = mvf − mv0

or

mvf = F ∆t + mv0

where mvf is the final momentum. Taking the direction in which the probe is traveling as the positive direction, we have that the initial momentum is mv0 = +7.5 × 107 kg⋅m/s and the force is F = −2.0 × 106 N . The force is negative, because it points opposite to the direction of the motion. With these data, we find that the final momentum after the retrorocket ceases to fire is

(

)

mvf = F ∆t + mv0 = −2.0 × 106 N (12 s ) + 7.5 × 107 kg ⋅ m/s = +5.1× 107 kg ⋅ m/s

8.

REASONING We will apply the impulse momentum theorem as given in Equation 7.4 to solve this problem. From this theorem we know that, for a given change in momentum, greater forces are associated with shorter time intervals. Therefore, we expect that the force in the stiff-legged case will be greater than in the knees-bent case. SOLUTION a. Assuming that upward is the positive direction, we find from the impulsemomentum theorem that

ΣF =

mvf − mv 0

∆t

=

( 75 kg )( 0 m/s ) − ( 75 kg )( −6.4 m/s ) 2.0 × 10

−3

s

= +2.4 × 105 N

Chapter 7 Problems

347

b. Again using the impulse-momentum theorem, we find that

ΣF =

mvf − mv 0

=

∆t

( 75 kg )( 0 m/s ) − ( 75 kg )( −6.4 m/s ) 0.10 s

= +4.8 × 103 N

c. The net average force acting on the man is ΣF = FGround + W , where FGround is the average upward force exerted on the man by the ground and W is the downward-acting weight of the man. It follows, then, that FGround = ΣF − W . Since the weight is W = −mg, we have Stiff − legged

FGround = ΣF − W

(

)

(

)

= +2.4 × 105 N − − ( 75 kg ) 9.80 m/s 2 = +2.4 × 105 N Knees − bent

FGround = ΣF − W = +4.8 × 103 N − − ( 75 kg ) 9.80 m/s 2 = +5.5 × 103 N

9.

REASONING AND SOLUTION

a. According to Equation 7.4, the impulse-momentum theorem,

( ΣF ) ∆t = mvf − mv0 .

Since the only horizontal force exerted on the puck is the force F exerted by the goalie, ΣF = F . Since the goalie catches the puck, vf = 0 m/s . Solving for the average force exerted on the puck, we have

F=

m( v f − v 0 ) ∆t

=

( 0.17 kg ) ( 0 m/s ) − ( +65 m/s ) 5.0 ×10

−3

s

= –2.2 ×103 N

By Newton’s third law, the force exerted on the goalie by the puck is equal in magnitude and opposite in direction to the force exerted on the puck by the goalie. Thus, the average force exerted on the goalie is +2.2 × 103 N . b. If, instead of catching the puck, the goalie slaps it with his stick and returns the puck straight back to the player with a velocity of –65 m/s, then the average force exerted on the puck by the goalie is F=

m( v f − v 0 ) ∆t

=

b0.17 kg g(–65 m / s) – (+65 m / s) 5.0 × 10 –3 s

= – 4.4 × 10 3 N

The average force exerted on the goalie by the puck is thus +4.4 × 103 N .

348 IMPULSE AND MOMENTUM

The answer in part (b) is twice that in part (a). This is consistent with the conclusion of Conceptual Example 3. The change in the momentum of the puck is greater when the puck rebounds from the stick. Thus, the puck exerts a greater impulse, and hence a greater force, on the goalie. 10. REASONING During the time interval ∆t, a mass m of water strikes the turbine blade. The incoming water has a momentum mv0 and that of the outgoing water is mvf. In order to

change the momentum of the water, an impulse ( ΣF ) ∆t is applied to it by the stationary

turbine blade. Now ( Σ F ) ∆t = F ∆t , since only the force of the blade is assumed to act on the water in the horizontal direction. These variables are related by the impulse-momentum theorem, F ∆t = mvf − mv0, which can be solved to find the average force F exerted on the water by the blade. SOLUTION Solving the impulse-momentum theorem for the average force gives F=

mv f − mv 0 ∆t

=

m ( v − v0 ) ∆t f

The ratio m / ( ∆t ) is the mass of water per second that strikes the blade, or 30.0 kg/s, so the average force is

F=

m ( v − v 0 ) = ( 30.0 kg/s ) ( −16.0 m/s ) − ( +16.0 m/s ) = −960 N ∆t f

The magnitude of the average force is 960 N .

11. REASONING We will divide this problem into two parts, because the forces acting on the student change abruptly at the instant of impact. In the first part, the student falls freely from rest, under the sole influence of the conservative gravitational force. Thus, the student’s total mechanical energy E is conserved up to the instant of impact. We will use the energy conservation principle to determine the student’s initial height H in terms of the student’s velocity vimpact at that instant. The second part of the student’s motion begins at impact, when the force Fground due to the ground overwhelms the gravitational force and brings the student to rest. The force of the ground is nonconservative, so instead of the energy conservation principle, we will apply the impulse-momentum theorem Fground ∆t = mvf − mv0 (Equation 7.4) to analyze the collision. Because this time interval

begins at impact, v0 is the student’s impact velocity: v0 = vimpact.

Chapter 7 Problems

SOLUTION We begin with the energy conservation principle 1 mv 2 f 2

1 mv 2 0 2

349

+ mgh0 =

+ mghf (Equation 6.9b) applied to the student’s fall to the ground. Falling from rest

implies v0 = 0 m/s, and the student’s final velocity is the impact velocity: vf = vimpact. Thus, we have

0 + m gh0 =

1 2

2 m vimpact

+ m ghf

or

g ( h0 − hf ) =

1 v2 2 impact

or

H=

2 vimpact

2g

(1)

H

For the student’s collision with the ground, the impulse-momentum theorem gives Fground ∆t = mvf − mv0 (Equation 7.4). The collision brings the student to rest, so we know that vf = 0 m/s, and Equation 7.4 becomes Fground ∆t = −mv0 . Solving for the impact speed v0, we obtain

v0 = vimpact = −

Fground ∆t

(2)

m

Substituting Equation (2) into Equation (1) yields 2

H=

2 vimpact

2g

Fground ∆t − Fground ∆t m = = 2g 2 gm2

(

)

2

( +18 000 N )( 0.040 s ) = = 6.7 m 2 2 9.80 m/s2 ( 63 kg ) 2

(

)

12. REASONING AND SOLUTION According to the impulse-momentum theorem (Equation 7.4)

( Σ F ) ∆t = m ( v f − v 0 )

(1)

Conservation of mechanical energy can be used to relate the velocities to the heights. If the floor is used to define the zero level for the heights, we have mgh0 = 21 mv B2

where h0 is the height of the ball when it is dropped and vB is the speed of the ball just before it strikes the ground. Solving for vB gives vB = 2 gh0

Similarly, mghf = 12 mvA2

(2)

350 IMPULSE AND MOMENTUM

where hf is the maximum height of the ball when it rebounds and vA is the speed of the ball just after it rebounds from the ground. Solving for vA gives

vA = 2 ghf

(3)

Substituting equations (2) and (3) into equation (1), where v0 = −vB, and vf = vA gives (taking "upward" as the positive direction)

( ΣF ) ∆t = m 2g + hf − ( − h0 )

(

)

= (0.500 kg) 2 9.80 m/s 2 + 0.700 m − ( − 1.20 m ) = +4.28 N ⋅ s Since the impulse is positive, it is directed upward .

13. SSM

WWW

REASONING The impulse applied to the golf ball by the floor can be

found from Equation 7.4, the impulse-momentum theorem: ( ΣF ) ∆t = mv f − mv 0 . Two

forces act on the golf ball, the average force F exerted by the floor, and the weight of the golf ball. Since F is much greater than the weight of the golf ball, the net average force ( ΣF ) is equal to F . Only the vertical component of the ball's momentum changes during impact with the floor. In order to use Equation 7.4 directly, we must first find the vertical components of the initial and final velocities. We begin, then, by finding these velocity components. SOLUTION The figures below show the initial and final velocities of the golf ball. Before impact

After impact vf sin 30.0°

v0 cos 30.0°

v0 30.0°

vf cos 30.0°

30.0° vf

v0 sin 30.0°

If we take up as the positive direction, then the vertical components of the initial and final velocities are, respectively, v 0y = −v0 cos 30.0° and vf y = +vf cos 30.0° . Then, from

Equation 7.4 the impulse is F ∆t = m( v fy − v 0y ) = m (+ vf cos 30.0°) – (– v0 cos 30.0°)

Chapter 7 Problems

351

Since v 0 = v f = 45 m / s , the impulse applied to the golf ball by the floor is

F∆t = 2mv0 cos 30.0° = 2(0.047 kg)(45 m/s)(cos 30.0°) = 3.7 N ⋅ s

14. REASONING This is a problem in vector addition, and we will use the component method for vector addition. Using this method, we will add the components of the individual momenta in the direction due north to obtain the component of the vector sum in the direction due north. We will obtain the component of the vector sum in the direction due east in a similar fashion from the individual components in that direction. For each jogger the momentum is the mass times the velocity. SOLUTION Assuming that the directions north and east are positive, the components of the joggers’ momenta are as shown in the following table:

Direction due east

Direction due north

85 kg jogger

b85 kg= 170 gb2.0kgm⋅ m/ s/gs

0 kg⋅m/s

55 kg jogger

b55 kg= g b3.0 m / sgcos 32 ° 140 kg ⋅ m / s

b55 kg= g b3.0 m / sgsin 32 ° 87 kg ⋅ m / s

Total

310 kg⋅m /s

87 kg⋅m/s

Using the Pythagorean theorem, we find that the magnitude of the total momentum is

b310 kg ⋅ m / sg+ b87 kg ⋅ m / sg = 2

2

322 kg ⋅ m / s

The total momentum vector points north of east by an angle θ, which is given by

θ = tan −1

F 87 kg ⋅ m / s I G H310 kg ⋅ m / s JK=

16 °

15. REASONING AND SOLUTION The excess weight of the truck is due to the force exerted on the truck by the sand. Newton's third law requires that this force be equal in magnitude to the force exerted on the sand by the truck. In time t, a mass m of sand falls into the truck bed and comes to rest. The impulse is

F∆t = m( vf − v 0 )

so

The sand gains a speed v0 in falling a height h so

F=

m( v f − v 0 ) ∆t

352 IMPULSE AND MOMENTUM

(

)

v0 = 2 gh = 2 9.80 m/s 2 ( 2.00 m ) = 6.26 m/s

The velocity of the sand just before it hits the truck is v0 = −6.26 m/s, where the downward direction is taken to be the negative direction. The final velocity of the sand is vf = 0 m/s. Thus, the average force exerted on the sand is

m F = ( vf − v 0 ) = ( 55.0 kg/s ) [( 0 m/s ) − ( −6.26 m/s ) ] = +344 N ∆t

16. REASONING The total momentum of the two-cart system is the sum of their individual momenta. In part a, therefore, we will use Pf = m1 v f1 + m2 v f2 to calculate the system’s total momentum Pf when both carts are rolling. In order to find the initial velocity v01 of the first cart in part b, we need to know the system’s initial momentum P0 = m1 v 01 + m2 v 02 . According to the principle of conservation of linear momentum, this is equal to the system’s final momentum Pf if the system is isolated. Let’s see if it is. The attractive magnetic forces felt by the carts are internal forces and do not, therefore, affect the application of the conservation principle. Friction is negligible, and the external gravitational force and normal forces balance out on the level track. Therefore, the net external force on the system is zero, and the system is indeed isolated. We conclude that the system’s initial momentum P0 is identical to the final momentum Pf found in part a. SOLUTION a. We calculate the system’s final momentum directly from the relation Pf = m1 v f1 + m2 v f2 , taking care to use the algebraic signs indicating the directions of the velocity and momentum vectors:

Pf = ( 2.3 kg )( +4.5 m/s ) + (1.5 kg )( −1.9 m/s ) = +7.5 kg ⋅ m/s b. The system’s initial momentum is the sum of the carts’ initial momenta: P0 = m1 v 01 + m2 v 02 . But the second cart is held initially at rest, so v02 = 0 m/s. Setting the initial momentum of the system equal to its final momentum, we have P0 = Pf = m1 v 01 . Solving for the first cart’s initial velocity, we obtain

v 01 =

Pf +7.5 kg ⋅ m/s = = +3.3 m/s m1 2.3 kg

Chapter 7 Problems

353

17. REASONING AND SOLUTION The collision is an inelastic one, with the total linear momentum being conserved: m1v1 = (m1 + m2)V The mass m2 of the receiver is

m2 =

m1v1 V

− m1 =

(115 kg ) ( 4.5 m/s ) − 115 kg = 2.6 m/s

84 kg

18. REASONING Since friction between the disks and the air-hockey table is negligible, and the weight of each disk is balanced by an upward-acting normal force, the net external force acting on the disks and spring is zero. Therefore, the two-disk system (including the spring) is an isolated system, and the total linear momentum of the system remains constant. With v01 and vf1 denoting the initial and final velocities of disk 1 and v02 and vf2 denoting the initial and final velocities of disk 2, the conservation of linear momentum states that

m1vf 1 + m2vf 2

=

Total momentum after spring is released

m1v01 + m2v02 Total momentum before spring is released

Solving this equation for vf2 yields the final velocity of disk 2. SOLUTION Using the fact that v01 = v02 = v0 (= +5.0 m/s), and remembering that disk 1 comes to a halt after the spring is released (vf1 = 0 m/s), we have vf 2 = =

19.

m1v01 + m2v02 − m1vf 1 m1v0 + m2v0 − m1vf1 = m2 m2

(1.2 kg )( +5.0 m/s ) + ( 2.4 kg )( +5.0 m/s ) − (1.2 kg )( 0 m/s ) = 2.4 kg

+7.5 m/s

SSM REASONING Let m be Al’s mass, which means that Jo’s mass is 168 kg – m. Since friction is negligible and since the downward-acting weight of each person is balanced by the upward-acting normal force from the ice, the net external force acting on the twoperson system is zero. Therefore, the system is isolated, and the conservation of linear momentum applies. The initial total momentum must be equal to the final total momentum. SOLUTION Applying the principle of conservation of linear momentum and assuming that the direction in which Al moves is the positive direction, we find

354 IMPULSE AND MOMENTUM

b gb

gb g b

gb

gb

m 0 m / s + 168 kg − m 0 m / s = m 0.90 m / s + 168 kg − m −1.2 m / s Initial total momentum

g

Final total momentum

Solving this equation for m, we find that

0 = m ( 0.90 m/s ) − (168 kg ) (1.2 m/s ) + m (1.2 m/s ) m=

(168 kg ) (1.2 m/s ) = 96 kg

0.90 m/s + 1.2 m/s

20. REASONING Friction is negligible and the downward-pointing weight of the wagon and its contents is balanced by the upward-pointing normal force from the ground. Therefore, the net external force acting on the wagon and its contents is zero, and the principle of conservation of linear momentum applies. The total linear momentum of the system, then, must remain the same before and after the rock is thrown. Let’s assume that m and v01 are, respectively the mass and initial velocity of the wagon, rider, and rock before the rock is thrown. In addition, let mrock be the mass of the rock, vf, rock be the final velocity of the rock after it is thrown, and vf be the final velocity of the wagon and rider (without the rock) after the rock is thrown. The conservation of linear momentum states that

( m − mrock ) vf + mrock vf, rock

=

m v01 Total momentum before rock is thrown

Total momentum after rock is thrown

Solving this equation for vf gives vf =

m v01 − mrock vf, rock m − mrock

SOLUTION In applying the momentum conservation principle, we assume that the forward direction is positive. Therefore the velocity of the wagon, rider, and rock before the rock is thrown is v01 = +0.500 m/s. Rock thrown forward: When the rock is thrown forward, its velocity is +16 m/s. The final velocity of the wagon is vf =

m v01 − mrock vf, rock m − mrock

=

( 95.0 kg )( +0.500 m/s ) − ( 0.300 kg )( +16.0 m/s ) = +0.451 m/s 95.0 kg − 0.300 kg

Chapter 7 Problems

355

The final speed of the wagon (which is the magnitude of its velocity) is 0.451 m/s . Rock thrown backward: When the rock is thrown backward, its velocity is −16.0 m/s. In this case the final velocity of the wagon is vf =

m v01 − mrock vf, rock m − mrock

=

( 95.0 kg )( +0.500 m/s ) − ( 0.300 kg )( −16.0 m/s ) = +0.552 m/s 95.0 kg − 0.300 kg

The final speed of the wagon is 0.552 m/s .

21. SSM REASONING The two-stage rocket constitutes the system. The forces that act to cause the separation during the explosion are, therefore, forces that are internal to the system. Since no external forces act on this system, it is isolated and the principle of conservation of linear momentum applies: m1 v f1 + m2 v f2 = ( m1 + m2 ) v 0 Total momentum after separation

Total momentum before separation

where the subscripts "1" and "2" refer to the lower and upper stages, respectively. This expression can be solved for vf1.

SOLUTION

v f1 = =

Solving for vf1 gives

( m1 + m2 ) v 0 − m2 v f2 m1 2400 kg + 1200 kg ( 4900 m / s) − (1200 kg)(5700 m / s) 2400 kg

= +4500 m / s

Since vf1 is positive, its direction is the same as the rocket before the explosion .

22. REASONING During the breakup¸ the linear momentum of the system is conserved, since the force causing the breakup is an internal force. We will assume that the +x axis is along the original line of motion (before the breakup), and the +y axis is perpendicular to this line and points upward. We will apply the conservation of linear momentum twice, once for the momentum components along the x axis and again for the momentum components along the y axis.

356 IMPULSE AND MOMENTUM

SOLUTION The mass of each piece of the rocket after breakup is m, and so the mass of the rocket before breakup is 2m. Applying the conservation of momentum theorem along the original line of motion (the x axis) gives mv1 cos 30.0° + mv2 cos 60.0° = 2mv0

or

v1 cos 30.0° + v2 cos 60.0° = 2v0

(1)

P0, x

Pf ,x

Applying the conservation of momentum along the y axis gives mv1 sin 30.0° − mv2 sin 60.0° = 0 P0, y

Pf , y

or

v2 =

v1 sin 30.0° sin 60.0°

(2)

a. To find the speed v1 of the first piece, we substitute the value for v2 from Equation (2) into Equation (1). The result is

v sin 30.0° v1 cos 30.0° + 1 cos 60.0° = 2v0 sin 60.0° Solving for v1 and setting v0 = 45.0 m/s yields v1 =

2v0 sin 30.0° cos 30.0° + cos 60.0° sin 60.0°

=

2 ( 45.0 m/s ) sin 30.0° cos 30.0° + cos 60.0° sin 60.0°

= 77.9 m/s

b. The speed v2 of the second piece can be found by substituting v1 = 77.9 m/s into Equation (2): v sin 30.0° ( 77.9 m/s ) sin 30.0° v2 = 1 = = 45.0 m/s sin 60.0° sin 60.0°

23. SSM REASONING No net external force acts on the plate parallel to the floor; therefore, the component of the momentum of the plate that is parallel to the floor is conserved as the plate breaks and flies apart. Initially, the total momentum parallel to the floor is zero. After the collision with the floor, the component of the total momentum parallel to the floor must remain zero. The drawing in the text shows the pieces in the plane parallel to the floor just after the collision. Clearly, the linear momentum in the plane parallel to the floor has two components; therefore the linear momentum of the plate must be conserved in each of these two mutually perpendicular directions. Using the drawing in the text, with the positive directions taken to be up and to the right, we have

Chapter 7 Problems

x direction

y direction

− m1 v 1 (sin 25.0 ° ) + m2 v 2 (cos 45.0 ° ) = 0

357

(1)

m1 v 1 (cos 25.0 ° ) + m2 v 2 (sin 45.0° ) – m3 v 3 = 0

(2)

These equations can be solved simultaneously for the masses m1 and m2. SOLUTION Using the values given in the drawing for the velocities after the plate breaks, we have, −m1 ( 3.00 m/s ) sin 25.0° + m2 (1.79 m/s ) cos 45.0° = 0 (1) m1 ( 3.00 m/s ) cos 25.0° + m2 (1.79 m/s ) sin 45.0° − (1.30 kg ) ( 3.07 m/s ) = 0 Subtracting (2) from (1), and noting that cos 45.0º = sin 45.0º, gives

(2)

m1 = 1.00 kg .

Substituting this value into either (1) or (2) then yields m2 = 1.00 kg .

24. REASONING We will divide the problem into two parts: (a) the motion of the freely falling block after it is dropped from the building and before it collides with the bullet, and (b) the collision of the block with the bullet. During the falling phase we will use an equation of kinematics that describes the velocity of the block as a function of time (which is unknown). During the collision with the bullet, the external force of gravity acts on the system. This force changes the momentum of the system by a negligibly small amount since the collision occurs over an extremely short time interval. Thus, to a good approximation, the sum of the external forces acting on the system during the collision is negligible, so the linear momentum of the system is conserved. The principle of conservation of linear momentum can be used to provide a relation between the momenta of the system before and after the collision. This relation will enable us to find a value for the time it takes for the bullet/block to reach the top of the building. SOLUTION Falling from rest (v0, block = 0 m/s), the block attains a final velocity vblock just before colliding with the bullet. This velocity is given by Equation 2.4 as vblock Final velocity of block just before bullet hits it

=

v0, block + a t Initial velocity of block at top of building

where a is the acceleration due to gravity (a = −9.8 m/s2) and t is the time of fall. The upward direction is assumed to be positive. Therefore, the final velocity of the falling block is vblock = a t (1)

358 IMPULSE AND MOMENTUM

During the collision with the bullet, the total linear momentum of the bullet/block system is conserved, so we have that

(m

bullet

+ mblock ) vf = mbullet vbullet + mblock vblock

(2)

Total linear momentum before collision

Total linear momentum after collision

Here vf is the final velocity of the bullet/block system after the collision, and vbullet and vblock are the initial velocities of the bullet and block just before the collision. We note that the bullet/block system reverses direction, rises, and comes to a momentary halt at the top of the building. This means that vf , the final velocity of the bullet/block system after the collision must have the same magnitude as vblock, the velocity of the falling block just before the bullet hits it. Since the two velocities have opposite directions, it follows that vf = −vblock. Substituting this relation and Equation (1) into Equation (2) gives

(m

bullet

+ mblock ) ( −a t ) = mbullet vbullet + mblock ( a t )

Solving for the time, we find that t=

− mbullet vbullet

a ( mbullet + 2mblock )

=

− ( 0.015 kg )( +810 m/s )

( −9.80 m/s ) ( 0.015 kg ) + 2 (1.8 kg ) 2

= 0.34 s

25. REASONING During the time that the skaters are pushing against each other, the sum of the external forces acting on the two-skater system is zero, because the weight of each skater is balanced by a corresponding normal force and friction is negligible. The skaters constitute an isolated system, so the principle of conservation of linear momentum applies. We will use this principle to find an expression for the ratio of the skater’s masses in terms of their recoil velocities. We will then obtain expressions for the recoil velocities by noting that each skater, after pushing off, comes to rest in a certain distance. The recoil velocity, acceleration, and distance are related by Equation 2.9 of the equations of kinematics.

SOLUTION While the skaters are pushing against each other, the total linear momentum of the two-skater system is conserved:

m1vf 1 + m2vf 2 = Total momentum after pushing

0 Total momentum before pushing

Solving this expression for the ratio of the masses gives

Chapter 7 Problems

m1 m2

=−

vf 2

359

(1)

vf 1

For each skater the (initial) recoil velocity vf , final velocity v, acceleration a, and displacement x are related by Equation 2.9 of the equations of kinematics: v 2 = vf2 + 2a x . Solving for the recoil velocity gives vf = ± v 2 − 2ax . If we assume that skater 1 recoils in the positive direction and skater 2 recoils in the negative direction, the recoil velocities are Skater 1

vf 1 = + v12 − 2a1 x1

Skater 2

vf 2 = − v22 − 2a2 x2

Substituting these expressions into Equation (1) gives − v22 − 2a2 x2 m1 = − = m2 v12 − 2a1 x1

v22 − 2a2 x2

(2)

v − 2a1 x1 2 1

Since the skaters come to rest, their final velocities are zero, so v1 = v2 = 0 m/s. We also

know that their accelerations have the same magnitudes. This means that a2 = −a1, where the minus sign denotes that the acceleration of skater 2 is opposite that of skater 1, since they are moving in opposite directions and are both slowing down. Finally, we are given that skater 1 glides twice as far as skater 2. Thus, the displacement of skater 1 is related to that of skater 2 by x1 = −2x2 , where, the minus sign denotes that the skaters move in opposite directions. Substituting these values into Equation (2) yields m1 = m2

( 0 m/s )

2

( 0 m/s )

2

− 2 ( −a1 )( x2 ) − 2a1 ( −2 x2 )

=

1 = 0.707 2

26. REASONING AND SOLUTION Since no net external force acts in the horizontal direction, the total horizontal momentum of the system is conserved regardless of which direction the mass is thrown. The momentum of the system before the mass is thrown off the wagon is mWvA, where mW and vA are the mass and velocity of the wagon, respectively. When ten percent of the wagon’s mass is thrown forward, the wagon is brought to a halt so its final momentum is zero. The momentum of the mass thrown forward is 0.1mW(vA + vM), where vM is the velocity of the mass relative to the wagon and (vA + vM) is the velocity of the mass relative to the ground. The conservation of momentum gives

360 IMPULSE AND MOMENTUM

0.1mW ( vA + vM ) = Momentum after mass is thrown forward

Solving for vM yields

mW vA Momentum before mass is thrown forward

vM = 9vA

(1)

When the direction in which the mass is thrown is reversed, the velocity of the mass relative to the ground is now (vA − vM). The momentum of the mass and wagon is, therefore, 0.1mW ( vA − vM ) + Momentum of the mass

0.9mW vB Momentum of the wagon

where vB is the velocity of the wagon. The conservation of momentum gives 0.1mW ( vA − vM ) + 0.9mW vB = Momentum after mass is thrown off

mW vA

(2)

Momentum before mass is thrown off

Substituting Equation (1) into Equation (2) and solving for vB/vA gives vB/vA = 2 .

27. SSM WWW REASONING The cannon and the shell constitute the system. Since no external force hinders the motion of the system after the cannon is unbolted, conservation of linear momentum applies in that case. If we assume that the burning gun powder imparts the same kinetic energy to the system in each case, we have sufficient information to develop a mathematical description of this situation, and solve it for the velocity of the shell fired by the loose cannon.

SOLUTION For the case where the cannon is unbolted, momentum conservation gives

m1vf1 + m2vf2 = Total momentum after shell is fired

0

(1)

Initial momentum of system

where the subscripts "1" and "2" refer to the cannon and shell, respectively. In both cases, the burning gun power imparts the same kinetic energy to the system. When the cannon is bolted to the ground, only the shell moves and the kinetic energy imparted to the system is 2 KE = 12 mshellvshell = 12 (85.0 kg)(551 m/s) 2 = 1.29 ×107 J

The kinetic energy imparted to the system when the cannon is unbolted has the same value and can be written using the same notation as in equation (1):

Chapter 7 Problems

2 2 KE = 12 m1vf1 + 12 m2vf2

361

(2)

Solving equation (1) for vf1, the velocity of the cannon after the shell is fired, and substituting the resulting expression into Equation (2) gives KE =

2 m22 vf2

2m1

2 + 12 m2vf2

(3)

Solving equation (3) for vf2 gives

vf2 =

2KE 2(1.29 × 107 J) = = +547 m/s m2 85.0 kg + 1 (85.0 kg) + 1 m2 3 m × 5.80 10 kg 1

28. REASONING Together, Ashley and Miranda constitute an isolated system, since their combined weight is balanced by an upward normal force, and friction is negligible. The total momentum of the system is, therefore, conserved when Miranda hops onto the tube. We will use the momentum conservation principle m1vf1 + m2 vf2 = m1v01 + m2 v02 (Equation 7.7b) to analyze this one-dimensional collision. We are ignoring the mass and momentum of the inner tube.

SOLUTION After Miranda (m2 = 58 kg) jumps onto the inner tube, she and Ashley (m1 = 71 kg) both have the same final velocity: vf = vf1 = vf2. Making this substitution in Equation 7.7b, and solving for their common final velocity, we obtain

m1vf + m2vf Total momentum after Miranda hops on

=

m1v01 + m2v02

or

vf =

Total momentum before Miranda hops on

m1v01 + m2v02 m1 + m2

Their common velocity after Miranda hops on is, therefore, vf =

m1v01 + m2 v02 ( 71 kg )( +2.7 m/s ) + ( 58 kg )( +4.5 m/s ) = = +3.5 m/s 71 kg + 58 kg m1 + m2

The common speed is the magnitude of this value or 3.5 m/s .

29. REASONING Since the collision is an elastic collision, both the linear momentum and kinetic energy of the two-vehicle system are conserved. The final velocities of the car and van are given in terms of the initial velocity of the car by Equations 7.8a and 7.8b.

362 IMPULSE AND MOMENTUM

SOLUTION a. The final velocity vf1 of the car is given by Equation 7.8a as m − m2 vf 1 = 1 v m + m 01 1 2

where m1 and m2 are, respectively, the masses of the car and van, and v01 is the initial velocity of the car. Thus,

715 kg − 1055 kg vf 1 = ( +2.25 m/s ) = −0.432 m/s 715 kg + 1055 kg b. The final velocity of the van is given by Equation 7.8b: 2m1 2 ( 715 kg ) vf 2 = v01 = ( +2.25 m/s ) = +1.82 m/s m +m + 715 kg 1055 kg 2 1

30. REASONING Since momentum is conserved, the total momentum of the two-object system after the collision must be the same as it was before the collision. Momentum is mass times velocity. Since one of the objects is at rest initially, the total initial momentum comes only from the moving object. Let m1 and v01 be, respectively, the mass and initial velocity of the moving object before the collision. In addition, m and vf are the total mass and final velocity of the two objects (which stick together) after the collision. The conservation of linear momentum can be written as

=

mvf Total momentum after the collision

m1v01 Total momentum before the collision

Solving this equation for vf, the final velocity of the two-object system gives vf =

m1v01 m

SOLUTION Large-mass object (8.0 kg) moving initially: Assume that, before the collision, the object is moving in the + direction so that v01 = +25 m/s. Then,

vf =

m1v01 (8.0 kg )( +25 m/s ) = = +18 m/s 3.0 kg + 8.0 kg m

363

Chapter 7 Problems

The final speed is 18 m/s . Small-mass object (3.0 kg) moving initially:

vf =

m1v01 ( 3.0 kg )( +25 m/s ) = = +6.8 m/s 3.0 kg + 8.0 kg m

The final speed is 6.8 m/s .

31. SSM REASONING We obtain the desired percentage in the usual way, as the kinetic energy of the target (with the projectile in it) divided by the projectile’s incident kinetic energy, multiplied by a factor of 100. Each kinetic energy is given by Equation 6.2 as

1 mv 2 , 2

where m and v are mass and speed, respectively. Data for the masses are given, but the speeds are not provided. However, information about the speeds can be obtained by using the principle of conservation of linear momentum.

SOLUTION We define the following quantities: KETP = kinetic energy of the target with the projectile in it KE0P = kinetic energy of the incident projectile mP = mass of incident projectile = 0.20 kg mT = mass of target = 2.50 kg vf = speed at which the target with the projectile in it flies off after being struck v0P = speed of incident projectile The desired percentage is

KE TP ×100 = Percentage = KE 0P

1 2

( mT + mP ) vf2 1 m v2 2 P 0P

×100 %

(1)

According to the momentum-conservation principle, we have

( mT + mP ) vf Total momentum of target and projectile after target is struck

=

0 + mP v0P Total momentum of target and projectile before target is struck

Note that the target is stationary before being struck and, hence, has zero initial momentum. Solving for the ratio vf /v0P, we find that

364 IMPULSE AND MOMENTUM

vf mP = v0P mT + mP Substituting this result into Equation (1) gives Percentage =

=

1 2

( mT + mP ) vf2 1 m v2 2 P 0P

×100 % =

1 2

( mT + mP ) 1 m 2 P

2

× 100 % mT + mP mP

mP 0.20 kg × 100 % = ×100 % = 7.4 % 2.50 kg + 0.20 kg mT + mP

32. REASONING The weight of each vehicle is balanced by the normal force exerted by the road. Assuming that friction and other resistive forces can be ignored, we will treat the twovehicle system as an isolated system and apply the principle of conservation of linear momentum.

SOLUTION Using v0, car and v0, SUV to denote the velocities of the vehicles before the collision and applying the principle of conservation of linear momentum, we have

0

= mcar v0, car + mSUV v0, SUV Total momentum before collision

Total momentum after collision

Note that the total momentum of both vehicles after the collision is zero, because the collision brings each vehicle to a halt. Solving this result for v0, SUV and taking the direction in which the car moves as the positive direction gives v0, SUV =

− mcar v0, car mSUV

=

− (1100 kg )( 32 m/s ) 2500 kg

= −14 m/s

This result is negative, since the velocity of the sport utility vehicle is opposite to that of the car, which has been chosen to be positive. The speed of the sport utility vehicle is the magnitude of v0, SUV or 14 m/s .

33. SSM WWW REASONING The system consists of the two balls. The total linear momentum of the two-ball system is conserved because the net external force acting on it is zero. The principle of conservation of linear momentum applies whether or not the collision is elastic.

Chapter 7 Problems

365

m1 v f1 + m2 v f2 = m1 v 01 + 0 Total momentum after collision

Total momentum before collision

When the collision is elastic, the kinetic energy is also conserved during the collision 1 2

m1 v f12 + 21 m2 v f22 = Total kinetic energy after collision

1 2

2 m1 v 01 +0

Total kinetic energy before collision

SOLUTION a. The final velocities for an elastic collision are determined by simultaneously solving the above equations for the final velocities. The procedure is discussed in Example 7 in the text, and leads to Equations 7.8a and 7.8b. According to Equation 7.8:

v f1 =

Fm − m Iv G Hm + m JK 1

2

01

1

and

v f2 =

2

F 2 m Iv G Hm + m JK 1

01

1

2

Let the initial direction of motion of the 5.00-kg ball define the positive direction. Substituting the values given in the text, these equations give

F 5.00 kg – 7 .50 kg I G H5.00 kg + 7.50 kg JK(2.00 m / s) = F 2(5.00 kg) IJ(2.00 m / s) = =G H5.00 kg + 7.50 kg K

5.00 - kg ball

v f1 =

–0.400 m / s

7.50 - kg ball

v f2

+1.60 m / s

The signs indicate that, after the collision, the 5.00-kg ball reverses its direction of motion, while the 7.50-kg ball moves in the direction in which the 5.00-kg ball was initially moving. b. When the collision is completely inelastic, the balls stick together, giving a composite body of mass m1 + m2 that moves with a velocity vf . The statement of conservation of linear momentum then becomes ( m1 + m2 ) v f = m1 v 01 + 0 Total momentum after collision

Total momentum before collision

The final velocity of the two balls after the collision is, therefore, vf =

m1 v 01 m1 + m2

=

(5.00 kg)(2.00 m / s) = +0.800 m / s 5.00 kg + 7.50 kg

366 IMPULSE AND MOMENTUM

34. REASONING The net external force acting on the two-puck system is zero (the weight of each ball is balanced by an upward normal force, and we are ignoring friction due to the layer of air on the hockey table). Therefore, the two pucks constitute an isolated system, and the principle of conservation of linear momentum applies. SOLUTION Conservation of linear momentum requires that the total momentum is the same before and after the collision. Since linear momentum is a vector, the x and y components must be conserved separately. Using the drawing in the text, momentum conservation in the x direction yields mA v0A = mA vfA ( cos 65° ) + mBvfB ( cos 37° )

(1)

while momentum conservation in the y direction yields 0 = mA vfA ( sin 65° ) − mBvfB ( sin 37° )

(2)

Solving equation (2) for vfB, we find that vfB =

mA vfA ( sin 65° ) mB ( sin 37° )

(3)

Substituting equation (3) into Equation (1) leads to m v ( sin 65° ) mA v0A = mA vfA ( cos 65° ) + A fA ( cos 37° ) sin 37°

a. Solving for vfA gives vfA =

v0A +5.5 m/s = = 3.4 m/s sin 65° sin 65° cos 65° + cos 65° + tan 37° tan 37°

b. From equation (3), we find that vfB =

( 0.025 kg )( 3.4 m/s )( sin 65° ) = ( 0.050 kg )( sin 37° )

2.6 m/s

35. SSM REASONING Batman and the boat with the criminal constitute the system. Gravity acts on this system as an external force; however, gravity acts vertically, and we are concerned only with the horizontal motion of the system. If we neglect air resistance and friction, there are no external forces that act horizontally; therefore, the total linear

Chapter 7 Problems

367

momentum in the horizontal direction is conserved. When Batman collides with the boat, the horizontal component of his velocity is zero, so the statement of conservation of linear momentum in the horizontal direction can be written as

(m1 + m2 )vf

=

Total horizontal momentum after collision

m1v01 + 0 Total horizontal momentum before collision

Here, m1 is the mass of the boat, and m2 is the mass of Batman. This expression can be solved for vf , the velocity of the boat after Batman lands in it. SOLUTION Solving for vf gives vf =

m1v01

m1 + m2

=

( 510 kg )( +11 m/s ) = 510 kg + 91 kg

+9.3 m/s

The plus sign indicates that the boat continues to move in its initial direction of motion.

36. REASONING According to the momentum-conservation principle, the final total momentum is the same as the initial total momentum. The initial total momentum is the vector sum of the two initial momentum vectors of the objects. One of the vectors (p0A) points due east and one (p0B) due north, so that they are perpendicular (see the drawing on the left). North

North

p0B

p0A

East

Pf

θ p0A

East

p0B

Based on the conservation principle, the direction of the final total momentum Pf must be the same as the direction of the initial total momentum. From the drawing on the right, we can see that the initial total momentum has a component p0A pointing due east and a component p0B pointing due north. The final total momentum Pf has these same components and, therefore, must point north of east at an angle θ.

368 IMPULSE AND MOMENTUM

SOLUTION Based on the conservation of linear momentum, we know that the magnitude of the final total momentum is the same as the magnitude of the initial total momentum. Using the Pythagorean theorem with the initial momentum vectors of the two objects, we find that the magnitude of the final total momentum is

( mAv0A )2 + ( mBv0B )2

Pf =

2 2 p0A + p0B =

Pf =

(17.0 kg )2 (8.00 m/s )2 + ( 29.0 kg )2 ( 5.00 m/s )2

= 199 kg ⋅ m/s

The direction is given by

p0B −1 mBv0B −1 ( 29.0 kg )( 5.00 m/s ) = tan = tan = 46.8° north of east (17.0 kg )(8.00 m/s ) p0A mAv0A

θ = tan −1

37. REASONING The velocity of the second ball just after the collision can be found from Equation 7.8b (see Example 7). In order to use Equation 7.8b, however, we must know the velocity of the first ball just before it strikes the second ball. Since we know the impulse delivered to the first ball by the pool stick, we can use the impulse-momentum theorem (Equation 7.4) to find the velocity of the first ball just before the collision. SOLUTION According to the impulse-momentum theorem, F ∆t = mvf − mv0 , and setting v0 = 0 m/s and solving for vf , we find that the velocity of the first ball after it is struck by the pool stick and just before it hits the second ball is

vf =

F ∆t +1.50 N ⋅ s = = 9 .09 m / s m 0.165 kg

Substituting values into Equation 7.8b (with v01 = 9.09 m/s), we have 2m1 2m vf2 = v = v = v = +9.09 m/s m + m 01 m + m 01 01 2 1

38. REASONING a. Since air resistance is negligible as the ball swings downward, the work done by nonconservative forces is zero, Wnc = 0 J. The force due to the tension in the wire is perpendicular to the motion and, therefore, does no work. Thus, the total mechanical energy, which is the sum of the kinetic and potential energies, is conserved (see Section 6.5). The

Chapter 7 Problems

369

conservation of mechanical energy will be used to find the speed of the ball just before it collides with the block. b. When the ball collides with the stationary block, the collision is elastic. This means that, during the collision, the total kinetic energy of the system is conserved. The horizontal or x-component of the total momentum is conserved, because the horizontal surface is frictionless, and so the net average external force acting on the ball-block system in the horizontal direction is zero. The total kinetic energy is conserved, since the collision is known to be elastic. SOLUTION a. As the ball falls, the total mechanical energy is conserved. Thus, the total mechanical energy at the top of the swing is equal to that at the bottom: m1 gL Total mechanical energy at top of swing, all potential energy

=

1 2

m1v 2

Total mechanical energy at bottom of swing, all kinetic energy

In this expression L is the length of the wire, m1 is the mass of the ball, g is the magnitude of the acceleration due to gravity, and v is the speed of the ball just before the collision. We have chosen the zero-level for the potential energy to be at ground level, so the initial potential energy of the ball is m1gL. Solving for the speed v of the ball gives v = 2 gL = 2 ( 9.80 m/s 2 ) (1.20 m ) = 4.85 m/s

At the bottom of the swing the ball is moving horizontally and to the right, which we take to be the +x direction (see the drawing in the text). Thus the velocity of the ball just before impact is vx = +4.85 m/s . b. The conservation of linear momentum and the conservation of the total kinetic energy can be used to describe the behavior of the system during the elastic collision. This situation is identical to that in Example 7 in Section 7.3, so Equation 7.8a applies:

m − m2 vf = 1 vx m + m 1 2 where vf is the final velocity of the ball after the collision, m1 and m2 are, respectively, the masses of the ball and block, and vx is the velocity of the ball just before the collision. Since vx = +4.85 m/s, we find that

370 IMPULSE AND MOMENTUM

1.60 kg − 2.40 kg vf = ( +4.85 m/s ) = −0.97 m/s 1.60 kg + 2.40 kg The minus sign indicates that the ball rebounds to the left after the collision.

39. REASONING AND SOLUTION The total linear momentum of the two-car system is conserved because no net external force acts on the system during the collision. We are ignoring friction during the collision, and the weights of the cars are balanced by the normal forces exerted by the ground. Momentum conservation gives (m1 + m2 )vf = m1v01 + m2v02 Total momentum after collision

Total momentum after collision

where v02 = 0 m/s since the 1900-kg car is stationary before the collision. a. Solving for vf , we find that the velocity of the two cars just after the collision is vf =

m1v01 + m2 v02 m1 + m2

=

( 2100 kg )( +17 m/s ) + (1900 kg )( 0 m/s ) = 2100 kg + 1900 kg

+8.9 m/s

The plus sign indicates that the velocity of the two cars just after the collision is in the same direction as the direction of the velocity of the 2100-kg car before the collision. b. According to the impulse-momentum theorem, Equation 7.4, we have F ∆t = (m1 + m2 )vfinal − (m1 + m2 )vafter Impulse due to friction

Final momentum when cars come to a halt

Total momentum just after collision

where vfinal = 0 m/s since the cars come to a halt, and vafter = vf = +8.9 m/s . Therefore, we have

F ∆t = ( 2100 kg + 1900 kg )( 0 m/s ) – ( 2100 kg + 1900 kg )( 8.9 m/s ) = – 3.6 × 104 N ⋅ s The minus sign indicates that the impulse due to friction acts opposite to the direction of motion of the locked, two-car system. This is reasonable since the velocity of the cars is decreasing in magnitude as the cars skid to a halt. c. Using the same notation as in part (b) above, we have from the equations of kinematics (Equation 2.9) that 2 2 vfinal = vafter + 2ax

Chapter 7 Problems

371

where vfinal = 0 m/s and vafter = vf = +8.9 m/s . From Newton's second law we have that

a = – f k / ( m1 + m2 ) , where f is the force of kinetic friction that acts on the cars as they skid k to a halt. Therefore, – fk 0 = v +2 x m + m 1 2 2 f

or

x=

( m1 + m2 ) vf2 2 fk

According to Equation 4.8, f k = µ k FN , where µk is the coefficient of kinetic friction and FN is the magnitude of the normal force that acts on the two-car system. There are only two vertical forces that act on the system; they are the upward normal force FN and the weight (m1 + m2)g of the cars. Taking upward as the positive direction and applying Newton's second law in the vertical direction, we have FN – (m1 + m2 ) g = (m1 + m2 )a y = 0 , or FN = (m1 + m2 ) g . Therefore, f k = µk (m1 + m2 ) g , and we have

x=

(m1 + m2 )vf2

2 µ k (m1 + m2 ) g

=

vf2

2µk g

=

(8.9 m/s) 2 = 5.9 m 2(0.68)(9.80 m/s 2 )

40. REASONING Considering the boat and the stone as a single system, there is no net external horizontal force acting on the stone-boat system, and thus the horizontal component of the system’s linear momentum is conserved: m1vf 1x + m2 vf2x = m1v01x + m2 v02x (Equation 7.9a). We have taken east as the positive direction.

SOLUTION We will use the following symbols in solving the problem: m1 = mass of the stone = 0.072 kg vf1 = final speed of the stone = 11 m/s v01 = initial speed of the stone = 13 m/s m2 = mass of the boat vf2 = final speed of the boat = 2.1 m/s v02 = initial speed of the boat = 0 m/s Because the boat is initially at rest, v02x = 0 m/s, and Equation 7.9a reduces to m1vf 1x + m2 vf2x = m1v01x . Solving for the mass m2 of the boat, we obtain

m2vf2x = m1v01x − m1vf1x

or

m2 =

m1 ( v01x − vf1x ) vf2x

372 IMPULSE AND MOMENTUM

As noted above, the boat’s final velocity is horizontal, so vf2x = vf2. The horizontal components of the stone’s initial and final velocities are, respectively, v01x = v01 cos 15° and vf1x = vf1 cos 12° (see the drawing). Thus, the mass of the boat must be

m2 =

m1 ( v01x − vf1x ) vf2

=

vf1 v01x

12°

vf1x

15°

v01 Initial and final velocities of the stone

( 0.072 kg ) (13 m/s ) cos15 − (11 m/s ) cos12 2.1 m/s

= 0.062 kg

41. SSM REASONING The two skaters constitute the system. Since the net external force acting on the system is zero, the total linear momentum of the system is conserved. In the x direction (the east/west direction), conservation of linear momentum gives Pf x = P0 x , or ( m1 + m2 ) v f cosθ = m1 v 01 y

Note that since the skaters hold onto each other, they move away with a common velocity vf. In the y direction, Pf y = P0 y , or

p

( m1 + m2 ) v f sin θ = m2 v 02

These equations can be solved simultaneously to obtain both the angle θ and the velocity vf.

p

1

SOLUTION a. Division of the equations above gives

θ = tan −1

Fm v I = tan L(70.0 kg)(7.00 m / s) O= 73.0° G Hm v JK M Q N(50.0 kg)(3.00 m / s) P 2

−1

02

1 01

b. Solution of the first of the momentum equations gives vf =

m1 v 01 ( m1 + m2 ) cosθ

=

(50.0 kg)(3.00 m / s) = 4.28 m / s (50.0 kg + 70.0 kg)(cos 73.0 ° )

2

x

θ

P

f

Chapter 7 Problems

373

42. REASONING +355 m/s a. The conservation of linear Block 1 Block 2 momentum can be applied to this three-object system (the bullet and (a) Before collisions the two blocks), even though the second collision occurs later than the first one. Since there is no vblock 2 +0.550 m/s friction between the blocks and the horizontal surface, air resistance is negligible, and the weight of each mblock 2 = 1530 g mblock 1 = 1150 g block is balanced by a normal mbullet = 4.00 g force, the net external force acting on this system is zero, and the (b) After collisions conservation of linear momentum applies. This principle will allow us to determine the velocity of the second block after the bullet imbeds itself. b. The total kinetic energy of the three-body system is not conserved. Both collisions are inelastic, and the collision with block 2 is completely inelastic since the bullet comes to rest within the block. As with any inelastic collision, the total kinetic energy after the collisions is less than that before the collisions. SOLUTION a. The conservation of linear momentum states that the total momentum of the system after the collisions [see part (b) of the drawing] is equal to that before the collisions [part (a) of the drawing]:

mblock 1vblock 1 + ( mblock 2 + mbullet ) vblock 2 = Total momentum after collisions

mbullet vbullet Total momentum before collisions

Solving for the velocity vblock 2 of block 2 after the collisions gives

vblock 2 =

mbullet vbullet − mblock 1vblock 1 mblock 2 + mbullet

4.00 × 10−3 kg ) ( +355 m/s ) ( =

− (1.150 kg )( +0.550 m/s )

1.530 kg + 4.00 × 10−3 kg

= +0.513 m/s

b. The ratio of the total kinetic energy (KE) after the collision to that before the collision is

374 IMPULSE AND MOMENTUM

KEafter = KE before

=

1m v2 2 block 1 block 1

+

1 2

2 ( mblock 2 + mbullet ) vblock 2

1m v2 2 bullet bullet

1 2

(1.530 kg + 4.00 × 10−3 kg ) ( 0.513 m/s)2 = 1.49 × 10−3 1 4.00 × 10−3 kg 355 m/s 2 ) )( 2(

(1.150 kg )( 0.550 m/s )2

+

1 2

43. REASONING The ratio of the kinetic energy of the hydrogen atom after the collision to that of the electron before the collision is KE hydrogen, after collision KE electron, before collison

=

1 2

2 mH vf,H

1 2

2 me v0,e

where vf,H is the final speed of the hydrogen atom, and v0,e is the initial speed of the electron. The ratio mH/me of the masses is known. Since the electron and the stationary hydrogen atom experience an elastic head-on collision, we can employ Equation 7.8b to determine how vf,H is related to v0,e.

SOLUTION According to Equation 7.8b, the final speed vf,H of the hydrogen atom after the collision is related to the initial speed v0,e of the electron by

2me vf,H = m + m H e

v0,e

Substituting this expression into the ratio of the kinetic energies gives

KE hydrogen, after collision KE electron, before collison

=

1 2

2 mH vf,H

1 2

2 me v0,e

2me mH m + m H e = me

2

The right hand side of this equation can be algebraically rearranged to give

KE hydrogen, after collision KE electron, before collison

m = H m e

2 m 1+ H me

2

2

2 −3 = (1837 ) 1 + 1837 = 2.175 × 10

Chapter 7 Problems +y

44. REASONING For use in our solution we define the following masses and initial speeds of the bullets: Mass

Initial Speed −3

Bullet 1

m1 = 4.50 × 10

Bullet 2

m2 = 4.50 × 10−3 kg v02 = 324 m/s

Bullet 3

kg v01 = 324 m/s

m3 = ?

375

m3 m1 30.0°

v03 v01

30.0° +x v02

v03 = 575 m/s

m2

The drawing shows the bullets just after they are fired from the guns. They collide at the origin of the x, y axes and form a stationary lump. Assuming that the momentumconservation principle applies, we can conclude that the total momentum of the three-bullet system is the same after the bullets collide as before they collide. Applying this principle will allow us to determine the unknown mass. SOLUTION Applying the momentum-conservation principle separately in terms of the x and y components of the total momentum, we have

= m1v01 cos 30.0° − m3v03 cos 30.0°

0

x Component

x component of total momentum after collision

= −m1v01 sin 30.0° − m3v03 sin 30.0° + m2v2

0

y Component

x component of total momentum before collision

y component of total momentum after collision

y component of total momentum before collision

Both the x and y components of the total momentum of the three-bullet system are zero after the collision, since the bullets form a stationary lump. The x component of the initial momentum of bullet 1 is positive and the y component is negative, because this bullet is fired to the right and downward in the drawing. The x and y components of the initial momentum of bullet 3 are negative, because this bullet is fired to the left and downward in the drawing. Either of the two equations presented above can be solved for the unknown mass m3. From the equation for the x component of the total momentum, we find that m3 =

m1v01 v03

4.50 × 10−3 kg ) ( 324 m/s ) ( = = 2.54 ×10−3 kg 575 m/s

45. SSM REASONING The two balls constitute the system. The tension in the wire is the only nonconservative force that acts on the ball. The tension does no work since it is perpendicular to the displacement of the ball. Since Wnc = 0 J, the principle of conservation of mechanical energy holds and can be used to find the speed of the 1.50-kg ball just before

376 IMPULSE AND MOMENTUM

the collision. Momentum is conserved during the collision, so the principle of conservation of momentum can be used to find the velocities of both balls just after the collision. Once the collision has occurred, energy conservation can be used to determine how high each ball rises. SOLUTION a. Applying the principle of energy conservation to the 1.50-kg ball, we have 1 2

mv f2 + mghf = 21 mv 02 + mgh0 Ef

E0

If we measure the heights from the lowest point in the swing, hf = 0 m, and the expression above simplifies to 1 mv f2 = 21 mv 02 + mgh0 2 Solving for vf , we have

v f = v 02 + 2 gh0 = ( 5.00 m / s) 2 + 2(9.80 m / s 2 )(0.300 m) = 5.56 m / s b. If we assume that the collision is elastic, then the velocities of both balls just after the collision can be obtained from Equations 7.8a and 7.8b: v f1 =

Fm − m Iv G Hm + m JK 1

2

01

1

and

v f2 =

2

F 2 m Iv G Hm + m JK 1

01

1

2

Since v01 corresponds to the speed of the 1.50-kg ball just before the collision, it is equal to the quantity vf calculated in part (a). With the given values of m1 = 1.50 kg and m2 = 4 .60 kg , and the value of v 01 = 5.56 m / s obtained in part (a), Equations 7.8a and 7.8b yield the following values:

v f1 = –2.83 m / s

and

v f2 = +2.73 m / s

The minus sign in vf1 indicates that the first ball reverses its direction as a result of the collision. c. If we apply the conservation of mechanical energy to either ball after the collision we have 1 mv f2 + mgh f = 21 mv 02 + mgh0 2 Ef

E0

Chapter 7 Problems

377

where v0 is the speed of the ball just after the collision, and hf is the final height to which the ball rises. For either ball, h0 = 0 m, and when either ball has reached its maximum height, vf = 0 m/s. Therefore, the expression of energy conservation reduces to

ghf = 21 v 02

or

hf =

v 02 2g

Thus, the heights to which each ball rises after the collision are 1.50 - kg ball

hf =

4.60 - kg ball

hf =

v 02 2g v 02 2g

=

(2.83 m / s) 2 = 0.409 m 2 (9.80 m / s 2 )

=

(2.73 m / s) 2 = 0.380 m 2 (9.80 m / s 2 )

46. REASONING AND SOLUTION During the elastic collision both the total kinetic energy and the total momentum of the balls are conserved. The conservation of momentum gives mvf1 + mvf2 = mv01 + mv02 where ball 1 has v01 = + 7.0 m/s and ball 2 has v02 = – 4.0 m/s. Hence, vf1 + vf2 = v01 + v02 = 3.0 m/s The conservation of kinetic energy gives 2 2 2 2 (1/2)mvf1 + (1/2)mvf2 = (1/2)mv01 + (1/2)mv02

or vf12 + vf22 = v012 + v022 = 65 m2/s2 Solving the first equation for vf1 and substituting into the second gives 2vf22 – (6.0 m/s)vf2 – 56 m2/s2 = 0 The quadratic formula yields two solutions, vf2 = 7.0 m/s and vf2 = – 4.0 m/s. The first equation now gives vf1 = – 4.0 m/s and vf1 = + 7.0 m/s. Hence, the 7.0-m/s-ball has a final velocity of −4.0 m/s , opposite its original direction.

378 IMPULSE AND MOMENTUM

The 4.0-m/s-ball has a final velocity of +7.0 m/s , opposite its original direction.

47. REASONING AND SOLUTION Initially, the ball has a total mechanical energy given by E0 = mgh0. After one bounce it reaches the top of its trajectory with an energy of E1 = 0.900 E0 = 0.900 mgh0 After two bounces it has energy E2 = (0.900)2mgh0 After N bounces it has a remaining energy of EN = (0.900)Nmgh0 In order to just reach the sill the ball must have EN = mgh where h = 2.44 m. Hence, N

(0.900) mgh0 = mgh

or

N

(0.900) = h/h0

Taking the log of both sides gives N log(0.900) = log(h/h0) Then

2.44 m log 6.10 m N= = 8.7 log (0.900) The ball can make 8 bounces and still reach the sill.

48. REASONING We will use the relation m1 x1 + m2 x2 xcm = (Equation 7.10) to determine m1 + m2 the mass of the smaller star. Let the line joining the centers of the two stars be the x axis, and let the position of the larger star’s center be x1 = 0 m (see the drawing). Then the position of the smaller star’s center is x2 = x1 + d = d, where d is the distance separating the centers of the stars.

d

x

xcm x1 = 0 m

x2

SOLUTION With the substitutions x1 = 0 m and x2 = d, Equation 7.10 simplifies to

Chapter 7 Problems

xcm =

379

0 + m2 d m2 d = m1 + m2 m1 + m2

Solving this relation for the mass m2 of the smaller star, we obtain

m1 + m2 =

m2d xcm

or

m2 =

m1 d −1 xcm

Thus, the mass of the smaller star is m2 =

m1 3.70 ×1030 kg = = 1.51×1030 kg 11 d 7.17 10 m × −1 −1 xcm 2.08 ×1011 m

49. SSM REASONING AND SOLUTION The velocity of the center of mass of a system is given by Equation 7.11. Using the data and the results obtained in Example 5, we obtain the following: a. The velocity of the center of mass of the two-car system before the collision is

( vcm )before =

m1v01 + m2v02

=

m1 + m2

(65 ×103 kg)( + 0.80 m/s) + (92 × 103 kg)( +1.2 m/s) 65 × 103 kg + 92 ×103 kg

= +1.0 m/s

b. The velocity of the center of mass of the two-car system after the collision is

( vcm )after =

m1vf + m2vf m1 + m2

= vf = +1.0 m/s

c. The answer in part (b) should be the same as the common velocity vf. Since the cars are coupled together, every point of the two-car system, including the center of mass, must move with the same velocity.

380 IMPULSE AND MOMENTUM

50. REASONING Using Equation 7.10, we can calculate the location of the center of mass of John and Barbara: m x + mB xB xcm = J J mJ + mB By calculating John and Barbara’s center of mass before and after they change positions, we can determine how far and in what direction their center of mass move as a result of the switch. SOLUTION

Before

xcm =

mJ xJ + mB xB (86 kg )( 9.0 m ) + ( 55 kg )( 2.0 m ) = = 6.3 m mJ + mB 86 kg + 55 kg

After

xcm =

mJ xJ + mB xB (86 kg )( 2.0 m ) + ( 55 kg )( 9.0 m ) = = 4.7 m 86 kg + 55 kg mJ + mB

The center of mass moves by an amount 6.3 m − 4.7 m = 1.6 m . Since it moves from the 6.3-m point to the 4.7-m point, the center of mass moves toward the origin.

51. REASONING AND SOLUTION The velocity of the center of mass is given by Equation 7.11, m v + m2v2 ( m1 / m2 ) v1 + v2 vcm = 1 1 = m1 + m2 ( m1 / m2 ) + 1 a.

When the masses are equal, m1 / m2 = 1 , and we have

vcm =

v1 + v2 2

=

9.70 m/s + (–11.8 m/s) = –1.05 m/s 2

b. When the mass of the ball moving at 9.70 m/s is twice the mass of the other ball, we have m1/m2 = 2, and the velocity of the center of mass is vcm =

2v1 + v2 2 +1

=

2(9.70 m/s) + ( –11.8 m/s) = +2.53 m/s 3

Chapter 7 Problems

381

52. REASONING The coordinates xcm and ycm of the molecule’s center of mass are given by m x + m2 x2 + m3 x3 m y + m2 y2 + m3 y3 xcm = 1 1 and ycm = 1 1 (Equation 7.10), where we make m1 + m2 + m3 m1 + m2 + m3 the following identifications: Atom Oxygen (left side)

Mass

Oxygen (right side) Sulfur

m1 = mO

x-coordinate x1 = −d sin 60.0°

y-coordinate y1 = +d cos 60.0°

m2 = mO

x2 = +d sin 60.0°

y2 = +d cos 60.0°

m3 = mS = 2mO

x3 = 0 nm

y3 = 0 nm

Here, we have used d = 0.143 nm to represent the distance between the center of the sulfur atom and the center of either oxygen atom. SOLUTION a. Making substitutions into xcm =

m1 x1 + m2 x2 + m3 x3

(Equation 7.10) from the data table m1 + m2 + m3 above, we obtain the x coordinate of the molecule’s center of mass: xcm =

− mOd sin 60.0 + mO d sin 60.0 + mS ( 0 ) = 0 nm mO + mO + mS

This result makes sense, because the sulfur atom’s x coordinate is x3 = 0 nm, and the oxygen atoms, which have equal masses, have x coordinates of equal magnitude but opposite signs: x1 = −x2. b. The y coordinate of the sulfur atom is zero (y3 = 0 nm), but the y coordinates of the two oxygen atoms are both positive, so the center of mass of the sulfur dioxide molecule has a positive y coordinate: ycm = =

+ mO d cos 60.0 + mO d cos 60.0 + mS ( 0 nm ) mO + mO + mS

+2 mO d cos 60.0 4 mO

=

=

+2mO d cos 60.0 2mO + 2mO

+ ( 0.143 nm ) cos 60.0 + d cos 60.0 = = +0.0358 nm 2 2

53. SSM WWW REASONING The system consists of the lumberjack and the log. For this system, the sum of the external forces is zero. This is because the weight of the system is balanced by the corresponding normal force (provided by the buoyant force of the water)

382 IMPULSE AND MOMENTUM

and the water is assumed to be frictionless. The lumberjack and the log, then, constitute an isolated system, and the principle of conservation of linear momentum holds. SOLUTION a. The total linear momentum of the system before the lumberjack begins to move is zero, since all parts of the system are at rest. Momentum conservation requires that the total momentum remains zero during the motion of the lumberjack.

m1vf1 + m2vf2 = Total momentum just before the jump

0 Initial momentum

Here the subscripts "1" and "2" refer to the first log and lumberjack, respectively. Let the direction of motion of the lumberjack be the positive direction. Then, solving for v1f gives v f1 = –

m2 v f2 m1

=–

(98 kg)(+3.6 m / s) = –1.5 m / s 230 kg

The minus sign indicates that the first log recoils as the lumberjack jumps off. b. Now the system is composed of the lumberjack, just before he lands on the second log, and the second log. Gravity acts on the system, but for the short time under consideration while the lumberjack lands, the effects of gravity in changing the linear momentum of the system are negligible. Therefore, to a very good approximation, we can say that the linear momentum of the system is very nearly conserved. In this case, the initial momentum is not zero as it was in part (a); rather the initial momentum of the system is the momentum of the lumberjack just before he lands on the second log. Therefore, m1 v f1 + m2 v f2 = m1 v 01 + m2 v 02 Total momentum just after lumberjack lands

Initial momentum

In this expression, the subscripts "1" and "2" now represent the second log and lumberjack, respectively. Since the second log is initially at rest, v 01 = 0 . Furthermore, since the lumberjack and the second log move with a common velocity, v f1 = v f2 = v f . The statement of momentum conservation then becomes m1 v f + m2 v f = m2 v 02

Solving for vf, we have vf =

m2 v 02 m1 + m2

=

(98 kg)(+3.6 m / s) = + 1.1 m / s 230 kg + 98 kg

The positive sign indicates that the system moves in the same direction as the original direction of the lumberjack's motion.

Chapter 7 Problems

383

54. REASONING Since all of the collisions are elastic, the total mechanical energy of the ball is conserved. However, since gravity affects its vertical motion, its linear momentum is not conserved. If hf is the maximum height of the ball on its final bounce, conservation of energy gives 1 mvf2 2

1 2

+ mghf = mv02 + mgh0 Ef

E0

Solving this equation for hf gives hf =

v02 − vf2 2g

+ h0

SOLUTION In order to use this expression, we must obtain the values for the velocities v0

and vf. The initial velocity has only a horizontal component, v0 = v0x .The final velocity also has only a horizontal component since the ball is at the top of its trajectory, vf = vf x . No

forces act in the horizontal direction so the momentum of the ball in this direction is conserved, hence v0 = vf . Therefore,

hf = h0 = 3.00 m

55. REASONING a. During the collision between the bullet and the wooden block, linear momentum is conserved, since no net external force acts on the bullet and the block. The weight of each is balanced by the tension in the suspension wire, and the forces that the bullet and block exert on each other are internal forces. This conservation law will allow us to find the speed of the bullet/block system immediately after the collision. b. Just after the collision, the bullet/block rise up, ultimately reaching a final height hf before coming to a momentary rest. During this phase, the tension in the wire (a nonconservative force) does no work, since it acts perpendicular to the motion. Thus, the work done by nonconservative forces is zero, and the total mechanical energy of the system is conserved. An application of this conservation law will enable us to determine the height hf. SOLUTION a. The principle of conservation of linear momentum states that the total momentum after the collision is equal to that before the collision.

(m

bullet

+ mblock ) vf = mbullet v0,bullet + mblock v0,block

Momentum after collision

Momentum before collision

384 IMPULSE AND MOMENTUM

Solving this equation for the speed vf of the bullet/block system just after the collision gives vf = =

mbullet v0,bullet + mblock v0,block mbullet + mblock

( 0.00250 kg )( 425 m/s ) + ( 0.215 kg )( 0 m/s ) = 0.00250 kg + 0.215 kg

4.89 m/s

b. Just after the collision, the total mechanical energy of the system is all kinetic energy, since we take the zero-level for the gravitational potential energy to be at the initial height of the block. As the bullet/block system rises, kinetic energy is converted into potential energy. At the highest point, the total mechanical energy is all gravitational potential energy. Since the total mechanical energy is conserved, we have

(m

bullet

+ mblock ) ghf =

Total mechanical energy at the top of the swing, all potential

1 2

(m

bullet

+ mblock ) vf2

Total mechanical energy at the bottom of the swing, all kinetic

Solving this expression for the height hf gives hf =

1 2

vf2 g

=

1 2

( 4.89 m/s )

2

9.80 m/s 2

= 1.22 m

56. REASONING AND SOLUTION According to the impulse-momentum theorem, Equation

7.4, ( ΣF ) ∆t = mv f − mv 0 , where ΣF is the net average force acting on the person. Taking the direction of motion (downward) as the negative direction and solving for the net average force ΣF , we obtain

ΣF =

m ( v f − v0 ) ∆t

=

( 62.0 kg ) [ −1.10 m/s – ( − 5.50 m/s)] 1.65 s

= +165 N

The plus sign indicates that the force acts upward . 57. REASONING AND SOLUTION The momentum is zero before the beat. Conservation of momentum requires that it is also zero after the beat; thus 0 = mpvp + mbvb so that vp = –(mb/mp)vb = –(0.050 kg/85 kg)(0.25 m/s) = −1.5 ×10−4 m/s

Chapter 7 Problems

385

58. REASONING The impulse that the wall exerts on the skater can be found from the impulse-momentum theorem, Equation 7.4. The average force F exerted on the skater by the wall is the only force exerted on her in the horizontal direction, so it is the net force; ΣF = F . SOLUTION From Equation 7.4, the average force exerted on the skater by the wall is

F=

mvf − mv 0 ∆t

=

( 46 kg ) ( −1.2 m/s ) − ( 46 kg ) ( 0 m/s ) 0.80 s

= −69 N

From Newton's third law, the average force exerted on the wall by the skater is equal in magnitude and opposite in direction to this force. Therefore, Force exerted on wall = +69 N The plus sign indicates that this force points opposite to the velocity of the skater.

59. REASONING Let the total amount of gas in the completely filled propulsion unit be mtotal, and the amount of gas ejected during the space-walk be mejected. The total mass of the gas is the sum of the mass mejected of the ejected gas and the mass mleft of the gas left over at the end of the space-walk: mtotal = mejected + mleft. The mass of the leftover gas, in turn, is the difference between the mass m1 = 165 kg of the astronaut with the partially full propulsion unit and the mass mempty = 146 kg of the astronaut with the completely empty propulsion unit: mleft = m1 − mempty = 165 kg − 146 kg = 19 kg. Thus, the percentage of gas propellant in the completely filled propulsion unit that is depleted during the space walk is

Percent depleted =

mejected mtotal

×100% =

mejected mejected + mleft

× 100%

(1)

We will find mejected by considering the space-walk. During the space-walk, the astronaut, the propulsion unit, and the gas form an isolated system. Although the propulsion unit and the ejected gas exert forces on one another, as do the astronaut and the propulsion unit, these are internal forces. The parts of the system interact only with each other, so the net external force on the system is zero. Hence, the total momentum of the system is conserved. We will apply the momentum conservation principle m1vf1 + m2 vf2 = m1v01 + m2 v02 (Equation 7.7b), with the following identifications: m1 = 165 kg (mass of astronaut and partially empty propulsion unit) m2 = mejected = ? (mass of ejected gas)

386 IMPULSE AND MOMENTUM

vf1 = −0.39 m/s (final velocity of astronaut and partially empty propulsion unit) vf2 = +32 m/s (final velocity of ejected gas) No part of the system is moving before the gas is ejected, so we also have v01 = v02 = 0 m/s. We will determine the unknown mass m2 = mejected of the ejected gas from Equation 7.7b. SOLUTION The momentum conservation principle yields the mass m2 = mejected of the ejected gas:

m1vf1 + m2vf2 = Total momentum after gas is ejected

0

m2 =

or

Total momentum before gas is ejected

− m1vf1 − (165 kg )( −0.39 m/s ) = = 2.0 kg vf2 ( +32 m/s )

Equation (1) then yields the percent of propellant gas depleted during the space-walk:

Percent depleted =

mejected mejected + mleft

× 100% =

2.0 kg ×100% = 9.5% 2.0 kg + 19 kg

60. REASONING AND SOLUTION Equation 7.10 gives the center of mass of this two-atom system as m x + mo xo xcm = c c mc + mo If we take the origin at the center of the carbon atom, then xc = 0 m, and we have xcm =

mc xc + mo xo mc + mo

=

xo

( mc / mo ) + 1

=

1.13 ×10−10 m = 6.46 × 10−11 m 0.750 / 1 + m m ( o o)

61. SSM REASONING AND SOLUTION The comet piece and Jupiter constitute an isolated system, since no external forces act on them. Therefore, the head-on collision obeys the conservation of linear momentum: ( mcomet + mJupiter ) v f Total momentum after collision

=

mcomet v comet + mJupiter v Jupiter Total momentum before collision

where vf is the common velocity of the comet piece and Jupiter after the collision. We assume initially that Jupiter is moving in the +x direction so v Jupiter = +1.3 × 10 4 m / s . The comet piece must be moving in the opposite direction so v comet = –6.0 × 10 4 m / s . The final

Chapter 7 Problems

387

velocity vf of Jupiter and the comet piece can be written as v f = v Jupiter + ∆v where ∆v is the change in velocity of Jupiter due to the collision. Substituting this expression into the conservation of momentum equation gives ( mcomet + mJupiter )( v Jupiter + ∆v ) =

mcomet v comet + mJupiter v Jupiter

Multiplying out the left side of this equation, algebraically canceling the term mJupiter v Jupiter from both sides of the equation, and solving for ∆v yields

∆v = =

mcomet ( v comet – v Jupiter) mcomet + mJupiter (4.0 × 10 12 kg) (–6.0 × 10 4 m / s – 1.3 × 10 4 m / s) = –1.5 × 10 –10 m / s 12 27 4.0 × 10 kg + 1.9 × 10 kg

The change in Jupiter's speed is 1.5 × 10 –10 m / s .

62. REASONING We consider the boy and the skateboard as a single system. Friction between the skateboard and the sidewalk is minimal, so there is no net horizontal force acting on the system. Therefore, the horizontal component of the system’s linear momentum remains constant while the boy pushes off from the skateboard: m1vf 1x + m2 vf2x = m1v01x + m2 v02x (Equation 7.9a). We will solve this equation to find the final velocity of the skateboard. SOLUTION The data given in the problem may be summarized as follows:

Horizontal velocity component Mass

Initial

Final

Boy

m1 = 40.0 kg

v01x = v0 = +5.30 m/s

vf1x = +(6.00 m/s)(cos 9.50°) = +5.92 m/s

Skateboard

m2 = 2.50 kg

v02x = v0 = +5.30 m/s

vf2x = ?

Solving equation 7.9a for the final horizontal component of the skateboard’s velocity, we obtain m ( v − v ) + m2v0 m2vf2x = m1v0 + m2v0 − m1vf 1x or vf2x = 1 0 f 1x m2 Thus, vf2x =

m1 ( v0 − vf 1x ) m2

+ v0 =

( 40.0 kg )( +5.30 m/s − 5.92 m/s ) + 5.30 m/s = 2.50 kg

−4.6 m/s

388 IMPULSE AND MOMENTUM

63. REASONING For the system consisting of the female character, the gun and the bullet, the sum of the external forces is zero, because the weight of each object is balanced by a corresponding upward (normal) force, and we are ignoring friction. The female character, the gun and the bullet, then, constitute an isolated system, and the principle of conservation of linear momentum applies. SOLUTION a. The total momentum of the system before the gun is fired is zero, since all parts of the system are at rest. Momentum conservation requires that the total momentum remains zero after the gun has been fired.

m1 v f1 + m2 v f2 = Total momentum after gun is fired

0 Total momentum before gun is fired

where the subscripts 1 and 2 refer to the woman (plus gun) and the bullet, respectively. Solving for vf1, the recoil velocity of the woman (plus gun), gives v f1 = –

m2 v f2 m1

=

– (0.010 kg)(720 m / s) = – 0.14 m / s 51 kg

b. Repeating the calculation for the situation in which the woman shoots a blank cartridge, we have m v – (5.0 × 10 –4 kg)(720 m / s) v f1 = – 2 f2 = = – 7.1 × 10 –3 m / s m1 51 kg In both cases, the minus sign means that the bullet and the woman move in opposite directions when the gun is fired. The total momentum of the system remains zero, because momentum is a vector quantity, and the momenta of the bullet and the woman have equal magnitudes, but opposite directions.

64. REASONING The mass of each part of the seated human figure will be treated as if it were all located at the corresponding center of mass point. See the drawing given in the text. In effect, then, the problem deals with the three-particle system shown in the drawing at the right. To determine the x and y coordinates of the center of mass of this system, we will employ equations analogous to Equation 7.10. The values for the masses are m1 = 41 kg, m2 = 17 kg, and m3 = 9.9 kg.

+y m1

0.39 m 0.17 m

m2

+x 0.26 m

0.43 m

m3

Chapter 7 Problems

389

SOLUTION The x and y coordinates of the center of mass for the three-particle system in the drawing are xcm = ycm =

m1x1 + m2 x2 + m3 x3

=

( 41 kg )( 0 m ) + (17 kg )( 0.17 m ) + ( 9.9 kg )( 0.43 m ) =

m1 y1 + m2 y2 + m3 y3

=

( 41 kg )( 0.39 m ) + (17 kg )( 0 m ) + ( 9.9 kg )( −0.26 m ) =

m1 + m2 + m3

m1 + m2 + m3

41 kg + 17 kg + 9.9 kg

0.11 m

41 kg + 17 kg + 9.9 kg

0.20 m

65. SSM REASONING We will define the system to be the platform, the two people and the ball. Since the ball travels nearly horizontally, the effects of gravity are negligible. Momentum is conserved. Since the initial momentum of the system is zero, it must remain zero as the ball is thrown and caught. While the ball is in motion, the platform will recoil in such a way that the total momentum of the system remains zero. As the ball is caught, the system must come to rest so that the total momentum remains zero. The distance that the platform moves before coming to rest again can be determined by using the expressions for momentum conservation and the kinematic description for this situation.

SOLUTION While the ball is in motion, we have MV + mv = 0

(1)

where M is the combined mass of the platform and the two people, V is the recoil velocity of the platform, m is the mass of the ball, and v is the velocity of the ball. The distance that the platform moves is given by x = Vt

(2)

where t is the time that the ball is in the air. The time that the ball is in the air is given by t=

L v −V

(3)

where L is the length of the platform, and the quantity (v – V) is the velocity of the ball relative to the platform. Remember, both the ball and the platform are moving while the ball is in the air. Combining equations (2) and (3) gives

V x= v −V

L

From equation (1) the ratio of the velocities is V / v = −m / M . Equation (4) then gives x=

(V / v ) L = ( −m / M ) L = − mL = − ( 6.0 kg )( 2.0 m ) = −0.097 m 1 − (V / v ) 1 + ( m / M ) M +m 118 kg + 6.0 kg

(4)

390 IMPULSE AND MOMENTUM

The minus sign indicates that displacement of the platform is in the opposite direction to the displacement of the ball. The distance moved by the platform is the magnitude of this displacement, or 0.097 m .

CHAPTER 8

ROTATIONAL KINEMATICS

ANSWERS TO FOCUS ON CONCEPTS QUESTIONS

1. (d) Using Equation 8.1 (θ = Arc length / Radius) to calculate the angle (in radians) that each object subtends at your eye shows that θMoon = 9.0 × 10−3 rad, θPea = 7.0 × 10−3 rad, and θDime = 25 × 10−3 rad. Since θPea is less than θMoon, the pea does not completely cover your view of the moon. However, since θDime is greater than θMoon, the dime does completely cover your view of the moon. 2. 2.20 cm 3. 38.2 s 4. (a) An angular acceleration of zero means that the angular velocity has the same value at all times, as in statements A or B. However, statement C is also consistent with a zero angular acceleration, because if the angular displacement does not change as time passes, then the angular velocity remains constant at a value of 0 rad/s. 5. (c) A non-zero angular acceleration means that the angular velocity is either increasing or decreasing. The angular velocity is not constant. 6. (b) Since values are given for the initial angular velocity ω0, the final angular velocity ω, and the time t, Equation 8.6 [ θ = 12 (ω0 + ω ) t ] can be used to calculate the angular displacement

θ. 7. 32 rad/s 8. 88 rad 9. (c) According to Equation 8.9 (vT = rω), the tangential speed is proportional to the radius r when the angular speed ω is constant, as it is for the earth. As the elevator rises, the radius, which is your distance from the center of the earth, increases, and so does your tangential speed. 10. (b) According to Equation 8.9 (vT = rω), the tangential speed is proportional to the radius r when the angular speed ω is constant, as it is for the merry-go-round. Thus, the angular 2.1 m speed of the second child is vT = ( 2.2 m/s ) . 1.4 m

392 ROTATIONAL KINEMATICS

11. 367 rad/s2 12. (e) According to Newton’s second law, the centripetal force is given by Fc = mac , where m is the mass of the ball and ac is the centripetal acceleration. The centripetal acceleration is given by Equation 8.11 as ac = rω2, where r is the radius and ω is the angular speed. Therefore, Fc = mrω 2 , and the centripetal force is proportional to the radius when the mass and the angular speed are fixed, as they are in this problem. As a result, 33 cm Fc = (1.7 N ) . 12 cm

13. (d) Since the angular speed ω is constant, the angular acceleration α is zero, according to Equation 8.4. Since α = 0 rad/s2, the tangential acceleration aT is zero, according to Equation 8.10. The centripetal acceleration ac, however, is not zero, since it is proportional to the square of the angular speed, according to Equation 8.11, and the angular speed is not zero. 14. 17.8 m/s2 15. (a) The number N of revolutions is the distance s traveled divided by the circumference 2πr of a wheel: N = s/(2πr). 16. 27.0 m/s

Chapter 8 Problems

393

CHAPTER 8 ROTATIONAL KINEMATICS PROBLEMS

1.

SSM REASONING AND SOLUTION Since there are 2π radians per revolution and it is stated in the problem that there are 100 grads in one-quarter of a circle, we find that the number of grads in one radian is

100 grad (1.00 rad ) 21π revrad 0.250 = 63.7 grad rev

2.

REASONING The average angular velocity ω has the same direction as θ − θ 0 , because θ − θ0 ω= according to Equation 8.2. If θ is greater than θ0, then ω is positive. If θ is t − t0 less than θ0, then ω is negative. SOLUTION The average angular velocity is given by Equation 8.2 as ω = t – t0 = 2.0 s is the elapsed time:

3.

(a )

ω=

(b)

ω=

(c)

ω=

(d )

ω=

θ − θ0 t − t0

θ − θ0 t − t0

θ − θ0 t − t0

θ − θ0 t − t0

=

0.75 rad − 0.45 rad = +0.15 rad /s 2.0 s

=

0.54 rad − 0.94 rad = −0.20 rad /s 2.0 s

=

4.2 rad − 5.4 rad = −0.60 rad /s 2.0 s

=

3.8 rad − 3.0 rad = +0.4 rad /s 2.0 s

θ − θ0 t − t0

, where

REASONING The average angular velocity ω is defined as the angular displacement ∆θ divided by the elapsed time ∆t during which the displacement occurs: ω = ∆θ / ∆t (Equation 8.2). This relation can be used to find the average angular velocity of the earth as it spins on its axis and as it orbits the sun.

394 ROTATIONAL KINEMATICS

SOLUTION a. As the earth spins on its axis, it makes 1 revolution (2π rad) in a day. Assuming that the positive direction for the angular displacement is the same as the direction of the earth’s rotation, the angular displacement of the earth in one day is ( ∆θ )spin = +2π rad . The average angular velocity is (converting 1 day to seconds):

ω=

( ∆θ )spin

=

( ∆t )spin

+2π rad = +7.3 × 10−5 rad/s 24 h 3600 s 1 day 1 day 1 h

(

)

b. As the earth orbits the sun, the earth makes 1 revolution (2π rad) in one year. Taking the positive direction for the angular displacement to be the direction of the earth’s orbital motion, the angular displacement in one year is ( ∆θ )orbit = +2π rad . The average angular velocity is (converting 365¼ days to seconds):

ω=

( ∆θ )orbit = ( ∆t )orbit

(

+2π rad 24 h 365 14 days 1 day

)

3600 s 1 h

= +2.0 ×10−7 rad/s

____________________________________________________________________________________________

4.

REASONING The average angular velocity of either mandible is given by ω = ∆θ ∆t (Equation 8.2), where θ is the angular displacement of the mandible and t is the elapsed time. In order to calculate the average angular velocity in radians per second, we will first convert the angular displacement θ from degrees to radians. SOLUTION Converting an angular displacement of 90° into radians, we find that the angular displacement of the mandible is

1 rev ∆θ = 90 degrees 360 degrees

(

)

2π rad π = rad 1 rev 2

The average angular velocity of the mandible is

π

rad ∆θ = 2 −4 = 1.2 × 10 4 rad/s ω= ∆t 1.3 × 10 s

(8.2)

Chapter 8 Problems

5.

395

SSM REASONING The average angular velocity is equal to the angular displacement divided by the elapsed time (Equation 8.2). Thus, the angular displacement of the baseball is equal to the product of the average angular velocity and the elapsed time. However, the problem gives the travel time in seconds and asks for the displacement in radians, while the angular velocity is given in revolutions per minute. Thus, we will begin by converting the angular velocity into radians per second. SOLUTION Since 2π rad = 1 rev and 1 min = 60 s, the average angular velocity ω (in rad/s) of the baseball is 330 rev 2 π rad 1 min ω = = 35 rad/s min 1 rev 60 s Since the average angular velocity of the baseball is equal to the angular displacement ∆θ divided by the elapsed time ∆t, the angular displacement is

∆θ = ω ∆t = ( 35 rad/s )( 0.60 s ) = 21 rad

6.

(8.2)

REASONING The jet is maintaining a distance of r = 18.0 km from the air traffic control tower by flying in a circle. The angle that the jet’s path subtends while its nose crosses over the moon is the same as the angular width θ of the moon. The corresponding distance the jet travels is the length of arc s subtended by the moon’s diameter. We will use the relation s = rθ (Equation 8.1) to determine the distance s. SOLUTION In order to use the relation s = rθ (Equation 8.1), the angle θ must be expressed in radians, as it is. The result will have the same units as r. Because s is required in meters, we first convert r to meters: 1000 m 4 r = 18.0 km = 1.8 × 10 m 1 km

(

)

Therefore, the distance that the jet travels while crossing in front of the moon is

(

)(

)

s = rθ = 1.80 ×104 m 9.04 ×10−3 rad = 163 m

7.

REASONING

α=

ω − ω0 t − t0

The average angular acceleration has the same direction as ω − ω0, because

, according to Equation 8.4. If ω is greater than ω0, α is positive. If ω is less

than ω0, α is negative.

396 ROTATIONAL KINEMATICS

SOLUTION The average angular acceleration is given by Equation 8.4 as α =

ω − ω0 t − t0

,

where t – t0 = 4.0 s is the elapsed time.

8.

(a )

α=

(b)

α=

(c)

α=

(d )

α=

ω − ω0 t − t0

ω − ω0 t − t0

ω − ω0 t − t0

ω − ω0 t − t0

=

+5.0 rad /s − 2.0 rad /s 2 = +0.75 rad /s 4.0 s

=

+2.0 rad /s − 5.0 rad /s = −0.75 rad /s 2 4.0 s

= =

−3.0 rad /s − ( −7.0 rad /s ) 4.0 s −4.0 rad /s − ( +4.0 rad /s ) 4.0 s

= +1.0 rad /s 2 = −2.0 rad /s 2

REASONING The relation between the final angular velocity ω, the initial angular velocity ω0, and the angular acceleration α is given by Equation 8.4 (with t0 = 0 s) as

ω = ω0 + α t If α has the same sign as ω0, then the angular speed, which is the magnitude of the angular

velocity ω, is increasing. On the other hand, If α and ω0 have opposite signs, then the angular speed is decreasing.

SOLUTION According to Equation 8.4, we know that ω = ω0 + α t. Therefore, we find:

(

) ( 2.0 s ) = + 18 rad /s . The angular speed is 18 rad/s .

(

)

(

)

(

) ( 2.0 s ) = − 18 rad /s. The angular speed is 18 rad/s .

(a) ω = + 12 rad /s + +3.0 rad /s 2

(b) ω = + 12 rad /s + −3.0 rad /s 2 ( 2.0 s ) = + 6.0 rad /s. The angular speed is 6.0 rad/s . (c) ω = − 12 rad /s + +3.0 rad /s 2 ( 2.0 s ) = − 6.0 rad /s. The angular speed is 6.0 rad /s . (d) ω = − 12 rad /s + −3.0 rad /s 2

Chapter 8 Problems

9.

397

SSM REASONING Equation 8.4 α = (ω − ω 0 ) / t indicates that the average angular acceleration is equal to the change in the angular velocity divided by the elapsed time. Since the wheel starts from rest, its initial angular velocity is ω0 = 0 rad/s. Its final angular velocity is given as ω = 0.24 rad/s. Since the average angular acceleration is given as α = 0.030 rad/s 2 , Equation 8.4 can be solved to determine the elapsed time t. SOLUTION Solving Equation 8.4 for the elapsed time gives

t=

ω − ω 0 0.24 rad/s − 0 rad/s = = 8.0 s α 0.030 rad/s 2

10. REASONING AND SOLUTION Using Equation 8.4 and the appropriate conversion factors, the average angular acceleration of the CD in rad/s2 is

α=

2

∆ω 210 rev/ min − 480 rev/ min 2π rad 1 min −3 2 = = – 6.4 × 10 rad/s ∆t 74 min 1 rev 60 s

The magnitude of the average angular acceleration is 6.4 × 10−3 rad/s2 . 11. REASONING The average angular velocity ω is defined as the angular displacement ∆θ divided by the elapsed time ∆t during which the displacement occurs: ω = ∆θ / ∆t (Equation 8.2). Solving for the elapsed time gives ∆t = ∆θ / ω . We are given ∆θ and can calculate ω from the fact that the earth rotates on its axis once every 24.0 hours. −3

SOLUTION The sun itself subtends an angle of 9.28 × 10 rad. When the sun moves a −3 distance equal to its diameter, it moves through an angle that is also 9.28 × 10 rad; thus, −3

∆θ = 9.28 × 10 rad. The average angular velocity ω at which the sun appears to move across the sky is the same as that of the earth rotating on its axis, ωearth , so ω = ωearth .

Since the earth makes one revolution (2π rad) every 24.0 h, its average angular velocity is

ωearth =

∆θ earth ∆tearth

=

2π rad = 24.0 h

2π rad = 7.27 ×10−5 rad/s 3600 s ( 24.0 h ) 1 h

The time it takes for the sun to move a distance equal to its diameter is ∆t =

∆θ

ωearth

=

9.28 ×10−3 rad = 128 s (a little over 2 minutes) 7.27 ×10−5 rad/s

398 ROTATIONAL KINEMATICS

____________________________________________________________________________________________

12. REASONING AND SOLUTION The angular displacements of the astronauts are equal. For A

θ = sA/rA

For B

θ = sB/rB

(8.1)

Equating these two equations for θ and solving for sB gives sB = (rB/rA)sA = [(1.10 × 103 m)/(3.20 × 102 m)](2.40 × 102 m) = 825 m

13. REASONING AND SOLUTION The people meet at time t. At this time the magnitudes of their angular displacements must total 2π rad.

θ1 + θ2 = 2π rad Then

ω1t + ω2t = 2π rad t=

2π rad 2π rad = = 1200 s − 3 ω1 + ω 2 1.7 × 10 rad/s + 3.4 × 10−3 rad/s

14. REASONING It does not matter whether the arrow is aimed closer to or farther away from the axis. The blade edge sweeps through the open angular space as a rigid unit. This means that a point closer to the axis has a smaller distance to travel along the circular arc in order to bridge the angular opening and correspondingly has a smaller tangential speed. A point farther from the axis has a greater distance to travel along the circular arc but correspondingly has a greater tangential speed. These speeds have just the right values so that all points on the blade edge bridge the angular opening in the same time interval. The rotational speed of the blades must not be so fast that one blade rotates into the open angular space while part of the arrow is still there. A faster arrow speed means that the arrow spends less time in the open space. Thus, the blades can rotate more quickly into the open space without hitting the arrow, so the maximum value of the angular speed ω increases with increasing arrow speed v. A longer arrow traveling at a given speed means that some part of the arrow is in the open space for a longer time. To avoid hitting the arrow, then, the blades must rotate more slowly. Thus, the maximum value of the angular speed ω decreases with increasing arrow length L.

Chapter 8 Problems

399

The time during which some part of the arrow remains in the open angular space is tArrow. The time it takes for the edge of a propeller blade to rotate through the open angular space between the blades is tBlade. The maximum angular speed is the angular speed such that these two times are equal. SOLUTION The time during which some part of the arrow remains in the open angular space is the time it takes the arrow to travel at a speed v through a distance equal to its own length L. This time is tArrow = L/v. The time it takes for the edge to rotate at an angular speed ω through the angle θ between the blades is tBlade = θ/ω. The maximum angular speed is the angular speed such that these two times are equal. Therefore, we have

L θ = ω v Arrow

Blade

In this expression we note that the value of the angular opening is θ = 60.0º, which is θ = 16 ( 2π ) rad = 13 π rad . Solving the expression for ω gives

ω=

θv L

=

πv 3L

Substituting the given values for v and L into this result, we find that a.

ω=

b.

ω=

c.

ω=

πv 3L

πv 3L

πv 3L

=

=

=

π ( 75.0 m/s ) 3 ( 0.71 m )

π ( 91.0 m/s ) 3 ( 0.71 m )

π ( 91.0 m/s ) 3 ( 0.81 m )

= 111 rad/s

= 134 rad/s

= 118 rad/s

15. REASONING AND SOLUTION The baton will make four revolutions in a time t given by t=

θ ω

Half of this time is required for the baton to reach its highest point. The magnitude of the initial vertical velocity of the baton is then

400 ROTATIONAL KINEMATICS

v0 = g

( 12 t ) = g 2θω

With this initial velocity the baton can reach a height of 2

h=

v0

2g

=

gθ

2

8ω 2

(9.80 m/s ) (8π rad ) 2

=

2

rev 2π rad 8 1.80 s 1 rev

2

= 6.05 m

16. REASONING The time required for the bullet to travel the distance d is equal to the time required for the discs to undergo an angular displacement of 0.240 rad. The time can be found from Equation 8.2; once the time is known, the speed of the bullet can be found using Equation 2.2.

SOLUTION From the definition of average angular velocity:

ω= the required time is ∆t =

∆θ

ω

=

∆θ ∆t

0.240 rad = 2.53 × 10 −3 s 95.0 rad/s

Note that ω = ω because the angular speed is constant. The (constant) speed of the bullet can then be determined from the definition of average speed:

v=

17.

∆x 0.850 m d = = = ∆t ∆t 2.53 × 10−3 s

336 m/s

SSM REASONING AND SOLUTION a. If the propeller is to appear stationary, each blade must move through an angle of 120° or 2π / 3 rad between flashes. The time required is t=

θ = ω

(2π / 3) rad –2 = 2.00 ×10 s 2 π rad (16.7 rev/s) 1 rev

b. The next shortest time occurs when each blade moves through an angle of 240°, or 4π / 3 rad, between successive flashes. This time is twice the value that we found in part a, or 4.00 × 10

−2

s.

Chapter 8 Problems

18. REASONING AND SOLUTION The figure at the right shows the relevant angles and dimensions for either one of the celestial bodies under consideration.

401

r θ

s

person on earth celestial body

a. Using the figure above

θ moon = θ sun =

s moon 3.48 × 10 6 m –3 = = 9.04 × 10 rad 8 rmoon 3.85 × 10 m s sun 1.39 × 109 m –3 = = 9.27 × 10 rad rsun 1.50 × 1011 m

b. Since the sun subtends a slightly larger angle than the moon, as measured by a person standing on the earth, the sun cannot be completely blocked by the moon. Therefore, a "total" eclipse of the sun is not really total . c. The relevant geometry is shown below. r sun

R sun s sun

R

r moon b

s

b

θ sun

moon

θ moon person on earth

The apparent circular area of the sun as measured by a person standing on the earth is given 2 by: Asun = π Rsun , where Rsun is the radius of the sun. The apparent circular area of the sun that is blocked by the moon is Ablocked = π Rb2 , where Rb is shown in the figure above. Also from the figure above, it follows that Rsun = (1/2) ssun

and Rb = (1/2) sb

Therefore, the fraction of the apparent circular area of the sun that is blocked by the moon is

402 ROTATIONAL KINEMATICS

Ablocked Asun

=

π Rb2 2 π Rsun

=

θ = moon θ sun

π ( sb / 2)2 π ( ssun

sb = 2 / 2) ssun

2

θ moon rsun = θ sun rsun

2

2

2

9.04 × 10−3 rad = = 0.951 9.27 × 10−3 rad

The moon blocks out 95.1 percent of the apparent circular area of the sun.

19. REASONING AND SOLUTION Since the ball spins at 7.7 rev/s, it makes (7.7 rev/s)t revolutions while in flight, where t is the time of flight and must be determined. The ball’s vertical displacement is y = 0 m since the ball returns to its launch point. The vertical component of the ball’s initial velocity is v0y = (19 m/s) sin 55°, assuming upward to be the positive direction. The acceleration due to gravity is ay = −9.80 m/s2. With these three 1 2

pieces of information at hand, we use y = v0 y t + a y t 2 (Equation 3.5b) to determine the time of flight. Noting that y = 0 m, we can solve this expression for t and find that

t=−

2v0 y ay

=−

2 (19 m/s ) sin 55° −9.80 m/s

2

=3.2 s

and

Number of = 7.7 rev/s 3.2 s = 25 rev )( ) revolutions (

____________________________________________________________________________________________

20. REASONING The angular displacement is given as θ = 0.500 rev, while the initial angular velocity is given as ω0 = 3.00 rev/s and the final angular velocity as ω = 5.00 rev/s. Since we seek the time t, we can use Equation 8.6 θ = 12 (ω0 + ω ) t from the equations of rotational kinematics to obtain it.

SOLUTION Solving Equation 8.6 for the time t, we find that t=

21.

2 ( 0.500 rev ) 2θ = = 0.125 s ω 0 + ω 3.00 rev/s + 5.00 rev/s

SSM REASONING AND SOLUTION a. From Equation 8.7 we obtain

θ = ω 0 t + 12 α t 2 = (5.00 rad/s)(4.00 s) + 12 (2.50 rad/s 2 )(4.00 s) 2 = 4.00 × 101 rad

Chapter 8 Problems

403

b. From Equation 8.4, we obtain

ω = ω 0 + α t = 5.00 rad/s + (2.50 rad/s 2 )(4.00 s) = 15.0 rad/s 22. REASONING We are given the turbine’s angular acceleration α, final angular velocity ω, and angular displacement θ. Therefore, we will employ ω 2 = ω02 + 2αθ (Equation 8.8) in

order to determine the turbine’s initial angular velocity ω0 for part a. However, in order to make the units consistent, we will convert the angular displacement θ from revolutions to radians before substituting its value into Equation 8.8. In part b, the elapsed time t is the only unknown quantity. We can, therefore, choose from among ω = ω0 + α t (Equation 8.4),

θ = 12 (ω0 + ω ) t (Equation 8.6), or θ = ω0t + 12 α t 2 (Equation 8.7) to find the elapsed time. Of the three, Equation 8.4 offers the least algebraic complication, so we will solve it for the elapsed time t. SOLUTION a. One revolution is equivalent to 2π radians, so the angular displacement of the turbine is

2π rad 4 = 1.80 ×10 rad 1 rev

θ = ( 2870 rev )

Solving ω 2 = ω02 + 2αθ (Equation 8.8) for the square of the initial angular velocity, we obtain ω02 = ω 2 − 2αθ , or

ω0 = ω 2 − 2αθ =

(137 rad/s )2 − 2 ( 0.140 rad/s2 )(1.80 ×104 rad ) = 117 rad/s

b. Solving ω = ω0 + α t (Equation 8.4) for the elapsed time, we find that t=

23.

ω − ω0 137 rad/s − 117 rad/s = = 140 s α 0.140 rad/s 2

SSM WWW REASONING AND SOLUTION

a.

ω = ω0 + α t = 0 rad/s + (3.00 rad/s2)(18.0 s) = 54.0 rad/s

b.

θ=

1 (ω + 0 2

ω)t =

1 (0 2

rad/s + 54.0 rad/s)(18.0 s) = 486 rad

404 ROTATIONAL KINEMATICS

24. REASONING The angular displacement is given by Equation 8.6 as the product of the average angular velocity and the time

θ =ωt =

1 2

(ω0 + ω )

t

Average angular velocity

This value for the angular displacement is greater than ω0t. When the angular displacement θ is given by the expression θ = ω0t, it is assumed that the angular velocity remains constant at its initial (and smallest) value of ω0 for the entire time, which does not, however, account for the additional angular displacement that occurs because the angular velocity is increasing.

The angular displacement is also less than ω t. When the angular displacement is given by the expression θ = ω t, it is assumed that the angular velocity remains constant at its final (and largest) value of ω for the entire time, which does not account for the fact that the wheel was rotating at a smaller angular velocity during the time interval. SOLUTION a. If the angular velocity is constant and equals the initial angular velocity ω0, then ω = ω0 and the angular displacement is

θ = ω 0 t = ( +220 rad /s )(10.0 s ) = +2200 rad b. If the angular velocity is constant and equals the final angular velocity ω, then ω = ω and the angular displacement is

θ = ω t = ( +280 rad /s )(10.0 s ) = +2800 rad c. Using the definition of average angular velocity, we have

θ=

1 2

(ω0 + ω ) t = 12 ( +220 rad /s + 280 rad /s )(10.0 s ) =

+2500 rad

(8.6)

25. REASONING a. The time t for the wheels to come to a halt depends on the initial and final velocities, ω0 and ω, and the angular displacement θ : θ =

time yields

t=

1 2

(ω0 + ω ) t

2θ ω0 + ω

(see Equation 8.6). Solving for the

Chapter 8 Problems

405

b. The angular acceleration α is defined as the change in the angular velocity, ω − ω0, divided by the time t: ω − ω0 α= (8.4) t SOLUTION a. Since the wheel comes to a rest, ω = 0 rad/s. Converting 15.92 revolutions to radians (1 rev = 2π rad), the time for the wheel to come to rest is

2π rad 2 ( +15.92 rev ) 2θ 1 rev = 10.0 s t= = +20.0 rad/s + 0 rad/s ω0 + ω b. The angular acceleration is

ω − ω0

0 rad/s − 20.0 rad/s = −2.00 rad/s 2 t 10.0 s ______________________________________________________________________________

α=

=

26. REASONING Equation 8.8 (ω 2 = ω 02 + 2αθ ) from the equations of rotational kinematics

can be employed to find the final angular velocity ω. The initial angular velocity is ω0 = 0 rad/s since the top is initially at rest, and the angular acceleration is given as α = 12 rad/s2. The angle θ (in radians) through which the pulley rotates is not given, but it can be obtained from Equation 8.1 (θ = s/r), where the arc length s is the 64-cm length of the string and r is the 2.0-cm radius of the top. SOLUTION Solving Equation 8.8 for the final angular velocity gives

ω = ± ω02 + 2αθ We choose the positive root, because the angular acceleration is given as positive and the top is at rest initially. Substituting θ = s/r from Equation 8.1 gives s

ω = + ω 02 + 2α = + r

27.

( 0 rad/s )

2

64 cm + 2 (12 rad/s 2 ) = 28 rad/s 2.0 cm

SSM REASONING The equations of kinematics for rotational motion cannot be used directly to find the angular displacement, because the final angular velocity (not the initial angular velocity), the acceleration, and the time are known. We will combine two of the equations, Equations 8.4 and 8.6 to obtain an expression for the angular displacement that contains the three known variables.

406 ROTATIONAL KINEMATICS

SOLUTION The angular displacement of each wheel is equal to the average angular velocity multiplied by the time

θ=

1 2

( ω0 + ω ) t

(8.6)

ω

The initial angular velocity ω0 is not known, but it can be found in terms of the angular acceleration and time, which are known. The angular acceleration is defined as (with t0 = 0 s)

α=

ω − ω0

or

t

ω0 = ω − α t

(8.4)

Substituting this expression for ω0 into Equation 8.6 gives

θ = 12 (ω − α t ) + ω t = ω t − 12 α t 2

ω0

= ( +74.5 rad /s )( 4.50 s ) −

1 2

( +6.70 rad /s ) ( 4.50 s ) 2

2

= +267 rad

______________________________________________________________________________ 28. REASONING AND SOLUTION circumstance.

The angular acceleration is found for the first

ω 2 − ω 02 ( 3.14 × 104 rad/s ) − (1.05 × 104 rad/s ) 4 2 α= = = 2.33 × 10 rad/s 4 2θ 2 (1.88 × 10 rad ) 2

2

For the second circumstance t=

ω − ω 0 7.85 × 104 rad/s − 0 rad/s = = 3.37 s 4 2 α 2.33 × 10 rad/s

____________________________________________________________________________________________

29. REASONING There are three segments to the propeller’s angular motion, and we will calculate the angular displacement for each separately. In these calculations we will remember that the final angular velocity for one segment is the initial velocity for the next segment. Then, we will add the separate displacements to obtain the total.

Chapter 8 Problems

407

SOLUTION For the first segment the initial angular velocity is ω0 = 0 rad/s, since the propeller starts from rest. Its acceleration is α = 2.90 × 10−3 rad/s2 for a time t = 2.10 × 103 s. Therefore, we can obtain the angular displacement θ1 from Equation 8.7 of the equations of rotational kinematics as follows: [First segment]

θ1 = ω 0t + 12 α t 2 = ( 0 rad/s ) ( 2.10 ×103 s ) + 12 ( 2.90 × 10−3 rad/s 2 )( 2.10 ×103 s )

2

= 6.39 ×103 rad The initial angular velocity for the second segment is the final velocity for the first segment, and according to Equation 8.4, we have

ω = ω 0 + α t = 0 rad/s + ( 2.90 ×10−3 rad/s 2 )( 2.10 × 103 s ) = 6.09 rad/s Thus, during the second segment, the initial angular velocity is ω0 = 6.09 rad/s and remains constant at this value for a time of t = 1.40 × 103 s. Since the velocity is constant, the angular acceleration is zero, and Equation 8.7 gives the angular displacement θ2 as [Second segment]

θ 2 = ω 0t + 12 α t 2 = ( 6.09 rad/s ) (1.40 ×103 s ) + 12 ( 0 rad/s 2 )(1.40 ×103 s ) = 8.53 ×103 rad 2

During the third segment, the initial angular velocity is ω0 = 6.09 rad/s, the final velocity is ω = 4.00 rad/s, and the angular acceleration is α = −2.30 × 10−3 rad/s2. When the propeller picked up speed in segment one, we assigned positive values to the acceleration and subsequent velocity. Therefore, the deceleration or loss in speed here in segment three means that the acceleration has a negative value. Equation 8.8 (ω 2 = ω 02 + 2αθ 3 ) can be used to find the angular displacement θ3. Solving this equation for θ3 gives

[Third segment]

ω 2 − ω 02 ( 4.00 rad/s ) − ( 6.09 rad/s ) = = 4.58 ×103 rad θ3 = 2 −3 2α 2 ( −2.30 ×10 rad/s ) 2

2

The total angular displacement, then, is

θ Total = θ1 + θ 2 + θ 3 = 6.39 ×103 rad + 8.53 × 103 rad + 4.58 ×103 rad = 1.95 × 104 rad 30. REASONING Since the time t and angular acceleration α are known, we will begin by using Equation 8.7 from the equations of kinematics to determine the angular displacement θ :

408 ROTATIONAL KINEMATICS

θ = ω0t + 12 α t 2 However, the initial angular velocity ω0 is not given. We can determine it by resorting to

another equation of kinematics, ω = ω0 + α t (Equation 8.4), which relates ω0 to the final angular velocity ω, the angular acceleration, and the time, all of which are known.

SOLUTION Solving Equation 8.4 for ω0 gives ω0 = ω − α t . Substituting this result into

θ = ω0t + 12 α t 2 gives

θ = ω0t + 12 α t 2 = (ω − α t ) t + 12 α t 2 = ω t − 12 α t 2 = ( +1.88 rad/s )(10.0 s ) − 12 ( −5.04 rad/s 2 ) (10.0 s ) = +2.71× 102 rad 2

31. REASONING According to Equation 3.5b, the time required for the diver to reach the water, assuming free-fall conditions, is t = 2 y / a y . If we assume that the "ball" formed

by the diver is rotating at the instant that she begins falling vertically, we can use Equation 8.2 to calculate the number of revolutions made on the way down. SOLUTION Taking upward as the positive direction, the time required for the diver to reach the water is 2(–8.3 m) t= 2 = 1.3 s –9.80 m/s

Solving Equation 8.2 for ∆ θ , we find ∆θ = ω ∆t = (1.6 rev/s)(1.3 s)= 2.1 rev

32. REASONING In addition to knowing the initial angular velocity ω0 and the acceleration α, we know that the final angular velocity ω is 0 rev/s, because the wheel comes to a halt. With values available for these three variables, the unknown angular displacement θ can be calculated from Equation 8.8 (ω 2 = ω 02 + 2αθ ) . When using any of the equations of rotational kinematics, it is not necessary to use radian measure. Any self-consistent set of units may be used to measure the angular quantities, such as revolutions for θ, rev/s for ω0 and ω, and rev/s2 for α. A greater initial angular velocity does not necessarily mean that the wheel will come to a halt on an angular section labeled with a greater number. It is certainly true that greater

Chapter 8 Problems

409

initial angular velocities lead to greater angular displacements for a given deceleration. However, remember that the angular displacement of the wheel in coming to a halt may consist of a number of complete revolutions plus a fraction of a revolution. In deciding on which number the wheel comes to a halt, the number of complete revolutions must be subtracted from the angular displacement, leaving only the fraction of a revolution remaining.

ω 2 − ω 02 SOLUTION Solving Equation 8.8 for the angular displacement gives θ = . 2α a. We know that ω0 = +1.20 rev/s, ω = 0 rev/s, and α = −0.200 rev/s2, where ω0 is positive since the rotation is counterclockwise and, therefore, α is negative because the wheel decelerates. The value obtained for the displacement is

ω 2 − ω02 ( 0 rev/s ) − ( +1.20 rev/s ) θ= = = +3.60 rev 2α 2 ( −0.200 rev/s 2 ) 2

2

To decide where the wheel comes to a halt, we subtract the three complete revolutions from this result, leaving 0.60 rev. Converting this value into degrees and noting that each angular section is 30.0º, we find the following number n for the section where the wheel comes to a halt: 360° 1 angular section n = ( 0.60 rev ) = 7.2 30.0° 1 rev A value of n = 7.2 means that the wheel comes to a halt in the section following number 7. Thus, it comes to a halt in section 8 . b. Following the same procedure as in part a, we find that

ω 2 − ω02 ( 0 rev/s ) − ( +1.47 rev/s ) θ= = = +5.40 rev 2α 2 ( −0.200 rev/s 2 ) 2

2

Subtracting the five complete revolutions from this result leaves 0.40 rev. Converting this value into degrees and noting that each angular section is 30.0º, we find the following number n for the section where the wheel comes to a halt: 360° 1 angular section n = ( 0.40 rev ) = 4.8 30.0° 1 rev A value of n = 4.8 means that the wheel comes to a halt in the section following number 4. Thus, it comes to a halt in section 5 .

410 ROTATIONAL KINEMATICS

33.

SSM WWW REASONING The angular displacement of the child when he catches the horse is, from Equation 8.2, θ c = ω c t . In the same time, the angular displacement of the

horse is, from Equation 8.7 with ω 0 = 0 rad/s, θ h = 12 α t 2 . If the child is to catch the horse θ c = θ h + (π / 2). SOLUTION Using the above conditions yields 1 αt2 2

or 1 (0.0100 2

− ωc t + 12 π = 0

rad/s 2 )t 2 − ( 0.250 rad/s ) t +

1 2

(π

rad ) = 0

The quadratic formula yields t = 7.37 s and 42.6 s; therefore, the shortest time needed to catch the horse is t = 7.37 s .

The angular acceleration α gives rise to a tangential acceleration aT according to aT = rα (Equation 8.10). Moreover, it is given that aT = g, where g is the magnitude of the acceleration due to gravity.

34. REASONING

SOLUTION Let r be the radial distance of the point from the axis of rotation. Then, according to Equation 8.10, we have g = rα aT

Thus, r=

g

α

=

9.80 m /s

2

12.0 rad /s

2

= 0.817 m

35. REASONING AND SOLUTION a.

ωA = v/r = (0.381 m/s)/(0.0508 m) = 7.50 rad/s

b.

ωB = v/r = (0.381 m/s)(0.114 m) = 3.34 rad/s α=

ω B −ω A t

=

3.34 rad/s − 7.50 rad/s 2.40 × 10 s

The angular velocity is decreasing .

3

= −1.73 × 10 −3 rad/s 2

Chapter 8 Problems

411

36. REASONING The tangential speed vT of a point on a rigid body rotating at an angular

speed ω is given by vT = rω (Equation 8.9), where r is the radius of the circle described by

the moving point. (In this equation ω must be expressed in rad/s.) Therefore, the angular speed of the bacterial motor sought in part a is ω = vT r . Since we are considering a point on the rim, r is the radius of the motor itself. In part b, we seek the elapsed time t for an angular displacement of one revolution at the constant angular velocity ω found in part a. We will use θ = ω0 t + 12 α t 2 (Equation 8.7) to calculate the elapsed time. SOLUTION a. The angular speed of the bacterial motor is, from ω = vT r (Equation 8.9),

ω=

vT r

=

2.3 × 10−5 m/s = 1500 rad/s 1.5 × 10 −8 m

b. The bacterial motor is spinning at a constant angular velocity, so it has no angular acceleration. Substituting α = 0 rad/s2 into θ = ω0t + 12 α t 2 (Equation 8.7), and solving for the elapsed time yields

(

)

θ = ω0t + 12 0 rad/s 2 t 2 = ω0t

or

t=

θ ω0

The fact that the motor has a constant angular velocity means that its initial and final angular velocities are equal: ω0 = ω = 1500 rad/s , the value calculated in part a, assuming a counterclockwise or positive rotation. The angular displacement θ is one revolution, or 2π radians, so the elapsed time is t=

37.

SSM

REASONING

θ 2π rad = = 4.2 × 10 −3 s ω0 1500 rad/s

The angular speed ω and tangential speed vT are related by

Equation 8.9 (vT = rω), and this equation can be used to determine the radius r. However, we must remember that this relationship is only valid if we use radian measure. Therefore, it will be necessary to convert the given angular speed in rev/s into rad/s.

412 ROTATIONAL KINEMATICS

SOLUTION Solving Equation 8.9 for the radius gives r=

vT

ω

54 m/s 2π rad ( 47 rev/s ) 1 rev

=

= 0.18 m

Conversion from rev/s into rad/s

where we have used the fact that 1 rev corresponds to 2π rad to convert the given angular speed from rev/s into rad/s. 38. REASONING The angular speed ω of the reel is related to the tangential speed vT of the fishing line by vT = rω (Equation 8.9), where r is the radius of the reel. Solving this

equation for ω gives ω = vT / r . The tangential speed of the fishing line is just the distance x

it travels divided by the time t it takes to travel that distance, or vT = x/t.

SOLUTION Substituting vT = x/t into ω = vT / r and noting that 3.0 cm = 3.0 × 10−2 m, we find that x 2.6 m vT t 9.5 s ω= = = = 9.1 rad/s r r 3.0 × 10 −2 m ______________________________________________________________________________ 39. REASONING The angular speed ω of the sprocket can be calculated from the tangential speed vT and the radius r using Equation 8.9 (vT = rω). The radius is given as r = 4.0 × 10−2 m. The tangential speed is identical to the linear speed given for a chain link at point A, so that vT = 5.6 m/s. We need to remember, however, that Equation 8.9 is only valid if radian measure is used. Thus, the value calculated for ω will be in rad/s, and we will have to convert to rev/s using the fact that 2π rad equals 1 rev.

SOLUTION Solving Equation 8.9 for the angular speed ω gives

ω=

vT r

=

5.6 m/s = 140 rad/s 4.0 ×10−2 m

Using the fact that 2π rad equals 1 rev, we can convert this result as follows: 1 rev = 22 rev/s 2π rad

ω = (140 rad/s )

Chapter 8 Problems

413

40. REASONING AND SOLUTION a. A person living in Ecuador makes one revolution (2π rad) every 23.9 hr (8.60 × 104 s). The angular speed of this person is ω = (2π rad)/(8.60 × 104 s) = 7.31 × 10−5 rad/s. According to Equation 8.9, the tangential speed of the person is, therefore,

( 6.38 ×106 m )( 7.31 × 10−5 rad/s ) =

vT = rω =

4.66 ×102 m/s

b. The relevant geometry is shown in the drawing at the right. Since the tangential speed is one-third of that of a person living in Ecuador, we have,

r θ

θ r

vT = rθ ω 3

θ

or rθ =

vT

3ω

=

(

4.66 × 102 m/s

3 7.31 × 10

−5

rad/s

)

= 2.12 × 106 m

The angle θ is, therefore, 6 rθ −1 2.12 × 10 m = cos = 70.6° 6 r 6.38 × 10 m

θ = cos −1

41.

SSM REASONING AND SOLUTION a. From Equation 8.9, and the fact that 1 revolution = 2π radians, we obtain

rev 2π rad = 1.25 m/s vT = r ω = (0.0568 m) 3.50 s 1 rev b. Since the disk rotates at constant tangential speed, v T1 = v T2

or

ω 1 r1 = ω 2 r2

Solving for ω2 , we obtain

ω2 =

ω 1 r1 r2

=

(3.50 rev/s)(0.0568 m) = 7.98 rev/s 0.0249 m

414 ROTATIONAL KINEMATICS

42. REASONING The linear speed v1 with which the bucket moves down the well is the same as the linear speed of the rope holding the bucket. The rope, in turn, is wrapped around the barrel of the hand crank, and unwinds without slipping. This ensures that the rope’s linear speed is the same as the tangential speed vT = r1ω (Equation 8.9) of a point on the surface of the barrel, where ω and r1 are the angular speed and radius of the barrel, respectively.

Therefore, we have v1 = r1ω . When applied to the linear speed v2 of the crank handle and the

radius r2 of the circle the handle traverses, Equation 8.9 yields v2 = r2ω . We are justified in

using the same symbol ω to represent the angular speed of the barrel and the angular speed of the hand crank, since both make the same number of revolutions in any given amount of time. Lastly, we note that the radii r1 of the crank barrel and r2 of the hand crank’s circular motion are half of the respective diameters d1 = 0.100 m and d2 = 0.400 m shown in the drawing provided in the text. SOLUTION Solving the relations v1 = r1ω and v2 = r2ω for the angular speed ω and the linear speed v1 of the bucket, we obtain

ω=

v1 r1

=

v2 r2

or

v1 =

v2 r1 r2

=

v2

( d )=v d 1 2 1

1d 2 2

2 1

d2

The linear speed of the bucket, therefore, is v1 =

(1.20 m/s) ( 0.100 m ) = 0.300 m/s ( 0.400 m )

43. REASONING AND SOLUTION The figure below shows the initial and final states of the system. m

L

L v

INITIAL CONFIGURATION

FINAL CONFIGURATION

Chapter 8 Problems

415

a. From the principle of conservation of mechanical energy: E0 = Ef Initially the system has only gravitational potential energy. If the level of the hinge is chosen as the zero level for measuring heights, then: E0 = mgh0 = mgL. Just before the object hits the floor, the system has only kinetic energy. Therefore 1 2

mgL = mv

2

Solving for v gives

v=

2 gL

From Equation 8.9, vT = rω. Solving for ω gives ω = vT/r. As the object rotates downward, it travels in a circle of radius L. Its speed just before it strikes the floor is its tangential speed. Therefore,

ω=

vT r

=

2 gL

v = L

L

b. From Equation 8.10:

=

2g = L

2

2(9.80 m/s ) = 3.61 rad/s 1.50 m

aT = rα

Solving for α gives α = aT/r. Just before the object hits the floor, its tangential acceleration is the acceleration due to gravity. Thus, 2

aT

g 9.80 m/s = = = α= r L 1.50 m

6.53 rad/s

2

44. REASONING AND SOLUTION The stone leaves the circular path with a horizontal speed v0 = vT = rω so ω = v0/r. We are given that r = x/30 so ω = 30v0/x. Kinematics gives x = v0t. With this substitution for x the expression for ω becomes ω = 30/t. Kinematics also gives for the 1 2

vertical displacement y that y = v0 y t + a y t 2 (Equation 3.5b). In Equation 3.5b we know 1 2

that v0y = 0 m/s since the stone is launched horizontally, so that y = a y t 2 or t = 2 y / a y . Using this result for t in the expression for ω and assuming that upward is positive, we find

416 ROTATIONAL KINEMATICS

ω = 30

ay 2y

−9.80 m/s 2 = 14.8 rad/s 2 ( −20.0 m )

= 30

45. SSM REASONING The magnitude ω of each car’s angular speed can be evaluated from ac = rω (Equation 8.11), where r is the radius of the turn and ac is the magnitude of the centripetal acceleration. We are given that the centripetal acceleration of each car is the same. In addition, the radius of each car’s turn is known. These facts will enable us to determine the ratio of the angular speeds. 2

SOLUTION Solving Equation 8.11 for the angular speed gives ω = ac / r . Applying this relation to each car yields: Car A:

ωA = ac, A / rA

Car B:

ωB = ac, B / rB

Taking the ratio of these two angular speeds, and noting that ac, A = ac, B, gives ac, A

ωA = ωB

rA ac, B

=

ac, A rB ac, B rA

=

36 m = 0.87 48 m

rB

46. REASONING Since the car is traveling with a constant speed, its tangential acceleration must be zero. The radial or centripetal acceleration of the car can be found from Equation 5.2. Since the tangential acceleration is zero, the total acceleration of the car is equal to its radial acceleration. SOLUTION a. Using Equation 5.2, we find that the car’s radial acceleration, and therefore its total acceleration, is v 2 (75.0 m/s) 2 a = aR = T = = 9.00 m/s 2 r 625 m

b. The direction of the car’s total acceleration is the same as the direction of its radial acceleration. That is, the direction is radially inward .

Chapter 8 Problems

47.

417

SSM REASONING AND SOLUTION a. The tangential acceleration of the train is given by Equation 8.10 as aT = r α = (2.00 × 10 m)(1.50 × 10 2

−3

2

rad/s ) = 0.300 m/s

2

The centripetal acceleration of the train is given by Equation 8.11 as 2

2

2

ac = r ω = (2.00 × 10 m)(0.0500 rad/s) = 0.500 m/s

2

The magnitude of the total acceleration is found from the Pythagorean theorem to be 2

2

a = aT + ac = 0.583 m/s

2

b. The total acceleration vector makes an angle relative to the radial acceleration of a T a c

θ = tan −1

=

tan −1

0.300 m/s 2

0.500

=

m/s 2

31.0°

48. REASONING a. According to Equation 8.2, the average angular speed is equal to the magnitude of the angular displacement divided by the elapsed time. The magnitude of the angular displacement is one revolution, or 2π rad. The elapsed time is one year, expressed in seconds. b. The tangential speed of the earth in its orbit is equal to the product of its orbital radius and its orbital angular speed (Equation 8.9). c. Since the earth is moving on a nearly circular orbit, it has a centripetal acceleration that is directed toward the center of the orbit. The magnitude ac of the centripetal acceleration is given by Equation 8.11 as ac = rω . 2

SOLUTION a. The average angular speed is

ω= ω=

2π rad ∆θ = = 1.99 × 10−7 rad /s 7 ∆t 3.16 × 10 s

(8.2)

b. The tangential speed of the earth in its orbit is

(

)(

vT = r ω = 1.50 × 10 m 1.99 × 10 11

−7

)

4

rad/s = 2.98 × 10 m/s

(8.9)

418 ROTATIONAL KINEMATICS

c. The centripetal acceleration of the earth due to its circular motion around the sun is

(

)(

a c = r ω 2 = 1.50 × 1011 m 1.99 × 10−7 rad /s

)

2

= 5.94 × 10−3 m /s2

(8.11)

The acceleration is directed toward the center of the orbit.

49. REASONING The centripetal acceleration ac at either corner is related to the angular speed

ω of the plate by ac = rω 2 (Equation 8.11), where r is the radial distance of the corner from the rotation axis of the plate. The angular speed ω is the same for all points on the plate, including both corners. But the radial distance rA of corner A from the rotation axis of the plate is different from the radial distance rB of corner B. The fact that the centripetal acceleration at corner A is n times as great as the centripetal acceleration at corner B yields the relationship between the radial distances:

(

rA ω 2 = n rB ω 2 Centripetal acceleration at corner A

)

rA = n rB

or

(1)

Centripetal acceleration at corner B

The radial distance rB at corner B is the length of the short side of the rectangular plate: rB = L1. The radial distance rA at corner A is the length of a straight line from the rotation axis to corner A. This line is the diagonal of the plate, so we obtain rA from the Pythagorean theorem (Equation 1.7): rA = L12 + L22 . SOLUTION Making the substitutions rA = L12 + L22 and rB = L1 in Equation (1) gives L12 + L22 = n L1 rA

(2)

rB

Squaring both sides of Equation (2) and solving for the ratio L1/L2 yields n 2 L12 = L12 + L22

or

( n2 − 1) L12 = L22

or

L1 L2

=

1 n2 − 1

Thus, when n = 2.00, the ratio of the lengths of the sides of the rectangle is

Chapter 8 Problems

419

L1 1 1 = = = 0.577 L2 3 22 − 1

50. REASONING

The centripetal acceleration of the trainee is given by ac = rω 2

(Equation 8.11), where ω is the angular speed (in rad/s) of the centrifuge, and r is the radius of the circular path the trainee follows. Because this radius is the length of the centrifuge arm, we will solve Equation 8.11 for r. In the second exercise, the trainee’s total acceleration gains a tangential component aT = rα (Equation 8.10) due to the angular acceleration α (in rad/s2) of the centrifuge. The angular speed ω and the length r of the centrifuge arm are both the same as in the first exercise, so the centripetal component of acceleration ac = rω 2 is unchanged in the second exercise. The two components of the trainee’s acceleration are perpendicular and, thus, are related to the trainee’s total acceleration a by the Pythagorean theorem: a 2 = ac2 + aT2 (Equation 1.7). We will solve the Pythagorean theorem for the trainee’s tangential acceleration aT, and then use aT = rα to

determine the angular acceleration α of the centrifuge.

SOLUTION a. Solving ac = rω 2 (Equation 8.11) yields the length r of the centrifuge arm: r=

ac

ω

2

=

(

3.2 9.80 m/s 2

( 2.5 rad/s )

2

) = 5.0 m

b. Solving a 2 = ac2 + aT2 for the tangential component of the trainee’s total acceleration, we obtain aT = a 2 − ac2 . Then, using aT = rα (Equation 8.10), we find that the angular acceleration of the centrifuge is

a α= T = r

51.

a 2 − ac2 r

=

(

)

2

(

)

2

( 4.8 ) 9.80 m/s 2 − ( 3.2 ) 9.80 m/s 2 = 7.0 rad/s 2 5.0 m

SSM REASONING a. The tangential speed vT of the sun as it orbits about the center of the Milky Way is related to the orbital radius r and angular speed ω by Equation 8.9, vT = rω. Before we use this relation, however, we must first convert r to meters from light-years.

420 ROTATIONAL KINEMATICS

b. The centripetal force is the net force required to keep an object, such as the sun, moving on a circular path. According to Newton’s second law of motion, the magnitude Fc of the centripetal force is equal to the product of the object’s mass m and the magnitude ac of its centripetal acceleration (see Section 5.3): Fc = mac. The magnitude of the centripetal acceleration is expressed by Equation 8.11 as ac = rω , where r is the radius of the circular 2

path and ω is the angular speed of the object.

SOLUTION a. The radius of the sun’s orbit about the center of the Milky Way is 9.5 × 1015 m 4 20 r = 2.3 × 10 light-years = 2.2 × 10 m 1 light-year

(

)

The tangential speed of the sun is

vT = rω = ( 2.2 × 1020 m )(1.1 × 10−15 rad/s ) = 2.4 × 105 m/s

(8.9)

b. The magnitude of the centripetal force that acts on the sun is

Fc

= mac = m r ω 2

Centripetal force

(

)(

)(

= 1.99 × 1030 kg 2.2 × 1020 m 1.1 × 10−15 rad /s

)

2

= 5.3 × 1020 N

52. REASONING The tangential acceleration and the centripetal acceleration of a point at a distance r from the rotation axis are given by Equations 8.10 and 8.11, respectively: aT = rα and a c = r ω 2 . After the drill has rotated through the angle in question, a c = 2 a T , or r ω 2 = 2rα

This expression can be used to find the angular acceleration α . Once the angular acceleration is known, Equation 8.8 can be used to find the desired angle. SOLUTION Solving the expression obtained above for α gives

α=

ω2 2

Chapter 8 Problems

421

Solving Equation 8.8 for θ (with ω0 = 0 rad/s since the drill starts from rest), and using the expression above for the angular acceleration α gives

ω 2 2 ω2 ω2 = 1.00 rad θ= = = 2α 2(ω 2 / 2) 2 ω 2 Note that since both Equations 8.10 and 8.11 require that the angles be expressed in radians, the final result for θ is in radians.

53.

SSM WWW REASONING AND SOLUTION acceleration of the motorcycle is a=

v − v0 t

=

From Equation 2.4, the linear

22.0 m/s − 0 m/s 2 = 2.44 m/s 9.00 s

Since the tire rolls without slipping, the linear acceleration equals the tangential acceleration of a point on the outer edge of the tire: a = aT . Solving Equation 8.13 for α gives

α=

aT 2.44 m/s 2 = = 8.71 rad/s2 r 0.280 m

54. REASONING AND SOLUTION The bike would travel with the same speed as a point on the wheel v = rω . It would then travel a distance

60 s x = v t = r ω t = ( 0.45 m )( 9.1 rad/s )( 35 min ) = 1 min

8.6 ×103 m

____________________________________________________________________________________________

55. REASONING The angular displacement θ of each wheel is given by Equation 8.7 (θ = ω0t + 12 α t 2 ) , which is one of the equations of rotational kinematics. In this expression

ω0 is the initial angular velocity, and α is the angular acceleration, neither of which is given

directly. Instead the initial linear velocity v0 and the linear acceleration a are given. However, we can relate these linear quantities to their analogous angular counterparts by means of the assumption that the wheels are rolling and not slipping. Then, according to Equation 8.12 (v0 = rω0), we know that ω0 = v0/r, where r is the radius of the wheels. Likewise, according to Equation 8.13 (a = rα), we know that α = a/r. Both Equations 8.12 and 8.13 are only valid if used with radian measure. Therefore, when we substitute the expressions for ω0 and α into Equation 8.7, the resulting value for the angular displacement θ will be in radians.

422 ROTATIONAL KINEMATICS

SOLUTION Substituting ω0 from Equation 8.12 and α from Equation 8.13 into Equation 8.7, we find that

v

a

θ = ω 0t + 12 α t 2 = 0 t + 12 t 2 r r 2 2 20.0 m/s 1 1.50 m/s = ( 8.00 s ) = 693 rad ( 8.00 s ) + 2 0.300 m 0.300 m

56. REASONING AND SOLUTION a. If the wheel does not slip, a point on the rim rotates about the axle with a speed vT = v = 15.0 m/s

For a point on the rim

ω = vT/r = (15.0 m/s)/(0.330 m) = 45.5 rad/s vT = rω = (0.175 m)(45.5 rad/s) = 7.96 m/s

b.

57. REASONING a. The constant angular acceleration α of the wheel is defined by Equation 8.4 as the change in the angular velocity, ω −ω0, divided by the elapsed time t, or α = (ω − ω0 ) / t . The time is known. Since the wheel rotates in the positive direction, its angular velocity is the same as its angular speed. However, the angular speed is related to the linear speed v of a wheel and its radius r by v = rω (Equation 8.12). Thus, ω = v / r , and we can write for the angular acceleration that v v0 ω − ω0 r − r v − v0 α= = = t t rt b. The angular displacement θ of each wheel can be obtained from Equation 8.7 of the equations of kinematics: θ = ω0t + 12 α t 2 , where ω0 = v0/r and α can be obtained as discussed in part (a). SOLUTION a. The angular acceleration of each wheel is

α=

v − v0 rt

=

2.1 m/s − 6.6 m/s = −1.4 rad/s 2 ( 0.65 m )( 5.0 s )

Chapter 8 Problems

423

b. The angular displacement of each wheel is v0 1 2 t + α t r 2

θ = ω0t + 12 α t 2 =

2 6.6 m/s ( 2 1 = 5.0 s ) + 2 ( −1.4 rad/s ) ( 5.0 s ) = +33 rad 0.65 m ______________________________________________________________________________

58. REASONING For a wheel that rolls without slipping, the relationship between its linear speed v and its angular speed ω is given by Equation 8.12 as v = rω, where r is the radius of a wheel. For a wheel that rolls without slipping, the relationship between the magnitude a of its linear acceleration and the magnitude α of the angular acceleration is given by Equation 8.13 as a = rα, where r is the radius of a wheel. The linear acceleration can be obtained using the equations of kinematics for linear motion, in particular, Equation 2.9.

SOLUTION a. From Equation 8.12 we have that

v = r ω = ( 0.320 m )( 288 rad /s ) = 92.2 m /s b. The magnitude of the angular acceleration is given by Equation 8.13 as α = a /r. The linear acceleration a is related to the initial and final linear speeds and the displacement x by v 2 − v02 Equation 2.9 from the equations of kinematics for linear motion; a = . Thus, the 2x magnitude of the angular acceleration is

(

)

v − v0 / ( 2 x ) a α= = r r 2

=

2

2

v − v0 2x r

2

2 2 92.2 m /s ) − ( 0 m /s ) ( = 2 ( 384 m )( 0.320 m )

= 34.6 rad /s

2

59. REASONING AND SOLUTION a. If the rope is not slipping on the cylinder, then the tangential speed of the teeth on the larger gear (gear 1) is 2.50 m/s. The angular speed of gear 1 is then

ω1 = v/r1 = (2.50 m/s)/(0.300 m) = 8.33 rad/s

424 ROTATIONAL KINEMATICS

The direction of the larger gear is counterclockwise . b. The gears are in contact and do not slip. This requires that the teeth on both gears move with the same tangential speed. vT1 = vT2 or

ω1r1 = ω2r2

So r

0.300 m

ω 2 = 1 ω1 = ( 8.33 rad/s ) = 14.7 rad/s 0.170 m r2 The direction of the smaller gear is clockwise .

60. REASONING The distance d traveled by the axle of a rolling wheel (radius = r) during one complete revolution is equal to the circumference (2π r) of the wheel: d = 2π r. Therefore, when the bicycle travels a total distance D in a race, and the wheel makes N revolutions, the total distance D is N times the circumference of the wheel:

D = Nd = N ( 2π r )

(1)

We will apply Equation (1) first to the smaller bicycle wheel to determine its radius r1. Equation (1) will then also determine the number of revolutions made by the larger bicycle wheel, which has a radius of r2 = r1 + 0.012 m.

SOLUTION Because 1 km = 1000 m, the total distance traveled during the race is D = (4520 km)[(1000 m)/(1 km)] = 4520×103 m. From Equation (1), then, the radius r1 of the smaller bicycle wheel is r1 =

4520 × 103 m D = = 0.330 m 2π N1 2π 2.18 × 106

(

)

The larger wheel, then, has a radius r2 = 0.330 m + 0.012 m = 0.342 m. Over the same distance D, this wheel would make N2 revolutions, where, by Equation (1), N2 =

D 4520 × 103 m = = 2.10 × 106 2π r2 2π ( 0.342 m )

Chapter 8 Problems

425

61. REASONING As a penny-farthing moves, both of its wheels roll without slipping. This means that the axle for each wheel moves through a linear distance (the distance through which the bicycle moves) that equals the circular arc length measured along the outer edge of the wheel. Since both axles move through the same linear distance, the circular arc length measured along the outer edge of the large front wheel must equal the circular arc length measured along the outer edge of the small rear wheel. In each case the arc length s is equal to the number n of revolutions times the circumference 2π r of the wheel (r = radius). SOLUTION Since the circular arc length measured along the outer edge of the large front wheel must equal the circular arc length measured along the outer edge of the small rear wheel, we have nRear 2π rRear = nFront 2π rFront Arc length for rear wheel

Solving for nRear gives nRear =

nFront rFront rRear

=

Arc length for front wheel

276 (1.20 m ) = 974 rev 0.340 m

62. REASONING While the ball is in the air, its angular speed ω is constant, and thus its angular displacement is given by θ = ω t (Equation 8.2). The angular speed of the ball is found by considering its rolling motion on the table top, because its angular speed does not change after it leaves the table. For rolling motion, the angular speed ω is related to the linear speed v by ω = v r (Equation 8.12), where r is the radius of the ball. In order to determine the time t the ball spends in the air, we treat it as a projectile launched horizontally. The vertical displacement of the ball is then given by y = v0 yt + 12 ayt 2 (Equation 3.5b), which we will use to determine the time t that elapses while the ball is in the air. SOLUTION Since the ball is launched horizontally in the projectile motion, its initial velocity v0 has no vertical component: v0y = 0 m/s. Solving y = v0 yt + 12 ayt 2 (Equation 3.5b) for the elapsed time t, we obtain

y = ( 0 m/s ) t + 12 a y t 2 = 12 a y t 2

or

t=

2y ay

(1)

426 ROTATIONAL KINEMATICS

Substituting Equation (1) and ω =

v (Equation 8.12) into θ = ω t (Equation 8.2) yields r

v 2y

θ = r ay ω t

(2)

We choose the upward direction to be positive. Once the ball leaves the table, it is in free fall, so the vertical acceleration of the ball is that due to gravity: ay = −9.80 m/s2. Further, the ball’s vertical displacement is negative as the ball falls to the floor: y = −2.10 m. The ball’s angular displacement while it is in the air is, from Equation (2),

θ=

63.

v r

2y 3.60 m/s = a y 0.200 m/s

SSM REASONING AND SOLUTION "axle" or the center of the moving quarter is

2 ( −2.10 m ) −9.80 m/s 2

= 11.8 rad

By inspection, the distance traveled by the

d = 2 π (2 r) = 4π r where r is the radius of the quarter. The distance d traveled by the "axle" of the moving quarter must be equal to the circular arc length s along the outer edge of the quarter. This arc length is s = rθ , where θ is the angle through which the quarter rotates. Thus,

4 π r = rθ so that θ = 4π rad . This is equivalent to

1 rev = 2 revolutions (4π rad) 2π rad 64. REASONING AND SOLUTION According to Equation 8.2, ω = ∆θ / ∆t . Since the angular speed of the sun is constant, ω = ω . Solving for ∆ t , we have ∆t =

∆θ

2π rad 1y 1 h 1 day = = 1.8 ×108 y − 15 ω 1.1× 10 rad/s 3600 s 24 h 365.25 day

Chapter 8 Problems

65.

427

SSM REASONING The tangential acceleration aT of the speedboat can be found by using Newton's second law, FT = maT , where FT is the net tangential force. Once the tangential acceleration of the boat is known, Equation 2.4 can be used to find the tangential speed of the boat 2.0 s into the turn. With the tangential speed and the radius of the turn known, Equation 5.2 can then be used to find the centripetal acceleration of the boat. SOLUTION a. From Newton's second law, we obtain

aT =

FT 550 N = = 2.5 m/s 2 m 220 kg

b. The tangential speed of the boat 2.0 s into the turn is, according to Equation 2.4, 2

1

v T = v 0T + aT t = 5.0 m/s + (2.5 m/s )(2.0 s) = 1.0 × 10 m/s

The centripetal acceleration of the boat is then v 2T

(1.0 × 10 1 m/s) 2 ac = = = 3.1 m/s 2 r 32 m

1 2

66. REASONING AND SOLUTION From Equation 8.6, θ = (ω 0 + ω ) t . Solving for t gives

t=

67.

2θ 2(85.1 rad) = = 5.22 s ω 0 + ω 18.5 rad/s +14.1 rad/s

SSM REASONING AND SOLUTION Since the angular speed of the fan decreases, the sign of the angular acceleration must be opposite to the sign for the angular velocity. Taking the angular velocity to be positive, the angular acceleration, therefore, must be a negative quantity. Using Equation 8.4 we obtain

ω 0 = ω − α t = 83.8 rad/s – (–42.0 rad/s 2 )(1.75 s) = 157.3 rad/s

68. REASONING The top of the racket has both tangential and centripetal acceleration components given by Equations 8.10 and 8.11, respectively: aT = rα and a c = r ω 2 . The total acceleration of the top of the racket is the resultant of these two components. Since these acceleration components are mutually perpendicular, their resultant can be found by using the Pythagorean theorem.

428 ROTATIONAL KINEMATICS

SOLUTION Employing the Pythagorean theorem, we obtain a=

aT2 + a 2c = (rα ) 2 + (rω 2 )2 = r α 2 + ω 4

Therefore,

a = (1.5 m) (160 rad/s2 )2 + (14 rad/s)4 = 380 m/s2

69. REASONING The length of tape that passes around the reel is just the average tangential speed of the tape times the time t. The average tangential speed vT is given by Equation 8.9

( vT = rω ) as the radius r times the average angular speed ω

in rad/s.

SOLUTION The length L of tape that passes around the reel in t = 13 s is L = vT t . Using Equation 8.9 to express the tangential speed, we find L = vT t = rω t = ( 0.014 m )( 3.4 rad/s )(13 s ) = 0.62 m

70. REASONING a. Since the angular velocity of the fan blade is changing, there are simultaneously a tangential acceleration aT and a centripetal acceleration ac that are oriented at right angles to each other. The drawing shows these two accelerations for a point on the tip of one of the blades (for clarity, the blade itself is not shown). The blade is rotating in the counterclockwise (positive) direction. The a=

magnitude ac2

+ aT2

of

the

total

acceleration

a

aT

φ ac

is

, according to the Pythagorean theorem.

The magnitude ac of the centripetal acceleration can be evaluated from ac = rω 2

(Equation 8.11), where ω is the final angular velocity. The final angular velocity can be determined from Equation 8.4 as ω = ω0 + α t . The magnitude aT of the tangential

acceleration follows from aT = rα (Equation 8.10).

b. From the drawing we see that the angle φ can be obtained by using trigonometry, φ = tan −1 ( aT / ac ) .

Chapter 8 Problems

429

SOLUTION a. Substituting ac = rω 2 (Equation 8.11) and aT = rα (Equation 8.10) into a = ac2 + aT2 gives

a = ac2 + aT2 =

( rω 2 )2 + ( rα )2

= r ω4 + α 2

The final angular velocity ω is related to the initial angular velocity ω0 by ω = ω0 + α t (see Equation 8.4). Thus, the magnitude of the total acceleration is

a = r ω4 +α 2 = r

(ω0 + α t )4 + α 2

= ( 0.380 m ) 1.50 rad/s + ( 2.00 rad/s 2 ) ( 0.500 s ) + ( 2.00 rad/s 2 ) = 2.49 m/s 2 4

2

b. The angle φ between the total acceleration a and the centripetal acceleration ac is (see the drawing above) aT ac

φ = tan −1

α −1 r α = tan −1 = tan 2 2 rω (ω0 + α t )

2.00 rad/s 2 = tan −1 = 17.7° 2 2 )( ( ) 1.50 rad/s + 2.00 rad/s 0.500 s where we have used the same substitutions for aT, ac, and ω as in part (a). ______________________________________________________________________________ 71.

SSM REASONING The tangential speed vT of a point on the “equator” of the baseball is given by Equation 8.9 as vT = rω, where r is the radius of the baseball and ω is its angular speed. The radius is given in the statement of the problem. The (constant) angular speed is related to that angle θ through which the ball rotates by Equation 8.2 as ω = θ /t, where we have assumed for convenience that θ0 = 0 rad when t0 = 0 s. Thus, the tangential speed of the ball is θ vT = r ω = r t The time t that the ball is in the air is equal to the distance x it travels divided by its linear speed v, t = x/v, so the tangential speed can be written as

430 ROTATIONAL KINEMATICS

θ θ rθ v vT = r = r = x x t v

SOLUTION The tangential speed of a point on the equator of the baseball is −2 r θ v ( 3.67 × 10 m ) ( 49.0 rad )( 42.5 m/s ) = = 4.63 m/s vT = x 16.5 m

72. REASONING The average angular velocity is defined as the angular displacement divided by the elapsed time (Equation 8.2). Therefore, the angular displacement is equal to the product of the average angular velocity and the elapsed time The elapsed time is given, so we need to determine the average angular velocity. We can do this by using the graph of angular velocity versus time that accompanies the problem.

Angular velocity

SOLUTION The angular displacement ∆θ is +15 rad/ s related to the average angular velocity ω and the elapsed time ∆t by Equation 8.2, ∆θ = ω ∆t . The elapsed time is given as 8.0 s. To obtain the average angular velocity, we need to extend the graph that accompanies this problem from a ω time of 5.0 s to 8.0 s. It can be seen from the +3.0 rad/s graph that the angular velocity increases by 0 +3.0 rad/s during each second. Therefore, when 3.0 s the time increases from 5.0 to 8.0 s, the angular velocity increases from +6.0 rad/s to 6 rad/s + 3×(3.0 rad/s) = +15 rad/s. A graph of the angular velocity from 0 to 8.0 s is shown at –9.0 rad/ s the right. The average angular velocity during this time is equal to one half the sum of the initial and final angular velocities:

Time (s) 8.0 s

ω = 12 (ω 0 + ω ) = 12 ( −9.0 rad/s + 15 rad/s ) = + 3.0 rad/s The angular displacement of the wheel from 0 to 8.0 s is ∆θ = ω ∆t = ( +3.0 rad/s )( 8.0 s ) = +24 rad

73. REASONING The time required for the change in the angular velocity to occur can be found by solving Equation 8.4 for t. In order to use Equation 8.4, however, we must know the initial angular velocity ω 0 . Equation 8.6 can be used to find the initial angular velocity.

431

Chapter 8 Problems

SOLUTION From Equation 8.6 we have 1 2

θ = (ω 0 + ω )t Solving for ω0 gives

ω0 =

2θ −ω t

Since the angular displacement θ is zero, ω0 = –ω. Solving ω = ω0 + α t (Equation 8.4) for t and using the fact that ω0 = –ω give

t=

2ω

α

=

2(−25.0 rad/s)

2 =

−4.00 rad/s

12.5 s

____________________________________________________________________________________________

74. REASONING The drawing shows a top view of the race car as it travels around the circular turn. Its acceleration a has two perpendicular components: a centripetal acceleration ac that arises because the car is moving on a circular path and a tangential acceleration aT due to the fact that the car has an angular acceleration and its angular velocity is increasing. We can determine the magnitude of the centripetal acceleration from Equation 8.11 as ac = rω2, since both r and ω are given in the statement of the problem. As the drawing shows, we can use trigonometry to determine the magnitude a of the total acceleration, since the angle (35.0°) between a and ac is given.

aT

a 35.0°

ac

Race car

SOLUTION Since the vectors ac and a are one side and the hypotenuse of a right triangle, we have that ac a= cos 35.0°

The magnitude of the centripetal acceleration is given by Equation 8.11 as ac = rω , so the magnitude of the total acceleration is 2

( 23.5 m )( 0.571 rad /s ) rω2 a= = = = 9.35 m /s 2 cos 35.0° cos 35.0° cos 35.0° ac

2

432 ROTATIONAL KINEMATICS

75. REASONING The golf ball must travel a distance equal to its diameter in a maximum time equal to the time required for one blade to move into the position of the previous blade. SOLUTION The time required for the golf ball to pass through the opening between two blades is given by ∆t = ∆θ / ω , with ω = 1.25 rad/s and ∆θ = (2π rad)/16 = 0.393 rad . Therefore, the ball must pass between two blades in a maximum time of

∆t =

0.393 rad = 0.314 s 1.25 rad/s

The minimum speed of the ball is

v=

∆x 4.50 × 10 –2 m = = 1.43 × 10–1 m/s ∆t 0.314 s

76. REASONING The wheels on both sides of the car have the same radius r = 0.350 m and undergo rolling motion, so we will use v = rω (Equation 8.12) to calculate their individual angular speeds:

ωleft =

vleft r

and

ωright =

vright

(1)

r

In Equations (1) the linear speeds vleft and vright at which the wheels on opposite sides of the car travel around the track differ. This is because the wheels on one side of the car are closer to the center of the track than are the wheels on the other side. As the car makes one complete lap of the track, therefore, both sets of wheels follow circular paths of different radii Rleft and Rright. The linear speed of each wheel is the circumference of its circular path divided by the elapsed time t, which is the same for both sets of wheels: vleft =

2π Rleft t

and

vright =

2π Rright

(2)

t

Substituting Equations (2) into Equations (1), we obtain

ωleft

2π Rleft 2π Rleft t = = r rt

2π Rright and

ωright =

t r

=

2π Rright rt

(3)

SOLUTION We will assume that the wheels on the left side of the car are closer to the center of the track than the wheels on the right side. We do not know the radii of the circular paths of either set of wheels, but the difference between them is Rright − Rleft = 1.60 m . We

Chapter 8 Problems

433

can now calculate the difference between the angular speeds of the wheels on the left and right sides of the car by subtracting ωleft from ωright [see Equations (3)]:

ωright − ωleft =

2π Rright rt

−

2π Rleft rt

=

(

2π Rright − Rleft rt

)=

2π (1.60 m )

( 0.350 m )(19.5 s )

= 1.47 rad/s

CHAPTER 9 ROTATIONAL DYNAMICS ANSWERS TO FOCUS ON CONCEPTS QUESTIONS ___________________________________________________________________________________________

1.

(d) A rigid body is in equilibrium if it has zero translational acceleration and zero angular acceleration. A body, such as a bicycle wheel, can be moving, but the translational and 2 2 angular accelerations must be zero (a = 0 m/s and α = 0 rad/s ).

2.

(e) As discussed in Section 9.2, a body is in equilibrium if the sum of the externally applied forces is zero and the sum of the externally applied torques is zero.

3.

(b) The torque τ3 is greater than τ2, because the lever arm for the force F3 is greater than that for F2. The lines of action for the forces F1 and F4 pass through the axis of rotation. Therefore, the lever arms for these forces are zero, and the forces produce no torque.

4.

(b) Since the counterclockwise direction is the positive direction for torque, the torque produced by the force F1 is τ1 = −(20.0 N)(0.500 m) and that produced by F2 is

τ2 = +(35.0 N)[(1.10 m)(cos 30.0°)]. The sum of these torques is the net torque. 5.

(e) The clockwise torque produced by F2 is balanced by the counterclockwise torque produced by F. The torque produced by F2 is (remembering that the counterclockwise direction is positive) τ2 = +F2[(80.0 cm − 20.0 cm)(sin 55.0°)], and the torque produced by F is

τ = −(175 N)(20.0 cm). Setting the sum of these torques equal to zero and solving for F2 gives the answer.

6.

(d) The sum of the forces (F − 2F + F) equals zero. Select an axis that passes through the center of the puck and is perpendicular to the screen. The sum of the torques [−FR + 2F(0) + FR ] equals zero, where R is the radius of the puck. Thus, the puck is in equilibrium.

7.

Magnitude of F1 = 12.0 N, Magnitude of F2 = 24.0 N

8.

(c) The horizontal component of F3 is balanced by F1, and the vertical component of F3 is balanced by F2. Thus, the net force and, hence, the translational acceleration of the box, is zero. For an axis of rotation at the center of the box and perpendicular to the screen, the forces F2 and F3 produce no torque, because their lines of action pass through the axis. The force F1 does produce a torque about the axis, so the net torque is not zero and the box will have an angular acceleration.

9.

Distance of center of gravity from support = 0.60 m.

Chapter 9 Answers to Focus on Concepts Questions

435

10. (b) The moment of inertia of each particle is given by Equation 9.6 as I = mr 2 , where m is its

mass and r is the perpendicular distance of the particle from the axis. Using this equation, the moment of inertia of each particle is: A: 10m0 r02 , B: 8m0 r02 , C: 9m0 r02 . 2

11. I = 1.7 kg⋅m

12. Magnitude α of the angular acceleration = 1.3 rad/s . 2

13. (a) According to Newton’s second law for rotational motion, Equation 9.7, the angular acceleration is equal to the torque exerted on the wheel (the product of the force magnitude and the lever arm) divided by the moment of inertia. Thus, the angular acceleration of the 2 smaller wheel is α = FR/(MR ) = F/(MR), while that of the larger wheel is F ( 2R ) α= = 12 ( F / MR ) , so the smaller wheel has twice the angular acceleration. 2 M ( 2R ) 14. Magnitude α of the angular acceleration = 12.0 rad/s2 15. (c) The translational kinetic energy is

1 2

Mv 2 , where v is the speed of the center of mass of

the wheel. The rotational kinetic energy is

1 2

I ω 2 , where I is the moment of inertia and ω is

the angular speed about the axis of rotation. Since I = MR and ω = v/R for rolling motion 2

2

( MR 2 ) Rv = 12 Mv2 , which is the same as the translational kinetic energy. Thus, the ratio of the two energies is 1. (See Equation 8.12), the rotational kinetic energy is

1 2

Iω 2 =

1 2

16. (c) As each hoop rolls down the incline, the total mechanical energy is conserved. Thus, the loss in potential energy is equal to the gain in the total kinetic energy (translational plus rotational). Because the hoops have the same mass and fall through the same vertical distance, they lose the same amount of potential energy. Moreover, both start from rest. Therefore, their total kinetic energies at the bottom are the same. 17. (d) As discussed in Section 9.6, the angular momentum a system is conserved (remains constant) if the net external torque acting on the system is zero. 18. (b) The rotational kinetic energy of a rotating body is KE R = 12 I ω 2 (see Equation 9.9),

where I is the moment of inertia and ω is the angular speed. We also know that her angular momentum, L = Iω (Equation 9.10), is conserved. Solving the last equation for I and L substituting the result into the first equation gives KE R = 12 I ω 2 = 12 ω 2 = 12 Lω . Since L ω is constant, the final rotational kinetic energy increases as ω increases.

19. I = 0.60 kg⋅m2

436 ROTATIONAL DYNAMICS

CHAPTER 9 ROTATIONAL DYNAMICS PROBLEMS 1.

REASONING AND SOLUTION

According to Equation 9.1, we have

Magnitude of torque = F where F is the magnitude of the applied force and is the lever arm. For an axis passing through the cable car at its center and perpendicular to the ground, the lever arm is one-half the length of the car or 4.60 m. Both people rotate the car in the same direction, so the torques that they create reinforce one another. Therefore, using Equation 9.1 to express the magnitude of each torque, we find that the magnitude of the net torque is Magnitude of net torque = (185 N )( 4.60 m ) + (185 N )( 4.60 m ) = 1.70 × 103 N ⋅ m

2.

REASONING The drawing shows the wheel as it rolls to the right, so the torque applied by the engine is assumed to be clockwise about the axis of rotation. The force of static friction that the ground applies to the wheel is labeled as fs. This force produces a counterclockwise torque τ about the axis of rotation, which is given by Equation 9.1 as τ = fs , where is the lever arm. Using this relation we can find the magnitude fs of the static frictional force.

Axis of rotation

r fs

SOLUTION The countertorque is given as τ = fs , where fs is the magnitude of the static is the lever arm. The lever arm is the distance between the line of frictional force and action of the force and the axis of rotation; in this case the lever arm is just the radius r of the tire. Solving for fs gives τ 295 N ⋅ m fs = = = 843 N 0.350 m

3.

SSM REASONING According to Equation 9.1, we have Magnitude of torque = F where F is the magnitude of the applied force and is the lever arm. From the figure in the text, the lever arm is given by = (0.28 m) sin 50.0° . Since both the magnitude of the torque and are known, Equation 9.1 can be solved for F.

Chapter 9 Problems

437

SOLUTION Solving Equation 9.1 for F, we have

4.

Magnitude of torque

=

45 N ⋅ m = 2.1×102 N (0.28 m) sin 50.0°

REASONING The net torque on the branch is the sum of the torques exerted by the children. Each individual torque τ is given by τ = F (Equation 9.1), where F is the magnitude of the force exerted on the branch by a child, and is the lever arm (see the diagram). The branch supports each child’s weight, so, by Newton’s third law, the magnitude F of the force exerted on the branch by each child has the same magnitude as the child’s weight: F = mg. Both forces are directed downwards. The lever arm for each force is the perpendicular distance between the axis and the force’s line of action, so we have = d cos θ (see the diagram).

d

Tree

F=

F

θ Axis of rotation

SOLUTION The mass of the first child is m1 = 44.0 kg. This child is a distance d1 = 1.30 m

from the tree trunk. The mass of the second child, hanging d2 = 2.10 m from the tree trunk, is m2 = 35.0 kg. Both children produce positive (counterclockwise) torques. The net torque exerted on the branch by the two children is then Στ = τ1 + τ 2 = F1 1 + F2

2

= m1 g d1 cos θ + m2 g d 2 cos θ = g cos θ ( m1d1 + m2 d1 ) F1

1

F2

2

Substituting the given values, we obtain

(

)(

)

Στ = 9.80 m/s 2 cos 27.0 ( 44.0 kg )(1.30 m ) + ( 35.0 kg )( 2.10 m ) = 1140 N ⋅ m

438 ROTATIONAL DYNAMICS

5.

axis

SSM REASONING In both parts of the problem, the magnitude of the torque is given by Equation 9.1 as the magnitude F of the force times the lever arm . In part (a), the lever arm is just the distance of 0.55 m given in the drawing. However, in part (b), the lever arm is less than the given distance and must be expressed using trigonometry as = ( 0.55 m ) sin θ . See the drawing at the right.

θ

0.55 m

lever arm 49 N

SOLUTION a. Using Equation 9.1, we find that

Magnitude of torque = F = ( 49 N )( 0.55 m ) = 27 N ⋅ m b. Again using Equation 9.1, this time with a lever arm of

= ( 0.55 m ) sin θ , we obtain

Magnitude of torque = 15 N ⋅ m = F = ( 49 N )( 0.55 m ) sin θ

6.

15 N ⋅ m ( 49 N )( 0.55 m )

15 N ⋅ m = 34° ( 49 N )( 0.55 m )

θ = sin −1

or

REASONING The maximum torque will occur when the force is applied perpendicular to the diagonal of the square as shown. The lever arm is half the length of the diagonal. From the Pythagorean theorem, the lever arm is, therefore, =

1 2

2

2

(0.40 m) + (0.40 m) = 0.28 m

Since the lever arm is now known, we can use Equation 9.1 to obtain the desired result directly. SOLUTION Equation 9.1 gives

τ = F = (15 N)(0.28 m) = 4.2 N ⋅ m

0.40 m

0.40 m

sin θ =

axis

F

439

Chapter 9 Problems

7.

SSM REASONING Each of the two forces produces a torque about the axis of rotation, one clockwise and the other counterclockwise. By setting the sum of the torques equal to zero, we will be able to determine the angle θ in the drawing.

F2 = 55.0 N

x

θ θ

Rod

F1 = 38.0 N 2

Hinge (axis of rotation)

90º

Table (Top view) SOLUTION The two forces act on the rod at a distance x from the hinge. The torque τ1 produced by the force F1 is given by τ1 = +F1

magnitude of the force and

1

1

(see Equation 9.1), where F1 is the

is the lever arm. It is a positive torque, since it tends to

produce a counterclockwise rotation. Since F1 is applied perpendicular to the rod,

1

= x.

The torque τ2 produced by F2 is τ2 = −F2 2 , where 2 = x sin θ (see the drawing). It is a negative torque, since it tends to produce a clockwise rotation. Setting the sum of the torques equal to zero, we have

+ F1

1

+ ( − F2

2

) =0

+F1 x − F2 ( x sin θ ) = 0

or

2

The distance x in this relation can be eliminated algebraically. Solving for the angle θ gives

sin θ =

8.

F1 F2

or

F1 −1 38.0 N = sin = 43.7° 55.0 N F2

θ = sin −1

REASONING We know that the torques generated by the two applied forces sum to zero. In order for this to be true, one of the forces must tend to produce a clockwise rotation about the pinned end of the meter stick, and the other must tend to produce a counterclockwise rotation about the pinned end. We will assume that the first force, applied perpendicular to

440 ROTATIONAL DYNAMICS

the length of the meter stick at its free end, tends to produce a counterclockwise rotation (see the drawing). Counterclockwise is the positive direction. F2

pinned end

d

θ θ F1

2

1

TOP VIEW

With this assumption, we obtain F1 1 − F2

2

The drawing shows that the lever arm stick:

1=

=0 1 of

or

F1

1

= F2

2

(1)

the force F1 is equal to the length of the meter

1.00 m. The force F2 is applied a distance d from the pinned end of the meter

stick. The lever arm

2

of the second force is 2

= d sin θ

(2)

SOLUTION Substituting Equation (2) into Equation (1) and solving for the distance d, we obtain F1

9.

1

= F2 ( d sin θ )

or

d=

F1 1 ( 2.00 N )(1.00 m ) = 0.667 m = F2 sin θ ( 6.00 N ) sin 30.0

REASONING AND SOLUTION The torque produced by each force of magnitude F is given by Equation 9.1, τ = F , where is the lever arm and the torque is positive since each force causes a counterclockwise rotation. In each case, the torque produced by the couple is equal to the sum of the individual torques produced by each member of the couple. a. When the axis passes through point A, the torque due to the force at A is zero. The lever arm for the force at C is L. Therefore, taking counterclockwise as the positive direction, we have τ = τ A + τ C = 0 + FL = FL

Chapter 9 Problems

441

b. Each force produces a counterclockwise rotation. The magnitude of each force is F and each force has a lever arm of L / 2 . Taking counterclockwise as the positive direction, we have L L τ = τ A + τ C = F + F = FL 2 2

c. When the axis passes through point C, the torque due to the force at C is zero. The lever arm for the force at A is L. Therefore, taking counterclockwise as the positive direction, we have τ = τ A + τ C = FL + 0 = FL Note that the value of the torque produced by the couple is the same in all three cases; in other words, when the couple acts on the tire wrench, the couple produces a torque that does not depend on the location of the axis.

10. REASONING AND SOLUTION The net torque about the axis in text drawing (a) is Στ = τ1 + τ2 = F1b − F2a = 0

Considering that F2 = 3F1, we have b – 3a = 0. The net torque in drawing (b) is then Στ = F1(1.00 m − a) − F2b = 0

or

1.00 m − a − 3b = 0

Solving the first equation for b, substituting into the second equation and rearranging, gives a = 0.100 m

and b = 0.300 m

11. REASONING Although this arrangement of body parts is vertical, we can apply Equation 9.3 to locate the overall center of gravity by simply replacing the horizontal position x by the vertical position y, as measured relative to the floor.

SOLUTION Using Equation 9.3, we have y cg = =

W1 y 1 + W2 y 2 + W3 y 3 W1 + W2 + W3 + 144 N g + 87 N g b438 N gb1.28 mgb b0.760 m gb b0.250 mg= 438 N + 144 N + 87 N

1.03 m

442 ROTATIONAL DYNAMICS

12. REASONING At every instant before the plank begins to tip, it is in equilibrium, and the net torque on it is zero: Στ = 0 (Equation 9.2). When the person reaches the maximum distance x along the overhanging part, the plank is just about to rotate about the right support. At that instant, the plank loses contact with the left support, which consequently exerts no force on it. This leaves only three vertical forces acting on the plank: the weight W of the plank, the force FR due to the right support, and the force P due to the person (see the free-body diagram of the plank). The force FR acts at the right support, which we take as the axis, so its lever arm is zero. The lever arm for the force P is the distance x. Since counterclockwise is the positive direction, Equation 9.2 gives Στ = W

W

− Px = 0

x=

or

W

W

(1)

P

FR d L/2

W

x

W Axis

P

Free-body diagram of the plank

SOLUTION The weight W = 225 N of the plank is known, and the force P due to the person is equal to the person’s weight: P = 450 N. This is because the plank supports the person against the pull of gravity, and Newton’s third law tells us that the person and the plank exert forces of equal magnitude on each other. The plank’s weight W acts at the center of the uniform plank, so we have (see the drawing) W

+ d = 12 L

or

W

= 12 L − d

(2)

where d = 1.1 m is the length of the overhanging part of the plank, and L = 5.0 m is the length of the entire plank. Substituting Equation (2) into Equation (1), we obtain

x=

W

( 12 L − d ) = ( 225 N ) 12 ( 5.0 m ) −1.1 m = 0.70 m P

450 N

Chapter 9 Problems

443

13. SSM REASONING The drawing shows the bridge and the four forces that act on it: the upward force F1 exerted on the left end by the support, the force due to the weight Wh of the hiker, the weight Wb of the bridge, and the upward force F2 exerted on the right side by the support. Since the bridge is in equilibrium, the sum of the torques about any axis of rotation must be zero ( Στ = 0 ) , and the sum of the forces in the vertical direction must be zero

( ΣFy = 0) . These two conditions will allow us to determine the magnitudes of F1 and F2. F1

+y

F2 Axis

Wh

+τ +x

1 5

L

1 2

L

Wb

SOLUTION a. We will begin by taking the axis of rotation about the right end of the bridge. The torque produced by F2 is zero, since its lever arm is zero. When we set the sum of the torques equal to zero, the resulting equation will have only one unknown, F1, in it. Setting the sum of the torques produced by the three forces equal to zero gives Στ = − F1L + Wh

( 45 L ) + Wb ( 12 L ) = 0

Algebraically eliminating the length L of the bridge from this equation and solving for F1 gives F1 = 54 Wh + 12 Wb = 54 ( 985 N ) + 12 ( 3610 N ) = 2590 N b. Since the bridge is in equilibrium, the sum of the forces in the vertical direction must be zero: Σ Fy = F1 − Wh − Wb + F2 = 0

Solving for F2 gives F2 = − F1 + Wh + Wb = −2590 N + 985 N + 3610 N = 2010 N

14. REASONING The net torque is the sum of the torques produced by the three forces: Στ = τA + τ B + τ D . The magnitude of a torque is the magnitude of the force times the lever arm of the force, according to Equation 9.1. The lever arm is the perpendicular distance between the line of action of the force and the axis. A torque that tends to produce a

444 ROTATIONAL DYNAMICS

counterclockwise rotation about the axis is a positive torque. Since the piece of wood is at equilibrium, the net torque is equal to zero. SOLUTION Let L be the length of the short side of the rectangle, so that the length of the long side is 2L. The counterclockwise torque produced by the force at corner B is + F B , and the clockwise torque produced by the force at corner D is − F D . Assuming that the force at A (directed along the short side of the rectangle) points toward corner B, the counterclockwise torque produced by this force is + FA A . Setting the net torque equal to zero gives: Στ = FA A + F B − F D = 0

FA L + (12 N )

( 12 L ) − (12 N ) L = 0

The length L can be eliminated algebraically from this result, which can then be solved for FA: FA = − (12 N )

( 12 ) + 12 N =

6.0 N (pointing toward corner B)

Since the value calculated for FA is positive, our assumption that FA points toward corner B must have been correct. Otherwise, the result for FA would have been negative.

15. REASONING Since the forearm is in equilibrium, the sum of the torques about any axis of rotation must be zero ( Στ = 0 ) . For convenience, we will take the elbow joint to be the axis of rotation. SOLUTION Let M and F be the magnitudes of the forces that the flexor muscle and test apparatus, respectively, exert on the forearm, and let M and F be the respective lever arms about the elbow joint. Setting the sum of the torques about the elbow joint equal to zero (with counterclockwise torques being taken as positive), we have Στ = M

Solving for M yields M=

F

F

=

M

−F

F

=0

(190 N )( 0.34 m ) = 1200 N

M

The direction of the force is to the left .

0.054 m

Chapter 9 Problems

16. REASONING The truck is subject to three vertical forces only (see the free-body diagram), and is in equilibrium. Therefore, the conditions ΣFy = 0 (Equation 4.9b) and

Free-body diagram of the truck

Στ = 0 (Equation 9.2) apply to the forces and torques acting on it. The ground exerts upward forces FR on the rear wheels and FF on the front wheels, and the earth exerts a downward weight force W on the truck’s center of gravity. The net vertical force on the truck must be zero, so we find FR + FF − W = 0

445

W

FF

FR

Rotation axis

(1)

F

R

where we have assumed upward to be the positive direction. To apply the zero net torque condition (Equation 9.2), we choose a rotation axis located on the ground, directly below the truck’s center of gravity (see the diagram). The weight force W has no lever arm about this axis, and so it generates no torque. The force FF exerts a counterclockwise torque about this axis, and the force FR exerts a clockwise torque about this axis. Since counterclockwise is the positive direction, we find from Equation 9.2 that Στ = FF

F

− FR

R

=0

or

FF

F

= FR

(2)

R

We will use Equations (1) and (2) to find the two unknown forces.

SOLUTION a. In order to find the force magnitude FF, we must eliminate the unknown force magnitude FR from Equations (1) and (2). Solving Equation (1) for FR yields FR = W − FF = mg − FF . Substituting this expression into Equation (2), we obtain

FF

F

= ( mg − FF )

R

= mg

R

− FF

R

or

FF (

F

+

R

) = mg

R

(3)

Solving Equation (3) for the force magnitude FF and noting from the drawing in the text that F

= 2.30 m and

R

FF =

= 0.63 m, we find that

mg F+

R R

=

( 7460 kg ) ( 9.80 m/s 2 ) ( 0.63 m ) 2.30 m + 0.63 m

= 1.57 × 104 N

446 ROTATIONAL DYNAMICS

b. Returning to Equation (1), we obtain the magnitude of the force on the rear wheels:

(

)

FR = W − FF = mg − FF = ( 7460 kg ) 9.80 m/s 2 − 1.57 × 10 4 N = 5.74 × 10 4 N

17. REASONING Multiple-Concept Example 8 discusses the static stability factor (SSF) and rollover. In that example, it is determined that the maximum speed v at which a vehicle can

negotiate a curve of radius r is related to the SSF according to v = rg ( SSF ) . No value is

given for the radius of the turn. However, by applying this result separately to the sport utility vehicle (SUV) and to the sports car (SC), we will be able to eliminate r algebraically and determine the maximum speed at which the sports car can negotiate the curve without rolling over. SOLUTION Applying v = rg ( SSF ) to each vehicle, we obtain vSUV = rg ( SSF )SUV

and

vSC = rg ( SSF )SC

Dividing these two expressions gives vSC vSUV

=

rg ( SSF )SC

rg ( SSF )SUV

or

vSC = vSUV

(SSF )SC = (18 m/s ) (SSF )SUV

1.4 = 24 m/s 0.80

18. REASONING Since the wheelbarrow is in equilibrium, the net torque acting on it must be zero: Στ = 0 (Equation 9.2). The magnitude of a torque is the magnitude of the force times the lever arm of the force (see Equation 9.1). The lever arm is the perpendicular distance between the line of action of the force and the axis. A torque that tends to produce a counterclockwise rotation about the axis is a positive torque.

SOLUTION The lever arms for the forces can be obtained from the distances shown in the text drawing for each design. Equation 9.1 can be used to obtain the magnitude of each torque. We will then write an expression for the zero net torque for each design. These expressions can be solved for the magnitude F of the man’s force in each case:

Chapter 9 Problems

Left design

447

b gb gb gb g b g + 60.0 N g b525 N gb0.400 mgb b0.600 mg= 189 N F=

Στ = − 525 N 0.400 m − 60.0 N 0.600 m + F 1.300 m = 0

1.300 m

Right design

b gb g b g b60.0 N gb0.600 mg= 27.7 N F=

Στ = − 60.0 N 0.600 m + F 1.300 m = 0

1.300 m

19. SSM REASONING AND SOLUTION The net torque about an axis through the contact point between the tray and the thumb is 2

2

Στ = F(0.0400 m) − (0.250 kg)(9.80 m/s )(0.320 m) − (1.00 kg)(9.80 m/s )(0.180 m) 2

− (0.200 kg)(9.80 m/s )(0.140 m) = 0

F = 70.6 N, up Similarly, the net torque about an axis through the point of contact between the tray and the finger is 2

2

Στ = T (0.0400 m) − (0.250 kg)(9.80 m/s )(0.280 m) − (1.00 kg)(9.80 m/s )(0.140 m) 2

− (0.200 kg)(9.80 m/s )(0.100 m) = 0

T = 56.4 N, down

20. REASONING The jet is in equilibrium, so the sum of the external forces is zero, and the sum of the external torques is zero. We can use these two conditions to evaluate the forces exerted on the wheels. SOLUTION a. Let Ff be the magnitude of the normal force that the ground exerts on the front wheel. Since the net torque acting on the plane is zero, we have (using an axis through the points of contact between the rear wheels and the ground) Στ = −W

where W is the weight of the plane, and respectively. Thus,

w

w

+ Ff

and

f

f

=0

are the lever arms for the forces W and Ff,

448 ROTATIONAL DYNAMICS

Στ = −(1.00 × 106 N)(15.0 m − 12.6 m) + Ff (15.0 m) = 0 Solving for Ff gives Ff = 1.60 × 10 5 N . b. Setting the sum of the vertical forces equal to zero yields ΣFy = Ff + 2Fr − W = 0 where the factor of 2 arises because there are two rear wheels. Substituting the data gives ΣFy = 1.60 × 105 N + 2Fr − 1.00 × 106 N = 0 Fr = 4.20 ×105 N

21. REASONING Since the beam is in equilibrium, the sum of the horizontal and vertical forces must be zero: ΣFx = 0 and ΣFy = 0 (Equations 9.4a and b). In addition, the net

torque about any axis of rotation must also be zero: Στ = 0 (Equation 9.2). SOLUTION The drawing shows the beam, as well as its weight W, the force P that the pin exerts on the right end of the beam, and the horizontal and vertical forces, H and V, applied to the left end of the beam by the hinge. Assuming that upward and to the right are the positive directions, we obtain the following expressions by setting the sum of the vertical and the sum of the horizontal forces equal to zero:

V H

P

Beam

θ

θ

W Brace

Pin (rotational axis)

Horizontal forces

P cosθ − H = 0

(1)

Vertical forces

P sin θ + V − W = 0

(2)

Using a rotational axis perpendicular to the plane of the paper and passing through the pin, and remembering that counterclockwise torques are positive, we also set the sum of the torques equal to zero. In doing so, we use L to denote the length of the beam and note that the lever arms for W and V are 12 L and L, respectively. The forces P and H create no torques relative to this axis, because their lines of action pass directly through it.

449

Chapter 9 Problems

W

Torques

c L h− VL = 0 1 2

( 3)

Since L can be eliminated algebraically, Equation (3) may be solved immediately for V:

V = 21 W =

1 2

b340 N g=

170 N

Substituting this result into Equation (2) gives

P sin θ + 21 W − W = 0 P=

340 N W = = 270 N 2 sin θ 2 sin 39 °

Substituting this result into Equation (1) yields

W I F G H2 sinθ JKcosθ − H = 0 H=

340 N W = = 210 N 2 tan θ 2 tan 39 °

22. REASONING AND SOLUTION The net torque about an axis through the elbow joint is

Στ = (111 N)(0.300 m) − (22.0 N)(0.150 m) − (0.0250 m)M = 0 or M = 1.20 ×103 N

23. SSM REASONING The drawing shows the forces acting on the board, which has a length L. The ground exerts the vertical normal force V on the lower end of the board. The maximum force of static friction has a magnitude of µsV and acts horizontally on the lower end of the board. The weight W acts downward at the board's center. The vertical wall applies a force P to the upper end of the board, this force being perpendicular to the wall since the wall is smooth (i.e., there is no friction along the wall). We take upward and to the right as our positive directions. Then, since the horizontal forces balance to zero, we have

µsV – P = 0

P

V

θ

W

µ sV

(1)

The vertical forces also balance to zero giving

V – W = 0

(2)

450 ROTATIONAL DYNAMICS

Using an axis through the lower end of the board, we express the fact that the torques balance to zero as L (3) PL sin θ − W cos θ = 0 2 Equations (1), (2), and (3) may then be combined to yield an expression for θ . SOLUTION Rearranging Equation (3) gives

tan θ =

W 2P

(4)

But, P = µsV according to Equation (1), and W = V according to Equation (2). Substituting these results into Equation (4) gives tan θ = Therefore,

V 2 µsV

=

1 2 µs

1 1 = tan −1 = 37.6° 2 µs 2(0.650)

θ = tan −1

24. REASONING When the wheel is resting on the ground it is in equilibrium, so the sum of the torques about any axis of rotation is zero ( Στ = 0 ) . This equilibrium condition will provide us with a relation between the magnitude of F and the normal force that the ground exerts on the wheel. When F is large enough, the wheel will rise up off the ground, and the normal force will become zero. From our relation, we can determine the magnitude of F when this happens. SOLUTION The free body diagram shows the forces acting on the wheel: its weight W, the normal force FN, the horizontal force F, and the force FE that the edge of the step exerts on the wheel. We select the axis of rotation to be at the edge of the step, so that the torque produced by FE is zero. Letting

r FN

F

Axis of Rotation

FE h W

N , W , and F represent the lever arms for the forces FN, W, and F, the sum of the torques is

Στ = − FN

r2 − (r − h) N

Solving this equation for F gives

2

+W

r2 − (r − h) W

2

− F (r − h) = 0 F

Chapter 9 Problems

(W − FN ) F=

r2 − (r − h)

451

2

r −h

When the bicycle wheel just begins to lift off the ground the normal force becomes zero (FN = 0 N). When this happens, the magnitude of F is

(W − FN ) F=

r 2 − ( r − h) r−h

2

=

( 25.0 N − 0 N ) ( 0.340 m )2 − ( 0.340 m − 0.120 m )2 0.340 m − 0.120 m

= 29 N

25. REASONING The drawing shows the beam and the five forces that act on it: the horizontal and vertical components Sx and Sy that the wall exerts on the left end of the beam, the weight Wb of the beam, the force due to the weight Wc of the crate, and the tension T in the cable. The beam is uniform, so its center of gravity is at the center of the beam, which is where its weight can be assumed to act. Since the beam is in equilibrium, the sum of the torques about any axis of rotation must be zero ( Στ = 0 ) , and the sum of the forces in the

(

)

horizontal and vertical directions must be zero ΣFx = 0, ΣFy = 0 . These three conditions will allow us to determine the magnitudes of Sx, Sy, and T.

T +y 50.0°

+τ

30.0° +x Wc Sy Axis

Wb 30.0° Sx

SOLUTION a. We will begin by taking the axis of rotation to be at the left end of the beam. Then the torques produced by Sx and Sy are zero, since their lever arms are zero. When we set the sum of the torques equal to zero, the resulting equation will have only one unknown, T, in it. Setting the sum of the torques produced by the three forces equal to zero gives (with L equal to the length of the beam)

452 ROTATIONAL DYNAMICS

Στ = −Wb

( 12 L cos 30.0°) − Wc ( L cos 30.0° ) + T ( L sin 80.0°) = 0

Algebraically eliminating L from this equation and solving for T gives T= =

Wb

( 12 cos 30.0°) + Wc ( cos 30.0°) sin 80.0°

(1220 N ) ( 12 cos 30.0° ) + (1960 N )( cos 30.0° ) sin 80.0°

= 2260 N

b. Since the beam is in equilibrium, the sum of the forces in the vertical direction is zero:

Σ Fy = + S y − Wb − Wc + T sin 50.0° = 0 Solving for Sy gives

S y = Wb + Wc − T sin 50.0° = 1220 N + 1960 N − ( 2260 N ) sin 50.0° = 1450 N The sum of the forces in the horizontal direction must also be zero: Σ Fx = + S x − T cos 50.0° = 0 so that

S x = T cos 50.0° = ( 2260 N ) cos 50.0° = 1450 N

26. REASONING If we assume that the system is in equilibrium, we know that the vector sum of all the forces, as well as the vector sum of all the torques, that act on the system must be zero. The figure below shows a free body diagram for the boom. Since the boom is assumed to be uniform, its weight WB is located at its center of gravity, which coincides with its geometrical center. There is a tension T in the cable that acts at an angle θ to the horizontal, as shown. At the hinge pin P, there are two forces acting. The vertical force V that acts on the end of the boom prevents the boom from falling down. The horizontal force H that also acts at the hinge pin prevents the boom from sliding to the left. The weight WL of the wrecking ball (the "load") acts at the end of the boom.

Chapter 9 Problems

453

θ = 32°

T φ–θ

WB

V θ = 32°

WL

P

φ = 48°

H

By applying the equilibrium conditions to the boom, we can determine the desired forces. SOLUTION The directions upward and to the right will be taken as the positive directions. In the x direction we have (1) ∑ Fx = H − T cos θ = 0 while in the y direction we have

∑ Fy = V − T sin θ − WL − WB = 0

(2)

Equations (1) and (2) give us two equations in three unknown. We must, therefore, find a third equation that can be used to determine one of the unknowns. We can get the third equation from the torque equation. In order to write the torque equation, we must first pick an axis of rotation and determine the lever arms for the forces involved. Since both V and H are unknown, we can eliminate them from the torque equation by picking the rotation axis through the point P (then both V and H have zero lever arms). If we let the boom have a length L, then the lever arm for WL is

L cos φ , while the lever arm for WB is ( L / 2) cos φ . From the figure, we see that the lever arm for T is L sin(φ – θ ) . If we take counterclockwise torques as positive, then the torque equation is L cos φ ∑ τ = −WB − WL L cos φ + TL sin (φ − θ ) = 0 2 Solving for T, we have 1 W +W L (3) T= 2 B cos φ sin(φ –θ )

a. From Equation (3) the tension in the support cable is T=

1 (3600 2

N) + 4800 N

sin(48° – 32°)

cos 48° = 1.6 × 104 N

454 ROTATIONAL DYNAMICS

b. The force exerted on the lower end of the hinge at the point P is the vector sum of the forces H and V. According to Equation (1),

(

)

H = T cos θ = 1.6 × 104 N cos 32° = 1.4 × 104 N

and, from Equation (2)

(

)

V = WL + WB + T sin θ = 4800 N + 3600 N + 1.6 × 104 N sin 32° = 1.7 × 104 N

Since the forces H and V are at right angles to each other, the magnitude of their vector sum can be found from the Pythagorean theorem:

FP = H 2 + V 2 = (1.4 ×104 N)2 + (1.7 ×104 N)2 = 2.2 ×104 N

27. SSM REASONING Since the man holds the ball motionless, the ball and the arm are in equilibrium. Therefore, the net force, as well as the net torque about any axis, must be zero. SOLUTION Using Equation 9.1, the net torque about an axis through the elbow joint is Στ = M(0.0510 m) – (22.0 N)(0.140 m) – (178 N)(0.330 m) = 0 Solving this expression for M gives M = 1.21× 103 N . The net torque about an axis through the center of gravity is Στ = – (1210 N)(0.0890 m) + F(0.140 m) – (178 N)(0.190 m) = 0 Solving this expression for F gives F = 1.01× 103 N . Since the forces must add to give a net force of zero, we know that the direction of F is downward .

28. REASONING Although the crate is in translational motion, it undergoes no angular acceleration. Therefore, the net torque acting on the crate must be zero: Στ = 0 (Equation 9.2). The four forces acting on the crate appear in the free-body diagram: its weight W, the kinetic friction force fk, the normal force FN, and the tension T in the strap. We will take the edge of the crate sliding along the floor as the rotation axis for applying Equation 9.2. Both the friction force and the normal force act at this point. These two forces, therefore, generate no torque about the axis of rotation, so the clockwise torque of the crate’s weight W must balance the counterclockwise torque of the tension T in the strap:

Chapter 9 Problems

455

T

L/2 H/2

θ

d

d θ

W

fk

25°

W

w

FN

Lever arm of the crate’s weight

Free-body diagram of the crate

T

T

=W

W

or

T=

W

W

=

mg

T

W

(1)

T

We will apply trigonometry to determine the lever arms W and T for the weight and the tension, respectively, and then calculate the magnitude T of the tension in the strap. SOLUTION The lever arm

above, and is given by

W

W

of the crate’s weight is shown in the right-hand diagram

(

)

= d cos θ + 25 , where d is the distance between the axis of

rotation (lower edge of the crate) and the crate’s center of gravity, and θ is the angle between that line and the bottom of the crate. The right triangle in the free-body diagram of the crate (drawing on the left) shows how we can use trigonometry to determine both the length d and the angle θ. The length d is the hypotenuse of that right triangle, and the other sides are the half-height (H/2 = 0.20 m) and half-length (L/2 = 0.45 m) of the crate, so by the Pythagorean theorem (Equation 1.7) we find that 2

2

H L d = + = 2 2

( 0.20 m )2 + ( 0.45 m )2

= 0.49 m

We can find the angle θ from the inverse tangent function:

H 2 −1 0.40 m = tan = 24 0.90 m L 2

θ = tan −1

456 ROTATIONAL DYNAMICS

The lever arm T

T

(

of the tension force is illustrated in the drawing below, where we see that

)

= L sin 61 − 25 = L sin 36 . Therefore, from Equation (1), the magnitude of the tension

in the strap is

T=

mg

W T

=

(

mgd cos 24 + 25 L sin 36

) = ( 72 kg ) ( 9.80 m/s2 ) ( 0.49 m ) cos 49 ( 0.90 m ) sin 36

= 430 N

T 61° L

25°

36°

T

Lever arm of the tension force

29. SSM WWW REASONING AND SOLUTION Consider the left board, which has a length L and a weight of (356 N)/2 = 178 N. Let Fv be the upward normal force exerted by the ground on the board. This force balances the weight, so Fv = 178 N. Let fs be the force of static friction, which acts horizontally on the end of the board in contact with the ground. fs points to the right. Since the board is in equilibrium, the net torque acting on the board through any axis must be zero. Measuring the torques with respect to an axis through the apex of the triangle formed by the boards, we have

L + (178 N)(sin 30.0°) + fs(L cos 30.0°) – Fv (L sin 30.0°) = 0 2 or

44.5 N + fs cos 30.0° – FV sin 30.0° = 0

so that fs =

(178 N)(sin 30.0°) – 44.5 N = 51.4 N cos 30.0°

Chapter 9 Problems

457

30. REASONING AND SOLUTION The weight W of the left side of the ladder, the normal force FN of the floor on the left leg of the ladder, the tension T in the crossbar, and the reaction force R due to the right-hand side of the ladder, are shown in the following figure. In the vertical direction −W + FN = 0, so that 2

FN = W = mg = (10.0 kg)(9.80 m/s ) = 98.0 N

R

In the horizontal direction it is clear that R = T. The net torque about the base of the ladder is

75.0º

Στ = – T [(1.00 m) sin 75.0o] – W [(2.00 m) cos 75.0o] + R [(4.00 m) sin 75.0o] = 0

W

Substituting for W and using R = T, we obtain

T=

FN

T

( 9.80 N )( 2.00 m ) cos 75.0° = 17.5 N ( 3.00 m ) sin 75.0°

31. REASONING The net torque Στ acting on the CD is given by Newton’s second law for rotational motion (Equation 9.7) as Στ = Ι α, where I is the moment of inertia of the CD and α is its angular acceleration. The moment of inertia can be obtained directly from Table 9.1, and the angular acceleration can be found from its definition (Equation 8.4) as the change in the CD’s angular velocity divided by the elapsed time. SOLUTION The net torque is Στ = Ι α. Assuming that the CD is a solid disk, its moment of inertia can be found from Table 9.1 as I = 12 MR 2 , where M and R are the mass and radius of the CD. Thus, the net torque is Στ = I α =

( 12 MR2 )α

The angular acceleration is given by Equation 8.4 as α = (ω − ω0 ) / t , where ω and ω0 are the final and initial angular velocities, respectively, and t is the elapsed time. Substituting this expression for α into Newton’s second law yields

Στ =

( 12 MR 2 )α = ( 12 MR2 ) (

)(

ω − ω0 t

)

2 21 rad/s − 0 rad/s −4 = 12 17 × 10−3 kg 6.0 × 10−2 m = 8.0 × 10 N ⋅ m 0.80 s

458 ROTATIONAL DYNAMICS

32. REASONING AND SOLUTION a. The net torque on the disk about the axle is Στ = F1R − F2R = (0.314 m)(90.0 N − 125 N) = −11 N ⋅ m b. The angular acceleration is given by α = Στ /I. From Table 9.1, the moment of inertia of the disk is 2

2

2

I = (1/2) MR = (1/2)(24.3 kg)(0.314 m) = 1.20 kg⋅m

α = (−11 N⋅m)/(1.20 kg⋅m2) = –9.2 rad / s 2

33. REASONING The moment of inertia of the stool is the sum of the individual moments of inertia of its parts. According to Table 9.1, a circular disk of radius R has a moment of inertia of I disk = 12 M disk R 2 with respect to an axis perpendicular to the disk center. Each thin rod is attached perpendicular to the disk at its outer edge. Therefore, each particle in a rod is located at a perpendicular distance from the axis that is equal to the radius of the disk. This means that each of the rods has a moment of inertia of Irod = Mrod R2. SOLUTION Remembering that the stool has three legs, we find that the its moment of inertia is I stool = I disk + 3 I rod = 21 M disk R 2 + 3 M rod R 2 =

1 2

b1.2 kg gb0.16 mg+ 3b0.15 kg gb0.16 mg= 2

2

0.027 kg ⋅ m 2

34. REASONING According to Table 9.1 in the text, the moment of inertia I of the disk is 1 2

I = MR 2 , where M is the disk’s mass and R is its radius. Thus, we can determine the mass from this expression, provided we can obtain a value for I. To obtain the value for I, we will use Newton’s second law for rotational motion, Στ = I α (Equation 9.7), where Στ is the net torque and α is the angular acceleration.

SOLUTION From the moment of inertia of the disk, we have 1 2

I = MR 2

or

M=

2I R2

Using Newton’s second law for rotational motion, we find for I that

(1)

Chapter 9 Problems

Στ = I α

or

I=

459

Στ

α

Substituting this expression for I into Equation (1) gives M=

2 I 2Στ = R 2 R 2α

(2)

The net torque Στ is due to the 45-N force alone. According to Equation 9.1, the magnitude of the torque that a force of magnitude F produces is F , where is the lever arm of the force. In this case, we have = R , since the force is applied tangentially to the disk and perpendicular to the radius. Substituting Στ = FR into Equation (2) gives

M=

2Στ Rα 2

=

2 FR Rα 2

=

2 ( 45 N ) 2F = = 5.0 kg Rα ( 0.15 m ) 120 rad/s 2

(

)

35. REASONING Newton’s second law for rotational motion (Equation 9.2) indicates that the net external torque is equal to the moment of inertia times the angular acceleration. To determine the angular acceleration, we will use Equation 8.7 from the equations of rotational kinematics. This equation indicates that the angular displacement θ is given by θ = ω0t + 12 α t 2 , where ω0 is the initial angular velocity, t is the time, and α is the angular acceleration. Since both wheels start from rest, ω0 = 0 rad/s for each. Furthermore, each

wheel makes the same number of revolutions in the same time, so θ and t are also the same for each. Therefore, the angular acceleration α must be the same for each. SOLUTION Using Equation 9.2, the net torque Στ that acts on each wheel is given by Στ = I α , where I is the moment of inertia and α is the angular acceleration. Solving

2 (θ − ω0t ) Equation 8.7 for the angular acceleration α = and substituting the result into t2 Equation 9.2 gives 2 (θ − ω0t ) Στ = I α = I t2

Table 9.1 indicates that the moment of inertia of a hoop is I hoop = MR 2 , while the moment of inertia of a disk is I disk = 21 MR 2 . The net external torques acting on the hoop and the disk are:

460 ROTATIONAL DYNAMICS

Hoop

2 (θ − ω0t ) Στ = I hoopα = MR 2 t2 2 2 (13 rad ) − 2 ( 0 rad/s )( 8.0 s ) = 0.20 N ⋅ m = ( 4.0 kg )( 0.35 m ) (8.0 s )2

Disk

2 (θ − ω0t ) Στ = I diskα = 12 MR 2 t2 =

1 2

2 (13 rad ) − 2 ( 0 rad/s )(8.0 s ) = 0.10 N ⋅ m (8.0 s )2

( 4.0 kg )( 0.35 m )2

36. REASONING The ladder is subject to three vertical forces: the P FN upward pull P of the painter on the top end of the ladder, the upward normal force FN that the ground exerts on the bottom end of the ladder, and the downward Axis of W force W of the ladder’s weight, rotation which acts at the ladder’s center of gravity, halfway between the Free-body diagram of the ladder ends (see the free-body diagram of the ladder). The bottom end serves as the axis of rotation. The normal force FN is applied at the axis, so it has no lever arm. The net torque acting on the ladder can be obtained with the aid of Equation 9.1:

Στ = P

( )

P − W W + FN 0 Torque Torque Torque due to P due to W due to F N

= PL − W

( 12 L) = PL − 12 mgL

(1)

where L is the length of the ladder and m is its mass. Once we know the net torque Σ τ acting on the ladder, we will use Newton’s second law for rotation, Στ = Iα (Equation 9.7) to determine the moment of inertia I of the ladder. SOLUTION a. From Equation (1), the net torque acting on the ladder is

(

)

Στ = PL − 12 mgL = ( 245 N )( 9.75 m ) − 12 ( 23.2 kg ) 9.80 m/s 2 ( 9.75 m ) = 1280 N ⋅ m

Chapter 9 Problems

461

b. The ladder’s moment of inertia is found from α = Στ I (Equation 9.7). Using the result found in part a, we obtain I=

Στ

=

α

1280 N ⋅ m = 711 kg ⋅ m 2 2 1.80 rad/s

37. SSM REASONING The rotational analog of Newton's second law is given by Equation 9.7, ∑τ = Iα . Since the person pushes on the outer edge of one section of the door with a force F that is directed perpendicular to the section, the torque exerted on the door has a magnitude of FL, where the lever arm L is equal to the width of one section. Once the moment of inertia is known, Equation 9.7 can be solved for the angular acceleration α. The moment of inertia of the door relative to the rotation axis is I = 4IP, where IP is the moment of inertia for one section. According to Table 9.1, we find I P = 13 ML2 , so that the rotational inertia of the door is I = 43 ML2 .

SOLUTION Solving Equation 9.7 for α, and using the expression for I determined above, we have

α=

FL = 4 ML2 3

4 3

68 N F = 4 = 0.50 rad/s 2 ML 3 (85 kg)(1.2 m)

462 ROTATIONAL DYNAMICS

38. REASONING The angular acceleration α that results from the application of a net external torque Στ to a rigid object with a moment of inertia I is given by Newton’s second law for Στ rotational motion: α = (Equation 9.7). By applying this equation to each of the two I situations described in the problem statement, we will be able to obtain the unknown angular acceleration. SOLUTION The drawings at the right illustrate the two types of rotational motion of the object. In applying Newton’s second law for rotational motion, we need to keep in mind that the moment of inertia depends on where the axis is. For axis 1 (see top drawing), piece B is rotating about one of its ends, so according to Table 9.1, the moment of inertia is

B Axis 1

1 3

I for axis 1 = M B L2B , where MB and LB are the mass

Axis 2

and length of piece B, respectively. For axis 2 (see bottom drawing), piece A is rotating about an axis through its midpoint. According to Table 9.1 the moment of inertia is I for axis 2 =

1 M A L2A . 12

A

Applying

the second law for each of the axes, we obtain

α for axis 1 =

Στ I for axis 1

and

α for axis 2 =

Στ I for axis 2

As given, the net torque Στ is the same in both expressions. Dividing the expression for axis 2 by the expression for axis 1 gives Στ α for axis 2 I for axis 2 I for axis 1 = = Στ I for axis 2 α for axis 1 I for axis 1 Using the expressions from Table 9.1 for the moments of inertia, this result becomes

α for axis 2 I for axis 1 = = α for axis 1 I for axis 2

1 M B L2B 3 1 M A L2A 12

Noting that M A = 2M B and LA = 2 LB , we find that

=

4M B L2B M A L2A

Chapter 9 Problems

463

α for axis 2 4M B L2B 4 M B L2B 1 = = = 2 2 α for axis 1 M A LA 2M ( 2 L ) 2 B B α for axis 2

1 4.6 rad/s 2 = α for axis 1 = = 2.3 rad/s 2 2 2

39. SSM REASONING The figure below shows eight particles, each one located at a different corner of an imaginary cube. As shown, if we consider an axis that lies along one edge of the cube, two of the particles lie on the axis, and for these particles r = 0. The next four particles closest to the axis have r = , where is the length of one edge of the cube. The remaining two particles have r = d , where d is the length of the diagonal along any one of the faces. From the Pythagorean theorem, d =

2

+

2

=

2.

Axis

d d

According to Equation 9.6, the moment of inertia of a system of particles is given by I = ∑ mr 2 .

SOLUTION Direct application of Equation 9.6 gives I = ∑ mr 2 = 4(m 2 ) + 2(md 2 ) = 4(m 2 ) + 2(2m 2 ) = 8m

2

or

I = 8(0.12 kg)(0.25 m) 2 = 0.060 kg ⋅ m2

40. REASONING The time t it takes to completely unwind the hose from the reel is related to the reel’s angular displacement θ and its angular acceleration α by θ = ω0t + 12 α t 2 (Equation 8.7). The reel is initially at rest, so ω0 = 0 rad/s. Substituting this into

θ = ω0t + 12 α t 2 , and solving for the elapsed time t, we obtain

464 ROTATIONAL DYNAMICS

θ = ( 0 rad/s ) t + 12 α t 2 = 12 α t 2

or

t=

2θ

(1)

α

The hose unwinds from the reel without slipping, so the total arc length s traversed by a point on the reel’s rim is equal to the total length L of the hose. The reel’s angular displacement θ, therefore, is related to the length of the hose and the radius R of the reel by s = L = Rθ (Equation 8.1). Thus, the reel’s angular displacement is given by θ = L/R, so that Equation (1) becomes

L 2 R t=

(2)

α

The angular acceleration α of the reel depends upon the net torque and the reel’s moment inertia I via α = Στ I (Equation 9.7). SOLUTION Assuming that the hose unwinds in the counterclockwise direction, the torque exerted on the reel by the tension in the hose is positive. Using Equation 9.1, we have τ = TR , where T is the magnitude of the tension in the hose and R is the radius of the reel. This torque is opposed by the frictional torque τf , so the net torque on the reel is Στ = TR − τ f . Thus, the angular acceleration of the reel is

α=

Στ TR − τ f = I I

(9.7)

Substituting Equation 9.7 into Equation (2) now yields the elapsed time:

t=

L 2 R = TR − τ f I

2 LI = R (TR − τ f )

(

2 (15.0 m ) 0.44 kg ⋅ m 2

)

( 0.160 m ) ( 25.0 N )( 0.160 m ) − 3.40 N ⋅ m

= 12 s

41. SSM REASONING The drawing shows the two identical sheets and the axis of rotation for each. Axis of Rotation

L1

= 0.40 m L2

= 0.20 m

Chapter 9 Problems

465

The time t it takes for each sheet to reach its final angular velocity depends on the angular acceleration α of the sheet. This relation is given by Equation 8.4 as t = (ω − ω0 ) / α , where

ω and ω0 are the final and initial angular velocities, respectively. We know that ω0 = 0 rad/s in each case and that the final angular velocities are the same. The angular acceleration can be determined by using Newton’s second law for rotational motion, Equation 9.7, as α = τ / I , where τ is the torque applied to a sheet and I is its moment of inertia. SOLUTION Substituting the relation α = τ / I into t = (ω − ω0 ) / α gives

t=

(ω − ω0 ) = (ω − ω0 ) = I (ω − ω0 ) α

τ I

τ

The time it takes for each sheet to reach its final angular velocity is:

tLeft =

I Left (ω − ω0 )

τ

and

tRight =

I Right (ω − ω0 )

τ

The moments of inertia I of the left and right sheets about the axes of rotation are given by the following relations, where M is the mass of each sheet (see Table 9.1 and the drawings above): I Left = 13 ML12 and I Right = 13 ML22 . Note that the variables M, ω, ω0, and τ are the same for both sheets. Dividing the time-expression for the right sheet by that for the left sheet gives I Right (ω − ω0 ) tRight I Right 13 ML22 L2 τ = = = = 22 1 ML2 tLeft I Left I Left (ω − ω0 ) L1 1 3 τ Solving this expression for tRight yields tRight

2 L22 0.20 m ) ( = tLeft 2 = ( 8.0 s ) = 2.0 s L ( 0.40 m )2 1

The final angular speed of the arm is ω = vT/r, where r = 0.28 m. The angular acceleration needed to produce this angular speed is α = (ω − ω0)/t.

42. REASONING AND SOLUTION

The net torque required is Στ = Iα. This torque is due solely to the force M, so that Στ = ML. Thus,

466 ROTATIONAL DYNAMICS

ω − ω0 I t ∑τ = M= L L

Setting ω0 = 0 rad/s and ω = vT/r, the force becomes v I T 0.065 kg ⋅ m 2 ( 5.0 m/s ) r t I vT M= = = = 460 N L Lrt ( 0.025 m )( 0.28 m )( 0.10 s )

(

)

43. REASONING a. The moment of inertia for the three-ball system is I = m1r12 + m2 r22 + m3r32 (Equation 9.6), where m1, m2, and m3 are the masses of the balls and r1, r2, and r3 are the distances from the axis. In system A, the ball whose mass is m1 does not contribute to the moment of inertia, because the ball is located on the axis and r1 = 0 m. In system B, the ball whose mass is m3 does not contribute to the moment of inertia, because it is located on the axis and r3 = 0 m. b. The magnitude of the torque is equal to the magnitude F of the force times the lever arm (see Equation 9.1). In system A the lever arm is = 3.00 m. In B the lever arm is = 0 m, since the line of action of the force passes through the axis of rotation. c. According to Newton’s second law for rotational motion, Equation 9.7, the angular acceleration α is given by α = ( Στ ) / I , where Στ is the net torque and I is the moment of inertia. The angular velocity ω is given by Equation 8.4 as ω = ω0 + α t, where ω0 is the initial angular velocity and t is the time.

SOLUTION a. The moment of inertia for each system is System A

I = m1r12 + m2 r22 + m3r32 = ( 9.00 kg )( 0 m ) + ( 6.00 kg )( 3.00 m ) + ( 7.00 kg )( 5.00 m ) = 229 kg ⋅ m 2 2

System B

2

2

I = m1r12 + m2 r22 + m3r32 = ( 9.00 kg )( 5.00 m ) + ( 6.00 kg )( 4.00 m ) + ( 7.00 kg )( 0 m ) = 321 kg ⋅ m 2 2

2

2

Chapter 9 Problems

467

b. A torque that tends to produce a counterclockwise rotation about the axis is a positive torque. The torque produced by the force has a magnitude that is equal to the product of the force magnitude and the lever arm: System A

τ = − F = − ( 424 N )( 3.00 m ) = −1270 N ⋅ m

The torque is negative because it produces a clockwise rotation about the axis. System B

τ = F = ( 424 N )( 0 m ) = 0 N ⋅ m

c. The final angular velocity ω is related to the initial angular velocity ω0, the angular

acceleration α, and the time t by ω = ω0 + α t (Equation 8.4). The angular acceleration is

given by Newton’s second law for rotational motion as α = ( Στ ) / I (Equation 9.7), where

Στ is the net torque and I is the moment of inertia. Since there is only one torque acting on each system, it is the net torque, so Στ = τ . Substituting this expression for α into Equation 8.4 yields

τ ω = ω0 + α t = ω0 + t I

In both cases the initial angular velocity is ω0 = 0 rad/s, since the systems start from rest. The final angular velocities after 5.00 s are:

System A

−1270 N ⋅ m τ ω = ω0 + t = ( 0 rad/s ) + ( 5.00 s ) = −27.7 rad/s 2 I

System B

τ ω = ω0 + t = ( 0 rad/s ) + I

229 kg ⋅ m

0 N⋅m ( 5.00 s ) = 0 rad/s 2 321 kg m ⋅

468 ROTATIONAL DYNAMICS

44. REASONING To determine the angular acceleration of the pulley and the tension in the cord attached to the block, we will apply Newton’s second law to the pulley and the block separately. Only one external forces acts on the pulley, as its free-body diagram at the right shows. This force T is due to the tension in the cord. The torque that results from this force is the net torque acting on the pulley and obeys Newton’s second law for rotational motion (Equation 9.7). Two external forces act on the block, as its free-body diagram at the right indicates. These are (1) the force T′ due to the tension in the cord and (2) the weight mg of the block. The net force that results from these forces obeys Newton’s second law for translational motion (Equation 4.2b).

R

+y +τ

T

+x Free-body diagram for pulley

T′

mg Free-body diagram for block

SOLUTION Applying Newton’s second law for rotational motion to the pulley gives Στ = +TR = I α

(1)

where we have written the torque as the magnitude of the tension force times the lever arm (the radius) as specified by Equation 9.1. In addition, we have assigned this torque a positive value, since it causes a counterclockwise rotation of the pulley. To obtain a value for T, we note that the tension has the same magnitude everywhere in the massless cord, so that T = T ′ . Thus, by applying Newton’s second law for translation (Equation 4.2b), we obtain ΣF = T ′ − mg = −ma or T = T ′ = mg − ma where we have used a to denote the magnitude of the vertical acceleration of the block and included the minus sign to account for the fact that the block is accelerating downward. Substituting this result for T into Equation (1) gives

( mg − ma ) R = Iα

(2)

To proceed further, we must deal with a. Note that the pulley rolls without slipping against the cord, so a and α are related according to a = Rα (Equation 8.13). With this substitution, Equation (2) becomes

mg − m ( Rα ) R = Iα Solving for α, we find that

or

mgR = mR 2α + Iα

Chapter 9 Problems

(

469

)

( 2.0 kg ) 9.80 m/s2 ( 0.040 m ) mgR α= = = 180 rad/s 2 2 2 3 2 − mR + I ( 2.0 kg )( 0.040 m ) + 1.1×10 kg ⋅ m The value for the tension can be now obtained by substituting this value for α into Equation (1): TR = Iα

or

(

)(

)

1.1×10−3 kg ⋅ m 2 180 rad/s 2 Iα T= = = 5.0 N R 0.040 m

45. SSM REASONING The angular acceleration of the bicycle wheel can be calculated from Equation 8.4. Once the angular acceleration is known, Equation 9.7 can be used to find the net torque caused by the brake pads. The normal force can be calculated from the torque using Equation 9.1.

SOLUTION The angular acceleration of the wheel is, according to Equation 8.4,

α=

ω − ω0 t

=

3.7 rad/s − 13.1 rad/s = −3.1 rad/s 2 3.0 s

If we assume that all the mass of the wheel is concentrated in the rim, we may treat the wheel as a hollow cylinder. From Table 9.1, we know that the moment of inertia of a hollow cylinder of mass m and radius r about an axis through its center is I = mr 2 . The net torque that acts on the wheel due to the brake pads is, therefore,

∑τ = Iα = (mr 2 )α

(1)

From Equation 9.1, the net torque that acts on the wheel due to the action of the two brake pads is (2) ∑τ = –2 f k where fk is the kinetic frictional force applied to the wheel by each brake pad, and = 0.33 m is the lever arm between the axle of the wheel and the brake pad (see the drawing in the text). The factor of 2 accounts for the fact that there are two brake pads. The minus sign arises because the net torque must have the same sign as the angular acceleration. The kinetic frictional force can be written as (see Equation 4.8) f k = µk FN

(3)

where µk is the coefficient of kinetic friction and FN is the magnitude of the normal force applied to the wheel by each brake pad.

470 ROTATIONAL DYNAMICS

Combining Equations (1), (2), and (3) gives

–2( µk FN ) = (mr 2 )α FN =

– mr 2α –(1.3 kg)(0.33 m)2 (–3.1 rad/s 2 ) = = 0.78 N 2µk 2(0.85)(0.33 m)

46. REASONING AND SOLUTION

The moment of inertia of a solid cylinder about an axis 2

coinciding with the cylinder axis which contains the center of mass is Icm = (1/2) MR . The 2

parallel axis theorem applied to an axis on the surface (h = R) gives I = Icm + MR , so that I=

3 2

MR 2

47. REASONING AND SOLUTION Newton's law applied to the 11.0-kg object gives 2

2

T2 − (11.0 kg)(9.80 m/s ) = (11.0 kg)(4.90 m/s )

or

T2 = 162 N

or

T1 = 216 N

A similar treatment for the 44.0-kg object yields 2

2

T1 − (44.0 kg)(9.80 m/s ) = (44.0 kg)(−4.90 m/s ) For an axis about the center of the pulley 2

T2r − T1r = I(−α) = (1/2) Mr (−a/r) Solving for the mass M we obtain 2

M = (−2/a)(T2 − T1) = [−2/(4.90 m/s )](162 N − 216 N) = 22.0 kg

Chapter 9 Problems

471

48. REASONING a. The kinetic energy is given by Equation 9.9 as KE R = 21 Iω 2 . Assuming the earth to be a uniform solid sphere, we find from Table 9.1 that the moment of inertia is I = 25 MR 2 . The mass and radius of the earth is M = 5.98 × 1024 kg and R = 6.38 × 106 m (see the inside of the text’s front cover). The angular speed ω must be expressed in rad/s, and we note that the earth turns once around its axis each day, which corresponds to 2π rad/day. b. The kinetic energy for the earth’s motion around the sun can be obtained from Equation 9.9 as KE R = 21 Iω 2 . Since the earth’s radius is small compared to the radius of the earth’s orbit (Rorbit = 1.50 × 1011 m, see the inside of the text’s front cover), the moment 2 . The angular speed ω of the earth as it goes around of inertia in this case is just I = MRorbit the sun can be obtained from the fact that it makes one revolution each year, which corresponds to 2π rad/year.

SOLUTION a. According to Equation 9.9, we have

KE R = 21 Iω 2 = =

1 2

2 5

1 2

c MR hω 2

2 5

c5.98 × 10

24

2

LF 1 day IF 1 h IO 2 π rad IF mh M G J G H3600 s JKP NH1 day KH24 h JKG Q 2

hc

kg 6.38 × 10

6

2

= 2 .57 × 10 29 J b. According to Equation 9.9, we have KE R = 21 Iω 2 = =

1 2

1 2

cMR hω

c5.98 × 10

2 orbit

24

hc

2

kg 1.50 × 10

LF 1 day IF 1 h IO 2 π rad IF 1 yr IF m hM G J G J G H3600 s JKP H1 yr KH365 day KH24 h JKG Q N 2

11

2

= 2 .67 × 10 33 J

49. SSM REASONING The kinetic energy of the flywheel is given by Equation 9.9. The moment of inertia of the flywheel is the same as that of a solid disk, and, according to Table 9.1 in the text, is given by I = 21 MR 2 . Once the moment of inertia of the flywheel is known, Equation 9.9 can be solved for the angular speed ω in rad/s. This quantity can then be converted to rev/min.

472 ROTATIONAL DYNAMICS

SOLUTION Solving Equation 9.9 for ω, we obtain,

ω=

2 ( KE R ) I

2 ( KE R )

=

MR 2

1 2

4(1.2 × 10 9 J) = 6.4 × 10 4 rad / s 2 (13 kg)(0.30 m)

=

Converting this answer into rev/min, we find that

c

ω = 6.4 × 10 4 rad / s

1 rev I F60 s I G hF H2π rad JKG H1 min JK=

6.1 × 10 5 rev / min

50. REASONING The kinetic energy of a rotating object is expressed as KE R = 21 Iω 2 (Equation 9.9), where I is the object’s moment of inertia and ω is its angular speed. According to Equation 9.6, the moment of inertia for rod A is just that of the attached particle, since the rod itself is massless. For rod A with its attached particle, then, the moment of inertia is I A = ML2 . According to Table 9.1, the moment of inertia for rod B is I B = 13 ML2 .

SOLUTION Using Equation 9.9 to calculate the kinetic energy, we find that Rod A

Rod B

cML hω b0.66 kg gb0.75 m gb4.2 rad / sg= ω I ω = c ML h b0.66 kg gb0.75 m gb4 .2 rad / sg=

KE R = 21 I A ω 2 = =

1 2

KE R =

1 2

=

1 6

2

1 2

2

2

2

B

1 2

1 3

2

2

3. 3 J

2

1.1 J

2

2

51. SSM REASONING AND SOLUTION a. The tangential speed of each object is given by Equation 8.9, v T = rω . Therefore, For object 1:

vT1 = (2.00 m)(6.00 rad/s) =

12.0 m / s

For object 2:

vT2 = (1.50 m)(6.00 rad/s) =

9.00 m / s

For object 3:

vT3 = (3.00 m)(6.00 rad/s) =

18.0 m / s

b. The total kinetic energy of this system can be calculated by computing the sum of the kinetic energies of each object in the system. Therefore,

Chapter 9 Problems

473

KE = 21 m1 v12 + 21 m2 v 22 + 21 m3 v 32 KE =

1 2

( 6.00 kg)(12.0 m / s) 2 + ( 4 .00 kg)(9.00 m / s) 2 + ( 3.00 kg)(18.0 m / s) 2 = 1.08 × 10 3 J

c. The total moment of inertia of this system can be calculated by computing the sum of the moments of inertia of each object in the system. Therefore, I = ∑ mr 2 = m1 r12 + m2 r22 + m3 r32

I = ( 6.00 kg)(2.00 m) 2 + ( 4 .00 kg)(1.50 m) 2 + ( 3.00 kg)(3.00 m) 2 = 60.0 kg ⋅ m 2 d. The rotational kinetic energy of the system is, according to Equation 9.9,

KE R = 21 Iω 2 = 21 (60.0 kg ⋅ m 2 )(6.00 rad / s) 2 = 1.08 × 10 3 J This agrees, as it should, with the result for part (b).

52. REASONING Each blade can be approximated as a thin rod rotating about an axis perpendicular to the rod and passing through one end. The moment of inertia of a blade is given in Table 9.1 as 13 ML2 , where M is the mass of the blade and L is its length. The total moment of inertia I of the two blades is just twice that of a single blade. The rotational kinetic energy KER of the blades is given by Equation 9.9 as KE R = 12 I ω 2 , where ω is the angular speed of the blades.

SOLUTION a. The total moment of inertia of the two blades is

I = 13 ML2 + 13 ML2 = 23 ML2 =

2 3

( 240 kg )( 6.7 m )2 =

7200 kg ⋅ m2

b. The rotational kinetic energy is KE R = 12 I ω 2 =

1 2

( 7200 kg ⋅ m2 ) ( 44 rad/s )2 = 7.0 × 106 J

53. REASONING The rotational kinetic energy of a solid sphere is given by Equation 9.9 as KE R = 12 I ω 2 , where I is its moment of inertia and ω its angular speed. The sphere has translational motion in addition to rotational motion, and its translational kinetic energy is KE T = 12 mv 2 (Equation 6.2), where m is the mass of the sphere and v is the speed of its

474 ROTATIONAL DYNAMICS

center of mass. The fraction of the sphere’s total kinetic energy that is in the form of rotational kinetic energy is KER/(KER + KET). SOLUTION The moment of inertia of a solid sphere about its center of mass is I = 25 mR 2 , where R is the radius of the sphere (see Table 9.1). The fraction of the sphere’s total kinetic energy that is in the form of rotational kinetic energy is 1 Iω 2 KE R 2 = = KE R + KE T 12 I ω 2 + 12 mv 2

(

1 2 mR 2 2 5 1 2 mR 2 ω 2 2 5

(

)

)ω2 + 12 mv 2

=

2 R 2ω 2 5 2 R 2ω 2 + v 2 5

Since the sphere is rolling without slipping on the surface, the translational speed v of the center of mass is related to the angular speed ω about the center of mass by v = Rω (see Equation 8.12). Substituting v = Rω into the equation above gives KE R

KE R + KE T

=

2 R 2ω 2 5 2 R 2ω 2 + v 2 5

=

2 5

2 5

R 2ω 2

R 2ω 2 + ( Rω )

2

=

2 7

54. REASONING Because we are ignoring frictional losses, the total mechanical energy E = 12 mv2 + 12 Iω 2 + mgh of both objects is conserved as they roll down the hill. We will apply this conservation principle twice: first, to determine the height h0 − hf of the hill, and second, to determine the translational speed vf of the frozen juice can at the bottom of the hill. Both the basketball and the frozen juice can roll without slipping, so the translational speed v of either one is related to its radius r and angular speed ω by v = rω (Equation 8.12).

SOLUTION a. Applying the energy conservation principle to the basketball, we obtain 1 2

mvf2 + 12 I ωf2 + mghf = 12 mv02 + 12 I ω02 + mgh0 Ef

(1)

E0

The basketball starts from rest, so we have v0 = 0 m/s and ω0 = 0 rad/s. The basketball is a thin-walled spherical shell, so its moment of inertia is given by I = 23 mr 2 (see Table 9.1 in the text). Substituting these values along with ωf = vf r (Equation 8.12) into Equation (1) yields 1 2

m vf2 + 12

(

2 3

)

2

v m r 2 f + m ghf = 0 + 0 + m gh0 r

or

1 v2 2 f

+ 13 vf2 = 65 vf2 = g ( h0 − hf )

Chapter 9 Problems

475

Solving for the height h0 − hf of the hill, we obtain

h0 − hf =

5vf2 6g

=

5 ( 6.6 m/s )

(

2

6 9.80 m/s 2

)

= 3.7 m

b. In Equation (1), we again substitute v0 = 0 m/s and ω0 = 0 rad/s, but this time we use

I = 12 mr 2 for the moment of inertia (see Table 9.1 in the text), because the frozen juice can is a solid cylinder: 1 2

m vf2 + 12

(

1 2

)

2

v m r 2 f + m ghf = 0 + 0 + m gh0 r

or

1 v2 2 f

+ 14 vf2 = 34 vf2 = g ( h0 − hf )

The final translational speed of the frozen juice can is

vf2

=

4 g ( h0 − hf ) 3

or

vf =

4 g ( h0 − hf ) 3

=

(

)

4 9.80 m/s 2 ( 3.7 m ) 3

= 7.0 m/s

55. SSM REASONING AND SOLUTION The only force that does work on the cylinders as they move up the incline is the conservative force of gravity; hence, the total mechanical energy is conserved as the cylinders ascend the incline. We will let h = 0 m on the horizontal plane at the bottom of the incline. Applying the principle of conservation of mechanical energy to the solid cylinder, we have mghs = 12 mv02 + 12 Isω02

(1)

where, from Table 9.1, Is = 12 mr 2 . In this expression, v0 and ω0 are the initial translational and rotational speeds, and hs is the final height attained by the solid cylinder. Since the

cylinder rolls without slipping, the rotational speed ω0 and the translational speed v0 are related according to Equation 8.12, ω0 = v0 / r . Then, solving Equation (1) for hs , we obtain hs =

3v02 4g

Repeating the above for the hollow cylinder and using I h = mr 2 we have

476 ROTATIONAL DYNAMICS

hh =

v02 g

The height h attained by each cylinder is related to the distance s traveled along the incline and the angle θ of the incline by h h ss = s and sh = h sin θ sin θ Dividing these gives ss = 3/4 sh

56. REASONING a. The kinetic energy of the rolling wheel is the sum of its translational rotational

( 12 mv2 ) and

( 12 Iω 2 ) kinetic energies. In these expressions m and I are, respectively, the mass

and moment of inertia of the wheel, and v and ω are, respectively, its linear and angular speeds. The sliding wheel only has translational kinetic energy, since it does not rotate.

b. As the wheels move up the incline plane, the total mechanical energy is conserved, since only the conservative force of gravity does work on each wheel. Thus, the initial kinetic energy at the bottom of the incline is converted entirely into potential energy when the wheels come to a momentary halt. The potential energy PE is given by PE = mgh (Equation 6.5), where h is the height of the wheel above an arbitrary zero level. SOLUTION a. Since the rolling wheel is a disk, its moment of inertia is I =

1 mR 2 2

(see Table 9.1),

where R is the radius of the disk. Furthermore, its angular speed ω is related to the linear speed v of its center of mass by Equation 8.12 as ω = v/R. Thus, the total kinetic energy of the rolling wheel is

Rolling Wheel

KE =

1 2

2

1 2

mv +

= 34 mv 2 =

3 4

I ω = mv + 2

1 2

2

(

1 1 2 2

( 2.0 kg )( 6.0 m/s )2 =

mR

2

54 J

The kinetic energy of the sliding wheel is Sliding Wheel

KE =

1 mv 2 2

=

1 2

( 2.0 kg )( 6.0 m/s )2 =

)

36 J

v R

2

Chapter 9 Problems

477

b. As each wheel rolls up the incline, its total mechanical energy is conserved. The initial kinetic energy KE at the bottom of the incline is converted entirely into potential energy PE when the wheels come to a momentary halt. Thus, the potential energies of the wheels have the values calculated in part a for the total kinetic energies. The potential energy of a wheel is given by Equation 6.5 as PE = mgh, where g is the acceleration due to gravity and h is the height relative to an arbitrary zero level. Therefore, the height reached by each wheel is as follows: Rolling Wheel

h=

PE 54 J = = 2.8 m mg ( 2.0 kg ) 9.80 m/s2

Sliding Wheel

h=

PE 36 J = = 1.8 m 2 mg 2.0 kg 9.80 m/s ( )

(

)

(

)

57. REASONING AND SOLUTION The conservation of energy gives 2

2

2

mgh + (1/2) mv + (1/2) Iω = (1/2) mv0 + (1/2) Iω0

2

If the ball rolls without slipping, ω = v/R and ω0 = v0/R. We also know that I = (2/5) mR2. Substitution of the last two equations into the first and rearrangement gives

v = v02 − 10 gh = 7

( 3.50 m/s )2 − 107 ( 9.80 m/s2 ) ( 0.760 m ) =

1.3 m/s

58. REASONING We first find the speed v0 of the ball when it becomes airborne using the conservation of mechanical energy. Once v0 is known, we can use the equations of kinematics to find its range x. SOLUTION When the tennis ball starts from rest, its total mechanical energy is in the form of gravitational potential energy. The gravitational potential energy is equal to mgh if we take h = 0 m at the height where the ball becomes airborne. Just before the ball becomes airborne, its mechanical energy is in the form of rotational kinetic energy and translational kinetic energy. At this instant its total energy is 12 mv02 + 12 I ω 2 . If we treat the tennis ball as a thin-walled spherical shell of mass m and radius r, and take into account that the ball rolls down the hill without slipping, its total kinetic energy can be written as 1 mv02 2

1 2

+ Iω = 2

1 mv02 2

+

v 1 2 ( mr 2 ) 0 2 3 r

2

5 2 = 6 mv0

478 ROTATIONAL DYNAMICS

Therefore, from conservation of mechanical energy, he have 5 6

mgh = mv02

or

v0 =

6 gh 5

The range of the tennis ball is given by x = v0 xt = v0 (cos θ ) t , where t is the flight time of the ball. From Equation 3.3b, we find that the flight time t is given by

t=

v − v0 y ay

=

(−v0 y ) − v0 y ay

=–

2v0 sin θ ay

Therefore, the range of the tennis ball is 2v sin θ x = v0 xt = v0 (cos θ ) – 0 ay

If we take upward as the positive direction, then using the fact that a y = – g and the expression for v0 given above, we find 2

2cos θ sin θ 2 2cos θ sin θ 6 gh 12 x= = h cos θ sin θ v0 = 5 g g 5 12 (1.8 m) (cos 35°)(sin 35°)= 2.0 m 5

=

59. SSM WWW REASONING Let the two disks constitute the system. Since there are no external torques acting on the system, the principle of conservation of angular momentum applies. Therefore we have Linitial = L final , or

I A ω A + I Bω B = ( I A + I B ) ω final This expression can be solved for the moment of inertia of disk B. SOLUTION Solving the above expression for I B , we obtain

IB = IA

Fω G Hω

final B

–ωA

– ω final

I = ( 3.4 kg ⋅ m ) L –2.4 rad / s – 7.2 rad / s O= 4.4 kg ⋅ m JK M N–9.8 rad / s – (–2.4 rad / s) P Q 2

2

Chapter 9 Problems

479

60. REASONING Before any sand strikes the disk, only the disk is rotating. After the sand has landed on the disk, both the sand and the disk are rotating. If the sand and disk are taken to be the system of objects under consideration, we note that there are no external torques acting on the system. As the sand strikes the disk, each exerts a torque on the other. However, these torques are exerted by members of the system, and, as such, are internal torques. The conservation of angular momentum states that the total angular momentum of a system remains constant (is conserved) if the net average external torque acting on the system is zero. We will use this principle to find the final angular velocity of the system. SOLUTION The angular momentum of the system (sand plus disk) is given by Equation 9.10 as the product of the system’s moment of inertia I and angular velocity ω, or L = Iω. The conservation of angular momentum can be written as Iω Final angular momentum

=

I 0ω0 Initial angular momentum

where ω and ω0 are the final and initial angular velocities, respectively, and I and I0 are final and initial moments of inertia. The initial moment of inertia is given, while the final moment of inertia is the sum of the values for the rotating sand and disk, I = Isand + I0. We note that the sand forms a thin ring, so its moment of inertia is given by (see Table 9.1) 2 Isand = M sand Rsand , where Msand is the mass of the sand and Rsand is the radius of the ring. Thus, the final angular velocity of the system is, then,

I0 I0 I0 = ω0 = ω0 2 + I I + M R I I sand 0 sand sand 0

ω = ω0

0.10 kg ⋅ m 2 = 0.037 rad/s = ( 0.067 rad/s ) ( 0.50 kg )( 0.40 m )2 + 0.10 kg ⋅ m 2

61. REASONING We consider a system consisting of the person and the carousel. Since the carousel rotates on frictionless bearings, no net external torque acts on this system. Therefore, its angular momentum is conserved, and we can set the total angular momentum of the system before the person hops on equal to the total angular momentum afterwards. SOLUTION The initial angular momentum of the carousel is I carousel ω0 (Equation 9.10) ,

where Icarousel is the moment of inertia of the carousel and ω0 is its initial angular velocity. After the person climbs aboard, the total angular momentum of the carousel and person is I carouselωf + I personωf , where ωf is the final angular velocity. According to Equation 9.6, the

480 ROTATIONAL DYNAMICS

person’s moment of inertial is I person = MR 2 , since he is at the outer edge of the carousel, which has a radius R. Applying the conservation of angular momentum, we have I carouselωf + I personωf

=

Initial total angular momentum

Final total angular momentum

ωf =

=

I carouselω0

I carouselω0 I carouselω0 = I carousel + I person I carousel + MR 2

(125 kg ⋅ m2 ) ( 3.14 rad/s)

125 kg ⋅ m2 + ( 40.0 kg ) (1.50 m )

2

= 1.83 rad/s

62. REASONING Once the motorcycle is in the air, it is subject to no net external torque, since gravity and air resistance are being ignored. Therefore, its total angular momentum is conserved: Lf = L0 . The total angular momentum of the motorcycle is the sum of the angular momentum LE = I EωE (Equation 9.10) of the engine and the angular momentum

LM = I MωM of the rest of the motorcycle (including the rider):

I EωEf + I MωMf = I EωE0 + I MωM0 Lf

(1)

L0

We will use Equation (1) to find the ratio IE/IM of the moments of inertia of the engine and the rest of the motorcycle. We note that, so long as all angular velocities are expressed in rev/min, there is no need to convert to SI units (rad/s).

SOLUTION Initially, only the engine is rotating, so the rest of the motorcycle has no angular velocity: ωM0 = 0 rad/s. Solving Equation (1) for the ratio IE/IM, we obtain

I EωEf + I MωMf = I EωE0 + I M ( 0)

or I E (ωEf − ωE0 ) = − I MωMf

or

−ωMf IE = I M ωEf − ωE0

As usual, clockwise rotation is negative, and counterclockwise is positive. The ratio of the moments of inertia is, then,

IE IM

=

−ωMf

ωEf − ωE0

=

− ( −3.8 rev/min )

( +12 500 rev/min ) − ( +7700 rev/min )

= 7.9 ×10−4

Chapter 9 Problems

481

63. REASONING The rod and bug are taken to be the system of objects under consideration, and we note that there are no external torques acting on the system. As the bug crawls out to the end of the rod, each exerts a torque on the other. However, these torques are internal torques. The conservation of angular momentum states that the total angular momentum of a system remains constant (is conserved) if the net average external torque acting on the system is zero. We will use this principle to find the final angular velocity of the system. SOLUTION The angular momentum L of the system (rod plus bug) is given by Equation 9.10 as the product of the system’s moment of inertia I and angular velocity ω, or L = Iω. The conservation of angular momentum can be written as

Iω Final angular momentum

=

I 0ω0 Initial angular momentum

where ω and ω0 are the final and initial angular velocities, respectively, and I and I0 are the final and initial moments of inertia. The initial moment of inertia is given. The initial moment of inertia of the bug is zero, because it is located at the axis of rotation. The final moment of inertia is the sum of the moment of inertia of the bug and that of the rod; I = Ibug + I0. When the bug has reached the end of the rod, its moment of inertia is 2

Ibug = mL , where m is its mass and L is the length of the rod. The final angular velocity of the system is, then, I0 I I0 = ω0 2 0 = ω0 mL + I I bug + I 0 I 0

ω = ω0

1.1×10−3 kg ⋅ m 2 = ( 0.32 rad/s ) 4.2 × 10−3 kg ( 0.25 m )2 + 1.1× 10−3 kg ⋅ m 2

(

)

(

)

= 0.26 rad/s

64. REASONING The carousel rotates on frictionless bearings and without air resistance, so no net external torque acts on the system comprised of the carousel and the person on it. Therefore, the total angular momentum L = Iω (Equation 9.10) of the system is conserved (Lf = L0). As the person moves closer to the center of the carousel, the person’s distance r from the rotation axis decreases from r0 = 1.50 m to rf = 0.750 m. Therefore, the moment of inertia I P = mr 2 (Equation 9.6) of the person also decreases, from I P0 = mr02 to I Pf = mrf2 . Consequently, the angular speed of the system increases from ω0 to ωf , preserving the system’s total angular momentum. We will use the angular-momentum-conservation principle to find the mass m of the person.

482 ROTATIONAL DYNAMICS

SOLUTION The person and the carousel have the same initial angular velocity ω0, and the

same final angular velocity ωf. Therefore, the conservation of angular momentum principle can be expressed as (1) I Pf ωf + I Cωf = I P0ω0 + I Cω0 Lf

L0

where IC is the moment of inertia of the carousel (without the person). Substituting

I P0 = mr02 and I Pf = mrf2 (Equation 9.6) for the person’s initial and final moments of inertia into Equation (1), we obtain mrf2 ωf + I Cωf = mr02 ω0 + I Cω0 I Pf

or

(

I C (ωf − ω0 ) = m r02ω0 − rf2ωf

)

I P0

Solving for the mass m of the person yields m=

I C (ωf − ω0 ) r02ω0 − rf2ωf

=

(125 kg ⋅ m2 ) ( 0.800 rad/s − 0.600 rad/s )

(1.50 m )2 ( 0.600 rad/s ) − ( 0.750 m )2 ( 0.800 rad/s )

= 28 kg

65. REASONING Let the space station and the people within it constitute the system. Then as the people move radially from the outer surface of the cylinder toward the axis, any torques that occur are internal torques. Since there are no external torques acting on the system, the principle of conservation of angular momentum can be employed.

SOLUTION Since angular momentum is conserved,

I final ω final = I 0 ω 0 Before the people move from the outer rim, the moment of inertia is 2 I 0 = I station + 500 mperson rperson

or I 0 = 3.00 × 10 9 kg ⋅ m 2 + ( 500)(70.0 kg)(82.5 m) 2 = 3.24 × 10 9 kg ⋅ m 2 If the people all move to the center of the space station, the total moment of inertia is I final = I station = 3.00 × 10 9 kg ⋅ m 2 Therefore,

I ω final 3.24 × 10 9 kg ⋅ m 2 = 1.08 = 0 = I final ω0 3.00 × 10 9 kg ⋅ m 2

Chapter 9 Problems

483

This fraction represents a percentage increase of 8 percent .

66. REASONING AND SOLUTION Since the change occurs without the aid of external torques, the angular momentum of the system is conserved: Ifωf = I0ω0. Solving for ωf gives

I0 If

ωf = ω0

where for a rod of mass M and length L, I 0 =

1 ML2 . 12

To determine If , we will treat the

arms of the "u" as point masses with mass M/4 a distance L/4 from the rotation axis. Thus, 1 M I f = 12 2

L 2

2

M L 2 1 ML2 + 2 = 4 4 24

and

1 ML2 = (7.0 rad/s)(2) = 14 rad/s ωf = ω0 121 ML2 24

67. REASONING AND SOLUTION After the mass has moved inward to its final path the centripetal force acting on it is T = 105 N.

r

105 N

Its centripetal acceleration is 2

ac = v /R = T/m Now v = ωR

so

2

R = T/(ω m)

484 ROTATIONAL DYNAMICS

The centripetal force is parallel to the line of action (the string), so the force produces no torque on the object. Hence, angular momentum is conserved. Iω = Ioωo

so that

ω = (Io/I)ωo = (Ro2/R2)ωo

Substituting and simplifying 3

4

2

R = (mRo ωo )/T, so that R =

0.573 m

68. REASONING AND SOLUTION The block will just start to move when the centripetal force on the block just exceeds f smax . Thus, if rf is the smallest distance from the axis at which the block stays at rest when the angular speed of the block is ωf, then

µs FN = mrf ωf2, or µs mg = mrf ωf2. Thus, µs g = rf ωf2

(1)

Since there are no external torques acting on the system, angular momentum will be conserved. I0 ω0 = If ωf 2

2

where I0 = mr0 , and If = mrf . Making these substitutions yields

r02 ω0 = rf2 ωf

(2)

Solving Equation (2) for ωf and substituting into Equation (1) yields:

µs g =

4 2 r0 rf ω0 4 rf

Solving for rf gives 1/3

ω2 r4 rf = 0 0 µ g s

1/3

(2.2 rad/s) 2 (0.30 m)4 = 2 (0.75)(9.80 m/s )

= 0.17 m

Chapter 9 Problems

485

69. REASONING To calculate the torques, we need to determine the lever arms for each of the forces. These lever arms are shown in the following drawings: Axis

Axis 32.0°

2.50 m

32.0°

2.50 m

T

= (2.50 m) cos 32.0°

T

W

W = (2.50 m) sin 32.0°

SOLUTION a. Using Equation 9.1, we find that the magnitude of the torque due to the weight W is

Magnitude of torque = W

W

= (10 200 N )( 2.5 m ) sin 32° = 13 500 N ⋅ m

b. Using Equation 9.1, we find that the magnitude of the torque due to the thrust T is

Magnitude of torque = T

T

= ( 62 300 N )( 2.5 m ) cos 32° = 132 000 N ⋅ m

70. REASONING The supernova explosion proceeds entirely under the influence of internal forces and torques. External forces and torques play no role. Therefore, the star’s angular momentum is conserved during the supernova and its subsequent transformation from a solid sphere into an expanding spherical shell: Lf = L0 . The star’s initial and final angular momenta are given by L = I ω (Equation 9.10), where I is the star’s moment of inertia, and ω is its angular velocity. Initially, the star is a uniform solid sphere with a moment of inertia given by I = 52 MR 2 (see Table 9.1 in the text). Following the supernova, the moment of

inertia of the expanding spherical shell is I = 23 MR 2 (see Table 9.1 in the text). We will use

the angular-momentum-conservation principle to calculate the final angular velocity ωf of the expanding supernova shell. SOLUTION Applying the angular-momentum-conservation principle yields

If ωf = I0ω0 Lf

L0

or

2 3

M

Rf2ωf

=

2 5

M R02ω0

or

ωf =

3R02ω0 5Rf2

Substituting R0 = R, Rf = 4.0R, and ω0 = 2.0 rev/d, we obtain the final angular velocity of the expanding shell:

486 ROTATIONAL DYNAMICS

ωf =

3 R 2 ω0

(

5 4.0 R

)

2

=

3ω0 80

=

3 ( 2.0 rev/d ) = 0.075 rev/d 80

71. REASONING According to Newton’s second law for rotational motion, Στ = Iα, the angular acceleration α of the blades is equal to the net torque Στ applied to the blades divided by their total moment of inertia I, both of which are known. SOLUTION The angular acceleration of the fan blades is

α=

Στ 1.8 N ⋅ m = = 8.2 rad/s 2 2 I 0.22 kg ⋅ m

72. REASONING The drawing shows the forces acting on the person. It also shows the lever arms for a rotational axis perpendicular to the plane of the paper at the place where the person’s toes touch the floor. Since the person is in equilibrium, the sum of the forces must be zero. Likewise, we know that the sum of the torques must be zero.

(9.7)

FHANDS

FFEET

Axis W W

= 0.840 m

HANDS =

1.250 m

SOLUTION Taking upward to be the positive direction, we have

FFEET + FHANDS − W = 0 Remembering that counterclockwise torques are positive and using the axis and the lever arms shown in the drawing, we find W FHANDS =

W

W

W

HANDS

− FHANDS =

HANDS

=0

b584 N gb0.840 m g= 392 N 1.250 m

Substituting this value into the balance-of-forces equation, we find

FFEET = W − FHANDS = 584 N − 392 N = 192 N

Chapter 9 Problems

487

The force on each hand is half the value calculated above, or 196 N . Likewise, the force on each foot is half the value calculated above, or 96 N .

73. REASONING a. The angular acceleration α is defined as the change, ω − ω0, in the angular velocity divided by the elapsed time t (see Equation 8.4). Since all these variables are known, we can determine the angular acceleration directly from this definition. b. The magnitude τ of the torque is defined by Equation 9.1 as the product of the magnitude F of the force and the lever arm . The lever arm is the radius of the cylinder, which is known. Since there is only one torque acting on the cylinder, the magnitude of the force can be obtained by using Newton’s second law for rotational motion, Στ = F = I α . SOLUTION a. From Equation 8.4 we have that α = (ω – ω0)/t. We are given that ω0 = 76.0 rad/s,

ω = 12 ω0 = 38.0 rad/s , and t = 6.40 s, so α=

ω − ω0 t

=

38.0 rad/s − 76.0 rad/s = −5.94 rad/s 2 6.40 s

The magnitude of the angular acceleration is 5.94 rad/s2 . b. Using Newton’s second law for rotational motion, we have that Στ = F = I α . Thus, the magnitude of the force is

F=

Iα

0.615 kg ⋅ m 2 )( 5.94 rad/s 2 ) ( = = 44.0 N 0.0830 m

488 ROTATIONAL DYNAMICS 0.61 m 74. REASONING The arm, being stationary, is in equilibrium, since it has no translational or angular acceleration. Therefore, the net 0.28 m external force and the net external torque 98 N acting on the arm are zero. Using the fact 0.069 m that the net external torque is zero will allow us to determine the magnitude of the force cg Axis M. The drawing at the right shows three 29° forces: M, the 47-N weight of the arm acting 47 N at the arm’s center of gravity (cg), and the 98-N force that acts upward on the right end M of the arm. The 98-N force is applied to the (0.070 m) sin 29° arm by the ring. It is the reaction force that arises in response to the arm pulling downward on the ring. Its magnitude is 98 N, because it supports the 98-N weight hanging from the pulley system. Other forces also act on the arm at the shoulder joint, but we can ignore them. The reason is that their lines of action pass directly through the axis at the shoulder joint, which is the axis that we will use to determine torques. Thus, these forces have zero lever arms and contribute no torque.

SOLUTION The magnitude of each individual torque is the magnitude of the force times the corresponding lever arm. The forces and their lever arms are as follows: Force

Lever Arm

98 N

0.61 m

47 N

0.28

M

(0.069 m) sin 29°

Each torque is positive if it causes a counterclockwise rotation and negative if it causes a clockwise rotation about the axis. Thus, since the net torque must be zero, we see that

( 98 N )( 0.61 m ) − ( 47 N )( 0.28 m ) − M ( 0.069 m ) sin 29° = 0 Solving for M gives M=

( 98 N )( 0.61 m ) − ( 47 N )( 0.28 m ) = 1400 N ( 0.069 m ) sin 29°

Chapter 9 Problems

489

75. SSM WWW REASONING AND SOLUTION The figure below shows the massless board and the forces that act on the board due to the person and the scales.

2.00 m person's feet

person's head

center of gravity of person

x 425 N

315 N

W

a. Applying Newton's second law to the vertical direction gives 315 N + 425 N – W = 0

or

W = 7.40 ×102 N, downward

b. Let x be the position of the center of gravity relative to the scale at the person's head. Taking torques about an axis through the contact point between the scale at the person's head and the board gives (315 N)(2.00 m) – (7.40 × 102 N)x = 0

or

x = 0.851 m

76. REASONING AND SOLUTION a. The net torque about an axis through the point of contact between the floor and her shoes is 2

Στ = − (5.00 × 10 N)(1.10 m)sin 30.0° + FN(cos 30.0°)(1.50 m) = 0

FN = 212 N b. Newton's second law applied in the horizontal direction gives Fh − FN = 0, so Fh = 212 N . c. Newton's second law in the vertical direction gives Fv − W = 0, so Fv = 5.00 × 102 N .

77. SSM REASONING When the modules pull together, they do so by means of forces that are internal. These pulling forces, therefore, do not create a net external torque, and the angular momentum of the system is conserved. In other words, it remains constant. We will use the conservation of angular momentum to obtain a relationship between the initial and

490 ROTATIONAL DYNAMICS

final angular speeds. Then, we will use Equation 8.9 (v = rω) to relate the angular speeds

ω0 and ωf to the tangential speeds v0 and vf.

SOLUTION Let L be the initial length of the cable between the modules and ω0 be the initial angular speed. Relative to the center-of-mass axis, the initial momentum of inertia of the two-module system is I0 = 2M(L/2)2, according to Equation 9.6. After the modules pull together, the length of the cable is L/2, the final angular speed is ωf, and the momentum of inertia is If = 2M(L/4)2. The conservation of angular momentum indicates that

I f ω f = I 0ω 0 Final angular momentum

Initial angular momentum

L L2 M F LI O FL I O 2 M GJ P ω =M G J Pω M M N H4 KP N H2 KP Q M Q 2

2

f

0

ω f = 4ω 0 According to Equation 8.9, ωf = vf/(L/4) and ω0 = v0/(L/2). With these substitutions, the result that ωf = 4ω0 becomes vf L/4

=4

v I F G HL / 2 JK or 0

b

g

v f = 2 v 0 = 2 17 m / s = 34 m / s

78. REASONING The drawing shows the rod in its initial (dashed lines) and final (solid lines) orientations. Since both friction and air resistance are absent, the total mechanical energy is conserved. In this case, the total mechanical energy is E =

1 2

Iω 2 +

Rotational kinetic energy

Center of mass

mgh Pivot

Gravitational potential energy

In this expression, m is the rod’s mass, I is its moment of inertia, ω is its angular speed, and h is the height of its center of mass above a reference level that we take to be the ground. Since the rod is uniform, its center of mass is located at the center of the rod. The energy-conservation principle will guide our solution.

hf

Center of mass h0

v0

Chapter 9 Problems

491

SOLUTION Conservation of the total mechanical energy dictates that 1 2

I ωf2 + mghf =

Final total mechanical energy

1 2

I ω02 + mgh0

(1)

Initial total mechanical energy

Since the rod is momentarily at rest in its final orientation, we know that ωf = 0 rad/s, so that Equation (1) becomes 1 2

mghf = Iω02 + mgh0

1 2

or

I ω02 = mg ( hf − h0 )

(2)

We can relate the initial angular speed ω0 to the initial linear speed v0 by using v Equation 8.9: ω0 = 0 . With this substitution, the fact that hf − h0 = L (see the drawing), L 1 3

and the fact that I = mL2 (see Table 9.1 in the text), Equation (2) indicates that 1 2

I ω02

= mg ( hf − h0 )

Solving for v0 gives

(

1 1 mL2 2 3

or

(

)

2

v0 = mgL L

or

1 2 v 6 0

= gL

)

v0 = 6 gL = 6 9.80 m/s2 ( 0.80 m ) = 6.9 m/s

79.

REASONING The drawing shows the forces acting on the board, which has a length L. Wall 2 exerts a normal force P2 on the lower end of the board. The maximum force of static friction that wall 2 can apply to the lower end of the board is µsP2 and is directed upward in the drawing. The weight W acts downward at the board's center. Wall 1 applies a normal force P1 to the upper end of the board. We take upward and to the right as our positive directions.

Wall 1

P1

Wall 2

1

µ s P2 W

θ

P2

SOLUTION Then, since the horizontal forces balance to zero, we have

P1 – P2 = 0

(1)

µsP2 – W = 0

(2)

The vertical forces also balance to zero:

492 ROTATIONAL DYNAMICS

Using an axis through the lower end of the board, we now balance the torques to zero:

L W ( cos θ ) − P1L ( sin θ ) = 0 2

(3)

Rearranging Equation (3) gives tan θ =

W 2 P1

(4)

But W = µsP2 according to Equation (2), and P2 = P1 according to Equation (1). Therefore, W = µsP1, which can be substituted in Equation (4) to show that tan θ = or

µs P1 2 P1

=

µs 2

=

0.98 2

θ = tan–1(0.49) = 26°

From the drawing at the right, cos θ =

d L

L

Therefore, the longest board that can be propped between the two walls is L=

d 1.5 m = = cos θ cos 26°

1.7 m

θ

d

Chapter 9 Problems

493

80. REASONING The drawing shows the drum, pulley, and the crate, as well as the tensions in the cord r2 T1

Pulley m2 = 130 kg

T2

T2

T1 r1

Crate m3 = 180 kg

Drum m1 = 150 kg

R

r1 = 0.76 m

Let T1 represent the magnitude of the tension in the cord between the drum and the pulley.

Then, the net torque exerted on the drum must be, according to Equation 9.7, Στ = I1α1,

where I1 is the moment of inertia of the drum, and α1 is its angular acceleration. If we assume that the cable does not slip, then Equation 9.7 can be written as aI c hF G Hr JK

− T1 r1 + τ = m1 r12 ∑τ

(1)

1

I1

α1

where τ is the counterclockwise torque provided by the motor, and a is the acceleration of the cord (a = 1.2 m/s2). This equation cannot be solved for τ directly, because the tension T1 is not known. We next apply Newton’s second law for rotational motion to the pulley in the drawing: + T1 r2 − T2 r2 = ∑τ

aI cm r hF G r J H K I 1 2

2 2 2

(2)

2

2

α2

where T2 is the magnitude of the tension in the cord between the pulley and the crate, and I2 is the moment of inertial of the pulley. Finally, Newton’s second law for translational motion (ΣFy = m a) is applied to the crate, yielding

494 ROTATIONAL DYNAMICS

+ T2 − m3 g = m3 a

(3)

∑ Fy SOLUTION Solving Equation (1) for T1 and substituting the result into Equation (2), then solving Equation (2) for T2 and substituting the result into Equation (3), results in the following value for the torque

τ = r1 a ( m1 + 12 m2 + m3 ) + m3 g

(

)

= (0.76 m) (1.2 m/s 2 ) 150 kg + 12 130 kg + 180 kg + (180 kg)(9.80 m/s 2 ) = 1700 N ⋅ m

CHAPTER

10 SIMPLE HARMONIC MOTION AND

ELASTICITY ANSWERS TO FOCUS ON CONCEPTS QUESTIONS

1. 0.12 m 2. (c) The restoring force is given by Equation 10.2 as F = −kx, where k is the spring constant (positive). The graph of this equation is a straight line and indicates that the restoring force has a direction that is always opposite to the direction of the displacement. Thus, when x is positive, F is negative, and vice versa.

1 m . Greater 2π k values for the mass m and smaller values for the spring constant k lead to greater values for the period.

3. (b) According to Equations 10.4 and 10.11, the period T is given by T =

4. (d) The maximum speed in simple harmonic motion is given by Equation 10.8 ( vmax = Aω ) . Thus, increases in both the amplitude A and the angular frequencyω lead to an increase in the maximum speed. 5. (e) The maximum acceleration in simple harmonic motion is given by Equation 10.10 ( amax = Aω 2 ) . A decrease in the amplitude A decreases the maximum acceleration, but this decrease is more than offset by the increase in the angular frequency ω, which is squared in Equation 10.10. 6. 1.38 m/s 7. (b) The velocity has a maximum magnitude at point A, where the object passes through the position where the spring is unstrained. The acceleration at point A is zero, because the spring is unstrained there and is not applying a force to the object. The velocity is zero at point B, where the object comes to a momentary halt and reverses the direction of its travel. The magnitude of the acceleration at point B is a maximum, because the spring is maximally stretched there and, therefore, applies a force of maximum magnitude to the object. 8. 0.061 m 9. +7.61 m/s2 10. 0.050 m

496 SIMPLE HARMONIC MOTION AND ELASTICITY

11. (c) The principle of conservation of mechanical energy applies in the absence of nonconservative forces, so that KE + PE = constant. Thus, the total energy is the same at all points of the oscillation cycle. At the equilibrium position, where the spring is unstrained, the potential energy is zero, and the kinetic energy is KEmax; thus, the total energy is KEmax. At the extreme ends of the cycle, where the object comes to a momentary halt, the kinetic energy is zero, and the potential energy is PEmax; thus, the total energy is also PEmax. Since both KEmax and PEmax equal the total energy, it must be true that KEmax = PEmax. 12. (e) In simple harmonic motion the speed and, hence, KE has a maximum value as the object passes through its equilibrium position, which is position 2. EPE has a maximum value when the spring is maximally stretched at position 3. GPE has a maximum value when the object is at its highest point above the ground, that is, at position 1. 13. (a) At the instant the top block is removed, the total mechanical energy of the remaining system is all elastic potential energy and is 12 kA2 (see Equation 10.13), where A is the amplitude of the previous simple harmonic motion. This total mechanical energy is conserved, because friction is absent. Therefore, the total mechanical energy of the ensuing simple harmonic motion is also 12 kA2 , and the amplitude remains the same as it was

k . Thus, when m the mass m attached to the spring decreases, the angular frequency increases.

previously. The angular frequency ω is given by Equation 10.11 as ω =

14. (b) The angular frequency ω of oscillation of a simple pendulum is given by Equation 10.16 g ω = . It depends only on the magnitude g of the acceleration due to gravity and the L length L of the pendulum. It does not depend on the mass. Therefore, the pendulum with the greatest length has the smallest frequency. 15. 1.7 s 16. (c) When the energy of the system is dissipated, the amplitude of the motion decreases. The motion is called damped harmonic motion. 17. (a) Resonance occurs when the frequency of the external force equals the frequency of oscillation of the object on the spring. The angular frequency of such a system is given by k Equation 10.11 ω = . Since the frequency of the force is doubled, the new frequency m must be 2ω = 2

k 8k k . The frequency of system A is, in fact, ω = =2 . m m 2m

Chapter 10 Answers to Focus on Concepts Questions

497

∆L 18. (c) According to Equation 10.17 F = Y A , the force F required to stretch a piece of L 0 material is proportional to Young’s modulus Y, the amount of stretch ∆L, and the crosssectional area A of the material, but is inversely proportional to the initial length L0 of the FL material. Solving this equation for the amount of stretch gives ∆ L = 0 . Thus, the greater YA the cross-sectional area, the smaller is the amount of stretch, for given values of Young’s modulus, the initial length, and the stretching force. Thus, B stretches more than A, because B has the smaller cross-sectional area of solid material.

19. 0.50 × 10−6 m 20. 0.0017

498 SIMPLE HARMONIC MOTION AND ELASTICITY

CHAPTER 10 SIMPLE HARMONIC MOTION AND

ELASTICITY PROBLEMS ______________________________________________________________________________ 1.

SSM REASONING AND SOLUTION Using Equation 10.1, we first determine the spring constant: FxApplied 89.0 N = = 4660 N/m k= x 0.0191 m Again using Equation 10.1, we find that the force needed to compress the spring by 0.0508 m is

FxApplied = kx = (4660 N/m)(0.0508 m) = 237 N ______________________________________________________________________________ 2.

REASONING AND SOLUTION According to Equation 10.1 and the data from the graph, the effective spring constant is Applied

Fx

160 N 2 = 6.7 × 10 N/m x 0.24 m ______________________________________________________________________________ k=

3.

=

REASONING The force required to stretch the spring is given by Equation 10.1 as FxApplied = kx, where k is the spring constant and x is the displacement of the stretched spring

from its unstrained length. Solving for the spring constant gives k = FxApplied / x . The force applied to the spring has a magnitude equal to the weight W of the board, so FxApplied = W . Since the board’s lower end just extends to the floor, the unstrained length x0 of the spring, plus the length L0 of the board, plus the displacement x of the stretched spring must equal the height h0 of the room, or x0 + L0 + x = h0. Thus, x = h0 − x0 − L0.

SOLUTION Substituting FxApplied = W and x = h0 − x0 − L0 into Equation 10.1, we find

k=

FxApplied x

=

W 104 N = = 650 N/m h0 − x0 − L0 2.44 m − 1.98 m − 0.30 m

____________________________________________________________________________________________

Chapter 10 Problems

4.

499

REASONING The weight of the person causes the spring in the scale to compress. The amount x of compression, according to Equation 10.1, depends on the magnitude FxApplied of the applied force and the spring constant k. SOLUTION a. Since the applied force is equal to the person’s weight, the spring constant is Applied

k=

Fx

=

x

670 N 0.79 × 10

−2

m

= 8.5 × 104 N / m

(10.1)

b. When another person steps on the scale, it compresses by 0.34 cm. The weight (or applied force) that this person exerts on the scale is

(

)(

)

FxApplied = k x = 8.5 × 104 N / m 0.34 × 10−2 m = 290 N

(10.1)

______________________________________________________________________________ 5.

SSM

REASONING AND SOLUTION According to Newton's second law, the force

required to accelerate the trailer is Fx = ma x (Equation 4.2a). The displacement x of the spring is given by Fx = −kx (Equation 10.2). Solving Equation 10.2 for x and using Fx = max , we obtain

x=−

Fx k

=−

ma x k

=−

( 92 kg ) ( 0.30 m/s2 ) 2300 N/m

= −0.012 m

The amount that the spring stretches is 0.012 m . ______________________________________________________________________________ 6.

REASONING The restoring force of the spring and the static frictional force point in opposite directions. Since the box is in equilibrium just before it begins to move, the net force in the horizontal direction is zero at this instant. This condition, together with the expression for the restoring force (Equation 10.2) and the expression for the maximum static frictional force (Equation 4.7), will allow us to determine how far the spring can be stretched without the box moving upon release. SOLUTION The drawing at the right shows the four forces that act on the box: its weight mg, the normal force FN, the restoring force Fx exerted by the spring, and the maximum MAX

. Since the box is not moving, it is static frictional force fs in equilibrium. Let the x axis be parallel to the table top. According to Equation 4.9a, the net force ΣFx in the x direction must be zero, ΣFx = 0.

+y FN MAX

Fx

fs

mg

+x

CHAPTER

11 FLUIDS

ANSWERS TO FOCUS ON CONCEPTS QUESTIONS ______________________________________________________________________________ 1.

(b) According to the relation P2 = P1 + ρ gh (Equation 11.4), the pressure P2 at the bottom of the container depends on the height h of the fluid above it. Since this height is the same for all three containers, the pressure at the bottom is the same for each container.

2.

(d) According to the relation P2 = P1 + ρ gh (Equation 11.4), the pressure at any point depends on the height h of the fluid above it. Since this height is the same for the ceiling of chamber 1 and the floor of chamber 2, the pressure at these two locations is the same.

3.

F = 3.0 × 105 N

4.

(c) The pressure at the top of each liquid is the same, since the U-tube is open at both ends. Also, the pressure at the location of the dashed line is the same in both the left and right sides of the U tube, since these two locations are at the same level. Thus, the pressure increment ρ1gh1 for liquid 1 must be equal to the pressure increment ρ2gh2 for liquid 2, where h1 and h2 are the heights of the liquids above the dashed line. Since h1 is greater than

h2, ρ1 must be less than ρ2.

5.

(b) The pressure at the top of each liquid is the same, since the U-tube is open at both ends. Also, the pressure at the location of the dashed line is the same in both the left and right sides of the U tube, since these two locations are at the same level. Thus, the pressure increment ρ1gh1

for liquid 1 must be equal to the pressure increment ρ2gh2 for liquid 2, where h1 and h2 are the heights of the liquids above the dashed line. Since h1 is 3 times as great as h2, ρ1 must be one-third that of ρ2.

6.

(b) According to the relation, P2 = P1 + ρ gh (Equation 11.4), a drop in the pressure P1 at the top of the pool produces an identical drop in the pressure P2 at the bottom of the pool.

7.

W = 14 000 N

8.

(a) According to Archimedes’ principle, the buoyant force equals the weight of the fluid that the object displaces. Both objects displace the same weight of fluid, since they have the same volume. The buoyant force does not depend on the depth of an object.

9.

(d) The buoyant force (19.6 N) is less than the weight (29.4 N) of the object. Therefore, the object sinks.

562 FLUIDS

(

)

10. (e) When an object floats, its weight ρobjectVobject g equals the buoyant force

( ρfluidVdisplaced g ) , where Vdisplaced is the volume of fluid displaced by the object. Thus, ρobjectVobject g = ρfluidVdisplaced g , so the density of the object is

(

)

ρobject = ρfluid Vdisplaced / Vobject . Thus, the density of the object is proportional to the ratio Vdisplaced / Vobject of the volumes. This ratio is greatest for object C and least for B. 11. (a) The beaker with the ball contains less water, because part of the ball is below the water line. According to Archimedes’ principle, the weight of this “missing” (or displaced) water is equal to the magnitude of the buoyant force that acts on the ball. Since the ball is floating, the magnitude of the buoyant force equals the weight of the ball. Thus, the weight of the missing water is exactly equal to the weight of the ball, so the two beakers weight the same. 12. (e) The volume flow rate is equal to the speed of the water times the cross-sectional area through which the water flows (see Equation 11.10). If the speed doubles and the cross-sectional area triples, the volume flow rate increases by a factor of six (2 × 3). 13. (c) Because water is incompressible and no water accumulates within the pipe, the volume of water per second flowing through the wide section is equal to that flowing through the narrow section. Thus, the volume flow rate is the same in both sections. 14. (b) Water is incompressible, so it cannot accumulate anywhere within the pipe. Thus, the volume flow rate is the same everywhere. Since the volume flow rate is equal to the speed of the water times the cross-sectional area of the pipe (see Equation 11.10), the speed is greatest where the cross-sectional area is smallest. 15. (b) The volume flow rate Q of the blood is the same everywhere. Since Q = Av, we have that A2v2 = A1v1, where the subscript 2 denotes the unblocked region of the artery and 1 the partially blocked region. Since the area of a circle is A = π r 2 , the speed v1 of the blood in

( 2.0 mm )2 r22 the partially blocked region is v1 = v2 2 = ( 0.40 m/s ) = 1.6 m/s. 2 r (1.0 mm ) 1 16. (a) Since blood is an incompressible fluid, the volume flow rate Q is the same everywhere within the artery. Q is equal to the cross-sectional area of the artery times the speed of the blood, so the blood slows down, or decelerates, as it moves from the narrow region into the wider region. Because the hemoglobin molecule decelerates, the direction of the net force acting on it must be opposite to its velocity. Therefore, the pressure ahead of the molecule is greater than that behind it, so the pressure in the wider region is greater than that in the narrow region. 17. P1 − P2 = 207 Pa

Chapter 11 Answers to Focus on Concepts Questions

563

18. (b) The pressure at C is greater than that at B. These two points are at the same elevation, but the fluid is moving slower at C since it has a greater cross-sectional area. Since the fluid is moving slower at C, its pressure is greater. The pressure at B is greater than that at A. The speed of the fluid is the same at both points, since the pipe has the same cross-sectional area. However, B is at the lower elevation and, consequently, has more water above it than A. The greater the height of fluid above a given point, the greater is the pressure at that point, provided the cross-sectional area does not change. 19. PB − PA = 12 000 Pa 20. (d) A longer pipe offers a greater resistance to the flow of a viscous fluid than a shorter pipe does. The volume flow rate depends inversely on the length of the pipe. For a given pipe radius and pressure difference between the ends of the pipe, the volume flow rate is less in longer pipes. In this case the longer pipe is twice as long, so its volume flow rate QB is one-half that of QA. 21. QB = 1.62 × 104 m3/s

564 FLUIDS

CHAPTER 11

FLUIDS

PROBLEMS 1.

SSM REASONING The weight W of the water bed is equal to the mass m of water times the acceleration g due to gravity; W = mg (Equation 4.5). The mass, on the other hand, is equal to the density ρ of the water times its volume V, or m = ρ V (Equation 11.1).

SOLUTION Substituting m = ρ V into the relation W = mg gives W = mg = ( ρ V ) g

(

)

(

)

= 1.00 × 103 kg/m 3 (1.83 m × 2.13 m × 0.229 m ) 9.80 m/s 2 = 8750 N

We have taken the density of water from Table 11.1. Since the weight of the water bed is greater than the additional weight that the floor can tolerate, the bed should not be purchased.

2.

m (Equation 11.1), where m is V the mass of the solvent and V is its volume. The solvent occupies a cylindrical tank of radius r and height h. Its volume V, therefore, is the product of the circular cross-sectional area π r 2 of the tank and the height h of the solvent: REASONING The density ρ of the solvent is given by ρ =

V = π r 2h SOLUTION Substituting Equation (1) into ρ =

(1)

m (Equation 11.1), we obtain the density V

of the solvent:

ρ=

3.

14 300 kg m m = 2 = = 824 kg/m3 V π r h π (1.22 m )2 ( 3.71 m )

SSM REASONING Equation 11.1 can be used to find the volume occupied by 1.00 kg of silver. Once the volume is known, the area of a sheet of silver of thickness d can be found from the fact that the volume is equal to the area of the sheet times its thickness.

Chapter 11 Problems

565

SOLUTION Solving Equation 11.1 for V and using a value of ρ = 10 500 kg/m3 for the density of silver (see Table 11.1), we find that the volume of 1.00 kg of silver is V=

m

ρ

=

1.00 kg = 9.52 ×10−5 m3 3 10 500 kg/m

The area of the silver, is, therefore,

A=

4.

V 9.52 ×10 –5 m3 = = 317 m 2 − 7 d 3.00 × 10 m

REASONING AND SOLUTION a. We will treat the neutron star as spherical in shape, so that its volume is given by the familiar formula, V = 43 π r 3 . Then, according to Equation 11.1, the density of the neutron

star described in the problem statement is

ρ=

3m 3(2.7 × 1028 kg) m m = = = = 3.7 × 1018 kg/m3 3 3 3 3 4 V 3πr 4π r 4π (1.2 × 10 m)

b. If a dime of volume 2.0 × 10 –7 m3 were made of this material, it would weigh W = mg = ρVg = (3.7 × 1018 kg/m3 )(2.0 × 10 –7 m3 )(9.80 m/s 2 )= 7.3 ×1012 N This weight corresponds to 1 lb 12 7.3 × 1012 N = 1.6 × 10 lb 4.448 N

5.

REASONING AND SOLUTION 14.0 karat gold is (14.0)/(24.0) gold or 58.3%. The weight of the gold in the necklace is then (1.27 N)(0.583) = 0.740 N. This corresponds to a volume given by V = M/ρ = W/(ρg) . Thus, V=

6.

0.740 N

(19 300 kg/m )(9.80 m/s ) 3

REASONING AND SOLUTION

2

= 3.91 × 10−6 m3

If the concrete were solid, it would have a mass of 3

3

M = ρV = (2200 kg/m )(0.025 m ) = 55 kg

566 FLUIDS

The mass of concrete removed to make the hole is then 55 kg − 33 kg = 22 kg. This corresponds to a volume V = (22 kg)/(2200 kg/m3) = 0.010 m3. Since the hole is spherical V = (4/3)π r3 so

r=

7.

3

3V = 4π

3

(

3 0.010 m3 4π

)

=

0.13 m

REASONING According to the definition of density ρ given in Equation 11.1, the mass m of a substance is m = ρV, where V is the volume. We will use this equation and the fact that the mass of the water and the gold are equal to find our answer. To convert from a volume in cubic meters to a volume in gallons, we refer to the inside of the front cover of the text to

find that 1 gal = 3.785 × 10

–3

3

m .

REASONING Using Equation 11.1, we find that

ρ Water VWater = ρ Gold VGold Using the fact that 1 gal = 3.785 × 10 Table 11.1, we find VWater =

–3

or 3

m

V Water =

ρ Gold VGold ρ Water

and densities for gold and water from

ρ GoldVGold ρ Water

c19 300 kg / m hb0.15 mgb0.050 mgb0.050 m gF 1 gal = G H3.785 × 10 c1000 kg / m h 3

−3

3

8.

m

3

IJ= K

1.9 gal

REASONING The total mass mT of the rock is the sum of the mass mG of the gold and the mass mQ of the quartz: mT = mG + mQ. Thus, the mass of the gold is mG = mT − mQ

(1)

The total volume VT of the rock is the sum of the volume VG of the gold and volume VQ of the quartz: VT = VG + VQ (2) The volume of a substance is related to the mass and the density of the substance according to the definition of density: ρ = m/V (Equation 11.1). These relations will enable us to find the mass of the gold in the rock.

Chapter 11 Problems

567

SOLUTION Substituting mQ = ρQVQ from Equation 11.1 into Equation (1) gives mG = mT − mQ = mT − ρQVQ

(3)

Solving Equation (2) for the volume VQ of the quartz and substituting the result into Equation (3) yields mG = mT − ρQVQ = mT − ρQ(VT − VG)

(4)

Substituting VG = mG/ρG from Equation 11.1 into Equation (4) gives m mG = mT − ρ Q (VT − VG ) = mT − ρ Q VT − G ρG

(5)

Solving Equation (5) for the mass of the gold, and using the densities for gold and quartz given in Table 11.1, gives VT − mG =

9.

mT

ρQ

1 1 − ρG ρQ

=

12.0 kg ( 4.00 ×10−3 m3 ) − 2660 kg/m3 1 1 − 3 19 300 kg/m 2660 kg/m3

= 1.6 kg

SSM WWW REASONING The period T of a satellite is the time for it to make one complete revolution around the planet. The period is the circumference of the circular orbit (2π R) divided by the speed v of the satellite, so that T = (2π R)/v (see Equation 5.1). In Section 5.5 we saw that the centripetal force required to keep a satellite moving in a circular orbit is provided by the gravitational force. This relationship tells us that the speed of the satellite must be v = GM / R (Equation 5.5), where G is the universal gravitational constant and M is the mass of the planet. By combining this expression for the speed with that for the period, and using the definition of density, we can obtain the period of the satellite.

SOLUTION The period of the satellite is

T=

2π R 2π R R3 = = 2π v GM GM R

568 FLUIDS

According to Equation 11.1, the mass of the planet is equal to its density ρ times its volume V. Since the planet is spherical, V = 43 π R 3 . Thus, M = ρ V = ρ

( 43 π R3 ) . Substituting this

expression for M into that for the period T gives T = 2π

3π R3 R3 = 2π = GM Gρ G ρ 43 π R 3

(

)

The density of iron is ρ = 7860 kg/m (see Table 11.1), so the period of the satellite is 3

3π = Gρ

T=

( 6.67 ×10

3π −11

)(

N ⋅ m 2 /kg 2 7860 kg/m3

)

= 4240 s

10. REASONING The total mass of the solution is the sum of the masses of its constituents. Therefore, ρsVs = ρwVw + ρgVg (1) where the subscripts s, w, and g refer to the solution, the water, and the ethylene glycol, respectively. The volume of the water can be written as Vw = Vs − Vg . Making this substitution for Vw , Equation (1) above can be rearranged to give Vg Vs

=

ρs − ρ w ρg − ρ w

(2)

Equation (2) can be used to calculate the relative volume of ethylene glycol in the solution. SOLUTION The density of ethylene glycol is given in the problem. The density of water is given in Table 11.1 as 1.000 ×103 kg/m3 . The specific gravity of the solution is given as 1.0730. Therefore, the density of the solution is

ρs = (specific gravity of solution) × ρ w = (1.0730)(1.000 ×103 kg/m3 ) = 1.0730 ×103 kg/m3 Substituting the values for the densities into Equation (2), we obtain Vg Vs

=

ρs − ρ w 1.0730 × 103 kg/m3 − 1.000 ×103 kg/m3 = = 0.63 ρg − ρ w 1116 kg/m3 − 1.000 × 103 kg/m3

Chapter 11 Problems

569

Therefore, the volume percentage of ethylene glycol is 63% .

11. SSM REASONING Since the inside of the box is completely evacuated; there is no air to exert an upward force on the lid from the inside. Furthermore, since the weight of the lid is negligible, there is only one force that acts on the lid; the downward force caused by the air pressure on the outside of the lid. In order to pull the lid off the box, one must supply a force that is at least equal in magnitude and opposite in direction to the force exerted on the lid by the outside air. SOLUTION According to Equation 11.3, pressure is defined as P = F / A ; therefore, the magnitude of the force on the lid due to the air pressure is F = (0.85 × 105 N/m 2 )(1.3 × 10 –2 m 2 ) = 1.1× 103 N

12. REASONING Pressure is the magnitude of the force applied perpendicularly to a surface divided by the area of the surface, according to Equation 11.3. The force magnitude, therefore, is equal to the pressure times the area. SOLUTION According to Equation 11.3, we have

c

F = PA = 8.0 × 10 4 lb / in. 2

hb6.1 in.gb2.6 in.g=

1.3 × 10 6 lb

13. REASONING According to Equation 11.3, the pressure P exerted on the ground by the stack of blocks is equal to the force F exerted by the blocks (their combined weight) divided by the area A of the block’s surface in contact with the ground, or P = F/A. Since the pressure is largest when the area is smallest, the least number of blocks is used when the surface area in contact with the ground is the smallest. This area is 0.200 m × 0.100 m. SOLUTION The pressure exerted by N blocks stacked on top of one another is

P=

F N Wone block = A A

(11.3)

where Wone block is the weight of one block. The least number of whole blocks required to 5

produce a pressure of two atmospheres (2.02 × 10 Pa) is

N=

PA Wone block

2.02 ×105 Pa ) ( 0.200 m × 0.100 m ) ( = = 24 169 N

570 FLUIDS

14. REASONING Since the weight is distributed uniformly, each tire exerts one-half of the weight of the rider and bike on the ground. According to the definition of pressure, Equation 11.3, the force that each tire exerts on the ground is equal to the pressure P inside the tire times the area A of contact between the tire and the ground. From this relation, the area of contact can be found. SOLUTION The area of contact that each tire makes with the ground is F A= = P

1 2

(Wperson + Wbike ) = 12 ( 625 N + 98 N ) = 4.76 ×10−4 m2

(11.3)

7.60 × 105 Pa

P

15. REASONING The cap is in equilibrium, so the sum of all the +y forces acting on it must be zero. There are three forces in the Foutside vertical direction: the force Finside due to the gas pressure inside the bottle, the force Foutside due to atmospheric pressure outside the bottle, and the force Fthread that the screw thread exerts on Finside F the cap. By setting the sum of these forces to zero, and using the thread relation F = PA, where P is the pressure and A is the area of the cap, we can determine the magnitude of the force that the screw threads exert on the cap. SOLUTION The drawing shows the free-body diagram of the cap and the three vertical forces that act on it. Since the cap is in equilibrium, the net force in the vertical direction must be zero. ΣFy = − Fthread + Finside − Foutside = 0 (4.9b) Solving this equation for Fthread, and using the fact that force equals pressure times area, F = PA (Equation 11.3), we have

Fthread = Finside − Foutside = Pinside A − Poutside A

(

)(

)

= ( Pinside − Poutside ) A = 1.80 ×105 Pa − 1.01×105 Pa 4.10 ×10−4 m 2 = 32 N

16. REASONING The power generated by the log splitter pump is the ratio of the work W done on the piston to the elapsed time t:

Power =

W t

(6.10a)

571

Chapter 11 Problems

The work done on the piston by the pump is equal to the magnitude F of the force exerted on the piston by the hydraulic oil, multiplied by the distance s through which the piston moves:

(

)

W = F cos0 s = Fs

(6.1)

We have used θ = 0° in Equation 6.1 because the piston moves in the same direction as the force acting on it. The magnitude F of the force applied to the piston is given by

F = PA

(11.3)

where A is the cross-sectional area of the piston and P is the pressure of the hydraulic oil. SOLUTION The head of the piston is circular with a radius r, so its cross-sectional area is given by A = π r 2 . Substituting Equation 11.3 into Equation 6.1, therefore, yields

W = PAs = Pπ r 2 s

(1)

Substituting Equation (1) into Equation 6.10a gives the power required to operate the pump:

(

)

2.0 × 107 Pa π ( 0.050 m ) W Pπ r 2 s = = Power = t t 25 s

2

( 0.60 m )

= 3.8 × 103 W

17. REASONING The pressure P due to the force FSonF that the suitcase exerts on the elevator F floor is given by P = SonF (Equation 11.3), where A is the area of the elevator floor A beneath the suitcase (equal to the product of the length and width of that region). According to Newton’s 3rd law, the magnitude FSonF of the downward force the suitcase exerts on the floor is equal to the magnitude FFonS of the upward force the floor exerts on the suitcase. We will use Newton’s 2nd law, ΣF = ma (Equation 4.1), to determine the magnitude FFonS of the upward force on the suitcase, which has a mass m and an upward acceleration of magnitude a = 1.5 m/s2, equal to that of the elevator. SOLUTION There are only two forces acting on the suitcase, the upward force FFonS that the floor exerts on the suitcase, and the downward weight W = mg (Equation 4.5) exerted by the earth, where g is the magnitude of the acceleration due to gravity. Taking upwards as the positive direction, Newton’s 2nd law yields

ΣF = FFonS − mg = ma

(1)

Solving Equation (1) for FFonS, and noting that by Newton’s 3rd law, FFonS = FSonF, we obtain

572 FLUIDS

FFonS = mg + ma = m( g + a ) = FSonF Substituting Equation (2) into P =

P=

FSonF A

=

FSonF A

(2)

(Equation 11.3), we find that

m ( g + a ) (16 kg ) ( 9.80 m/s 2 + 1.5 m/s2 ) = = 2400 Pa ( 0.15 m )( 0.50 m ) A

18. The tension forces T acting on the edges of the square section of the bladder wall add up to give a total inward force Fin. Because the section is in equilibrium, the net force acting on it must be zero. Therefore, the inward force Fin balances the outward force Fout due to the internal pressure P of the bladder: Fin = Fout. The magnitude Fout of the outward force is

found from Fout = P A (Equation 11.3), where A is the area of the outer surface of the bladder wall. Thus, we have that Fin = Fout = PA θ

Fout

(1)

T Thoriz θ

θ

T

θ

T

θ

T

T

Tin

Fin

SOLUTION We note that the horizontal components Thoriz = T cos θ of the tension forces acting on opposite edges of the section balance each other out. Thus, there is no net horizontal force on the section. The inward components Tin of the four tension forces are equal to one another, and each is given by Tin = T sin θ The magnitude Fin of the net inward force is, therefore, Fin = 4Tin = 4T sin θ

or

T=

Substituting Fin = PA [Equation (1)] into Equation (2) yields

Fin 4 sin θ

(2)

Chapter 11 Problems

T=

PA 4sin θ

573

(3)

The area A of the square is the product of the lengths (l = 0.010 m) of two of its sides: A = l 2 . Making this substitution into Equation (3), we obtain

( 3300 Pa )( 0.010 m ) = 0.95 N PA Pl 2 T= = = 4sin θ 4sin θ 4sin 5.0 2

19. SSM REASONING AND SOLUTION Both the cylinder and the hemisphere have circular cross sections. According to Equation 11.3, the pressure exerted on the ground by the hemisphere is W W P = h = h2 Ah π rh where Wh and rh are the weight and radius of the hemisphere. Similarly, the pressure exerted on the ground by the cylinder is P=

Wc

=

Ac

Wc

π rc2

where Wc and rc are the weight and radius of the cylinder. Since each object exerts the same pressure on the ground, we can equate the right-hand sides of the expressions above to obtain Wh W = c2 2 π rh π rc Solving for rh 2 , we obtain Wh

rh2 = rc2

(1)

Wc

The weight of the hemisphere is

Wh = ρgV h = ρg

1 2

cπ r h= 4 3

3 h

2 3

ρgπ rh3

where ρ and V h are the density and volume of the hemisphere, respectively. The weight of the cylinder is Wc = ρgV c = ρgπ rc2 h

574 FLUIDS

where ρ and Vc are the density and volume of the cylinder, respectively, and h is the height of the cylinder. Substituting these expressions for the weights into Equation (1) gives rh2 = rc2

Wh Wc

= rc2

ρg π rh3 ρg π rc2 h

2 3

Solving for rh gives

rh = 23 h = 23 ( 0.500 m ) = 0.750 m

20. REASONING The magnitude of the force that would be exerted on the window is given by Equation 11.3, F = PA , where the pressure can be found from Equation 11.4: P2 = P1 + ρgh . Since P1 represents the pressure at the surface of the water, it is equal to atmospheric pressure, Patm . Therefore, the magnitude of the force is given by F = ( Patm + ρgh ) A

where, if we assume that the window is circular with radius r, its area A is given by A = π r . 2

SOLUTION a. Thus, the magnitude of the force is

F = 1.013 × 10 5 Pa + (1025 kg / m 3 )( 9.80 m / s 2 )( 11 000 m) π (0.10 m) 2 = 3.5 × 10 6 N b. The weight of a jetliner whose mass is 1.2 × 10 5 kg is W = mg = (1.2 × 10 5 kg)(9.80 m / s 2 ) = 1.2 × 10 6 N

Therefore, the force exerted on the window at a depth of 11 000 m is about three times greater than the weight of a jetliner!

21. REASONING AND SOLUTION The pump must generate an upward force to counteract the weight of the column of water above it. Therefore, F = mg = (ρhA)g. The required pressure is then P = F/A = ρgh = (1.00 × 103 kg/m3)(9.80 m/s2)(71 m) = 7.0 × 105 Pa

22. REASONING The atmospheric pressure outside the tube pushes the sauce up the tube, to the extent that the smaller pressure in the bulb allows it. The smaller the pressure in the bulb, the higher the sauce will rise. The height h to which the sauce rises is related to the

575

Chapter 11 Problems

atmospheric pressure PAtmospheric outside the tube, the pressure PBulb in the bulb, and the density ρ of the sauce by PAtmospheric = PBulb + ρ gh (Equation 11.4).

SOLUTION a. Solving Equation (11.4) for the absolute pressure in the bulb when the height of the sauce is 0.15 m, we find that

PBulb = PAtmospheric − ρ gh

(

)(

)

= 1.013 ×105 Pa − 1200 kg/m3 9.80 m/s 2 ( 0.15 m ) = 9.95 ×104 Pa b. When the height of the sauce is 0.10 m, the absolute pressure in the bulb is PBulb = PAtmospheric − ρ gh

(

)(

)

= 1.013 ×105 Pa − 1200 kg/m3 9.80 m/s 2 ( 0.10 m ) = 1.001×105 Pa

23. SSM REASONING As the depth h increases, the pressure increases according to Equation 11.4 (P2 = P1 + ρgh). In this equation, P1 is the pressure at the shallow end, P2 is 3

3

the pressure at the deep end, and ρ is the density of water (1.00 × 10 kg/m , see Table 11.1). We seek a value for the pressure at the deep end minus the pressure at the shallow end. SOLUTION Using Equation 11.4, we find PDeep = PShallow + ρgh

or

PDeep − PShallow = ρgh

The drawing at the right shows that a value for h can be obtained from the 15-m length of the pool by using the tangent of the 11° angle:

tan11° =

h 15 m

or

15 m 11°

h = (15 m ) tan11°

PDeep − PShallow = ρ g (15 m ) tan11°

(

)(

)

= 1.00 × 103 kg/m3 9.80 m/s 2 (15 m ) tan11° = 2.9 × 104 Pa

h

576 FLUIDS

24. REASONING AND SOLUTION The gauge pressure of the solution at the location of the vein is 3

2

3

P = ρgh = (1030 kg/m )(9.80 m/s )(0.610 m) = 6.16 × 10 Pa Now 1.013 × 105 Pa = 760 mm Hg

so

1 Pa = 7.50 × 10–3 mm Hg

Then 7.50 × 10−3 mm Hg P = 6.16 × 103 Pa = 46.2 mm Hg 1 Pa

(

)

25. REASONING Since the diver uses a snorkel, the pressure in her lungs is atmospheric pressure. If she is swimming at a depth h below the surface, the pressure outside her lungs is atmospheric pressure plus that due to the water. The water pressure P2 at the depth h is

related to the pressure P1 at the surface by Equation 11.4, P2 = P1 + ρ gh, where ρ is the density of the fluid, g is the magnitude of the acceleration due to gravity, and h is the depth. This relation can be used directly to find the depth.

SOLUTION We are given that the maximum difference in pressure between the outside and inside of the lungs is one-twentieth of an atmosphere, or P2 − P1 =

1 20

(1.01×105 Pa ) .

Solving Equation 11.4 for the depth h gives

(

)

1 1.01× 105 Pa P2 − P1 20 = = 0.50 m h= ρg 1025 kg/m3 9.80 m/s 2

(

)(

)

26. REASONING The pressure difference across the diver’s eardrum is Pext − Pint , where Pext is the external pressure and Pint is the internal pressure. The diver’s eardrum ruptures when Pext − Pint = 35 kPa . The exterior pressure Pext on the diver’s eardrum increases with the

depth h of the dive according to Pext = Patm + ρ gh (Equation 11.4), where Patm is the atmospheric pressure at the surface of the water, ρ is the density of seawater, and g is the magnitude of the acceleration due to gravity. Solving Equation 11.4 for the depth h yields h=

Pext − Patm

ρg

(1)

SOLUTION When the diver is at the surface, the internal pressure is equalized with the external pressure: Pint = Patm . If the diver descends slowly from the surface, and takes steps to equalize the exterior and interior pressures, there is no pressure difference across the eardrum, regardless of depth: Pext = Pint . But if the diver sinks rapidly to a depth h without

Chapter 11 Problems

577

equalizing the pressures, then the internal pressure is still atmospheric pressure. In this case, Patm in Equation (1) equals Pint, so the eardrum is at risk for rupture at the depth h given by h=

35 kPa

=

35 × 103 Pa

(1025 kg/m3 )( 9.80 m/s2 ) (1025 kg/m3 )( 9.80 m/s2 )

= 3.5 m

27. SSM REASONING AND SOLUTION a. The pressure at the level of house A is given by Equation 11.4 as P = Patm + ρ gh . Now the height h consists of the 15.0 m and the diameter d of the tank. We first calculate the radius of the tank, from which we can infer d. Since the tank is spherical, its full mass is given by M = ρV = ρ[(4 / 3)π r 3 ] . Therefore,

3M r = 4π ρ 3

1/ 3

or

3M r = 4π ρ

1/3

3(5.25 × 105 kg) = 3 3 4π (1.000 ×10 kg/m )

= 5.00 m

Therefore, the diameter of the tank is 10.0 m, and the height h is given by h = 10.0 m + 15.0 m = 25.0 m

According to Equation 11.4, the gauge pressure in house A is, therefore,

P − Patm = ρ gh = (1.000 × 103 kg/m3 )(9.80 m/s 2 )(25.0 m) = 2.45 × 105 Pa b. The pressure at house B is P = Patm + ρ gh , where h = 15.0 m + 10.0 m − 7.30 m = 17.7 m

According to Equation 11.4, the gauge pressure in house B is

P − Patm = ρ gh = (1.000 × 103 kg/m3 )(9.80 m/s 2 )(17.7 m) = 1.73 × 105 Pa

28. REASONING In each case the pressure at point A in Figure 11.11 is atmospheric pressure and the pressure in the tube above the top of the liquid column is P. In Equation 11.4 (P2 = P1 + ρgh), this means that P2 = Patmosphere and P1 = P. With this identification of the

pressures and a value of 13 600 kg/m3 for the density of mercury (see Table 11.1), Equation 11.4 provides a solution to the problem.

578 FLUIDS

SOLUTION Rearranging Equation 11.4, we have P2 – P1 = ρgh. Applying this expression to each setup gives P2 − P1 Mercury = PAtmosphere − P = ρgh Mercury

c h cP − P h 2

1

Unknown

= PUnknown

b g −P=b ρgh g

Unknown

Since the left side of each of these equations is the same, we have

ρ Mercury gh Mercury = ρ Unknown gh Unknown ρ Unknown = ρ Mercury

Fh G Hh

Mercury

Unknown

I = c13 600 kg / m hFh G JK H16h 3

Mercury Mercury

I= JK

850 kg / m 3

29. SSM REASONING Let the length of the tube be denoted by L, and let the length of the liquid be denoted by . When the tube is whirled in a circle at a constant angular speed about an axis through one end, the liquid collects at the other end and experiences a centripetal force given by (see Equation 8.11, and use F = ma ) F = mrω 2 = mLω 2 . Since there is no air in the tube, this is the only radial force experienced by the liquid, and it results in a pressure of F mLω 2 P= = A A where A is the cross-sectional area of the tube. The mass of the liquid can be expressed in terms of its density ρ and volume V: m = ρV = ρA . The pressure may then be written as

P=

ρ A Lω 2 A

= ρ Lω 2

(1)

If the tube were completely filled with liquid and allowed to hang vertically, the pressure at the bottom of the tube (that is, where h = L ) would be given by

P = ρgL

(2)

SOLUTION According to the statement of the problem, the quantities calculated by Equations (1) and (2) are equal, so that ρ Lω 2 = ρgL . Solving for ω gives

ω=

g

=

9.80 m / s 2 = 31.3 rad / s 0.0100 m

Chapter 11 Problems

579

30. REASONING According to Equation 11.4, the pressure Pmercury at a point 7.10 cm below the ethyl alcohol-mercury interface is Pmercury = Pinterface + ρ mercury gh mercury

where Pinterface is the pressure at the alcohol-mercury interface, and

(1) h mercury = 0.0710 m .

The pressure at the interface is Pinterface = Patm + ρ ethyl ghethyl

(2)

Equation (2) can be used to find the pressure at the interface. This value can then be used in Equation (1) to determine the pressure 7.10 cm below the interface.

SOLUTION Direct substitution of the numerical data into Equation (2) yields Pinterface = 1.01 × 10 5 Pa + (806 kg / m 3 )( 9.80 m / s 2 )( 1.10 m) = 1.10 × 10 5 Pa Therefore, the pressure 7.10 cm below the ethyl alcohol-mercury interface is

Pmercury = 1.10 × 10 5 Pa + (13 600 kg / m 3 )( 9.80 m / s 2 )( 0.0710 m) = 1.19 × 10 5 Pa

31. REASONING Pressure is defined as the magnitude of the force acting perpendicular to a surface divided by the area of the surface. Thus, the magnitude of the total force acting on the vertical dam surface is the pressure times the area A of the surface. But exactly what pressure should we use? According to Equation 11.4, the pressure at any depth h under the water is P = Patm + ρ gh , where Patm is the pressure of the atmosphere acting on the water at the top of the reservoir. Clearly, P has different values at different depths. As a result, we need to use an average pressure. In Equation 11.4, it is only the term ρgh that depends on depth, and the dependence is linear. In other words, the value of ρgh is proportional to h. Therefore, the average value of this term is ρ g water in the full reservoir and

1 2

( H ) , where H is the total depth of the 1 2

H is the average depth. The average pressure acting on the

vertical surface of the dam in contact with the water is, then, P = Patm + ρ g

( H) 1 2

(1)

SOLUTION According to the definition of pressure given in Equation 11.3, the magnitude Ftotal of the total force acting on the vertical surface of the dam is Ftotal = PA , where P is given by Equation (1). Using Equation (1) to substitute for P gives

580 FLUIDS

Ftotal = PA = Patm + ρ g

( H ) A 1 2

The area is A = (120 m)(12 m), and Patm = 1.01×105 Pa . Table 11.1 gives the density of water as ρ = 1.00 ×103 kg/m3 . With these values, we find that Ftotal = Patm + ρ g

( H ) A 1 2

(

)(

= 1.01×105 Pa + 1.00 × 103 kg/m3 9.80 m/s 2

) 12 (12 m ) (120 m )(12 m ) = 2.3 ×108 N

32. REASONING AND SOLUTION The force exerted on the top surface is F2 = P2A2 = Patmπ R2

2

The force exerted on the bottom surface is F1 = P1A1 = (Patm + ρgh)π R12. Equating and rearranging yields R22 = R12(1 + ρgh/Patm) or 2

2

R2 = 1.485 R1

(1)

Consider a right triangle formed by drawing a vertical line from a point on the circumference of the bottom circle to the plane of the top circle so that two sides are equal to R2 – R1 and h. Then tan 30.0° = (R2 – R1)/h. a. Now,

R1 = R2 −h tan 30.0° = R2 – 2.887 m

(2)

Substituting (2) into (1) and rearranging yields 0.485 R22 – 8.57 R2 + 12.38 = 0 which has two roots, namely, R2 = 16.1 m and 1.59 m. The value R2 = 1.59 m leads to a negative value for R1. Clearly, a radius cannot be negative, so we can eliminate the root R2 = 1.59 m, and we conclude that R2 = 16.1 m . b. Now that R2 is known, Equation (2) gives R1 = 13.2 m .

Chapter 11 Problems

581

33. SSM REASONING According to Equation 11.4, the initial pressure at the bottom of the

c h+ ρgh , while the final pressure is P = cP h+ ρgh .

pool is P0 = Patm

f

0

atm

Therefore, the

f

change in pressure at the bottom of the pool is

∆P = Pf − P0 =

cP h+ ρgh − cP h+ ρgh = cP h− cP h atm

atm

f

atm

0

f

atm

0

According to Equation 11.3, F = PA , the change in the force at the bottom of the pool is

∆F = ( ∆P ) A =

cP h− cP h A atm

f

atm

0

SOLUTION Direct substitution of the data given in the problem into the expression above yields 133 Pa ∆F = 765 mm Hg − 755 mm Hg (12 m)(24 m) = 3.8 × 10 5 N 1.0 mm Hg

b

F G H

g

IJ K

Note that the conversion factor 133 Pa = 1.0 mm Hg is used to convert mm Hg to Pa.

34. REASONING Equation 11.5 gives the force F2 of the output plunger in terms of the force F1 applied to the input piston as F2 = F1(A2/A1), where A2 and A1 are the corresponding areas. In this problem the chair begins to rise when the output force just barely exceeds the weight, so F2 = 2100 N. We are given the input force as 55 N. We seek the ratio of the radii, so we will express the area of each circular cross section as π r2 when we apply Equation 11.5. SOLUTION According to Equation 11.5, we have

A2 A1

=

F2

or

F1

π r22 F2 = π r12 F1

Solving for the ratio of the radii yields r2 r1

=

F2 F1

=

2100 N = 6 .2 55 N

35. REASONING We label the input piston as “2” and the output plunger as “1.” When the bottom surfaces of the input piston and output plunger are at the same level, Equation 11.5, F2 = F1 ( A2 / A1 ) , applies. However, this equation is not applicable when the bottom surface of the output plunger is h = 1.50 m above the input piston. In this case we must use Equation

582 FLUIDS

11.4, P2 = P1 + ρ gh , to account for the difference in heights. In either case, we will see that the input force is less than the combined weight of the output plunger and car. SOLUTION a. Using A = π r2 for the circular areas of the piston and plunger, the input force required to support the 24 500-N weight is

(

−3 A2 π 7.70 × 10 m F2 = F1 = ( 24 500 N ) 2 A1 π ( 0.125 m )

)

2

= 93.0 N

(11.5)

b. The pressure P2 at the input piston is related to the pressure P1 at the bottom of the output plunger by Equation 11.4, P2 = P1 + ρgh, where h is the difference in heights. Setting

( )

( )

P2 = F2 / A2 = F2 / π r22 , P1 = F1 / π r12 , and solving for F2, we have (11.4)

π r2 F2 = F1 22 + ρ gh π r22 πr 1

( )

(

−3 π 7.70 × 10 m = ( 24 500 N ) 2 π ( 0.125 m )

(

)(

)

2

)

(

+ 8.30 × 102 kg/m3 9.80 m/s 2 (1.30 m ) π 7.70 × 10−3 m

)

2

= 94.9 N

36. REASONING When a force of magnitude Fp is applied to the piston, the hydraulic fluid exerts a force of magnitude Fv against the safety valve. The two forces are related by Equation 11.5: Ap (11.5) Fp = Fv A v where Ap is the cross-sectional area of the piston and Av is the cross-sectional area of the valve opening. In order for the valve to open, the outward force Fv of the hydraulic fluid must be at least as great as the inward force Fs exerted on the valve by the compressed spring: Fv = Fs . The magnitude of the spring force, in turn, is given by Fs = kx (Equation 10.2 without the minus sign), where k is the spring constant, and x is the amount by which the spring is compressed. Therefore, we have

Chapter 11 Problems

Fv = Fs = kx

583

(1)

SOLUTION Substituting Equation (1) into Equation 11.5, we obtain Ap Fp = kx A v

(2)

Both the piston and the valve have circular cross-sections, so the area A of either one is calculated from its radius r according to A = π r 2 . Substituting A = π r 2 into Equation (2), we obtain the force Fp on the piston needed to open the safety valve: 2

π rp2

2 rp 0.0281 m Fp = kx 885 N/m 0.0100 m = = kx ( )( ) = 165 N π rv2 0.00650 m rv

37. REASONING AND SOLUTION From Pascal's principle, the pressure in the brake fluid at the master cylinder is equal to the pressure in the brake fluid at the plungers: PC = PP, or FC AC

=

FP AP

or

r π r2 = FC P2 = FC P FP = FC r AC π rC C AP

2

The torque on the pedal is equal to the torque that is applied to the master cylinder so that

F = FC

C

or

FC = F C

Combining the expression for FC with the expression for FP above, we have

FP = F C

rp rC

2

2

0.150 m 1.90 × 10 –2 m = (9.00 N) = 108 N –3 0.0500 m × 9.50 10 m

38. REASONING The magnitude F1 of the force the spring exerts on the piston is found from

F1 = kx (Equation 10.3, without the minus sign), where k is the spring constant of the spring, and x is the amount by which the spring is compressed from its unstrained position. F The piston and the plunger are at the same height, so the fluid pressure P = A (Equation 11.3) at the piston is equal to the fluid pressure at the plunger. Therefore, the

584 FLUIDS

A magnitude of the force F2 that the rock exerts on the plunger is given by F2 = F1 2 A 1 (Equation 11.5) where A1 is the area of the piston and A2 is the area of the plunger. The magnitude F2 of the force the rock exerts on the plunger is equal to the magnitude W = mg (Equation 4.5) of the rock’s weight, where m is the rock’s mass and g is the magnitude of the acceleration due to gravity.

SOLUTION Solving F1 = kx (Equation 10.3, without the minus sign) for x, we obtain

x=

F1

(1)

k

A Solving F2 = F1 2 (Equation 11.5) for F1 and substituting F2 = mg yields A 1 A A F1 = F2 1 = mg 1 A2 A2

(2)

Substituting Equation (2) into Equation (1), we find that x=

F1 mg A1 ( 40.0 kg ) ( 9.80 m/s 2 ) 15 cm 2 = = 5.7 × 10 −2 m = 2 k k A2 1600 N/m 65 cm

39. SSM REASONING The pressure P ′ exerted on the bed of the truck by the plunger is P ′ = P − Patm . According to Equation 11.3, F = P ′A , so the force exerted on the bed of the

c

h

truck can be expressed as F = P − Patm A . If we assume that the plunger remains perpendicular to the floor of the load bed, the torque that the plunger creates about the axis shown in the figure in the text is

c

h c

h

τ = F = P − Patm A = P − Patm ( π r 2 ) SOLUTION Direct substitution of the numerical data into the expression above gives

c

h

τ = 3.54 × 10 6 Pa − 1.01 × 10 5 Pa π ( 0.150 m ) 2 ( 3.50 m ) = 8.50 × 10 5 N ⋅ m

40. REASONING The ice with the bear on it is floating, so that the upward-acting buoyant force balances the downward-acting weight Wice of the ice and weight Wbear of the bear. The magnitude FB of the buoyant force is the weight WH

2

O

of the displaced water, according to

Chapter 11 Problems

Archimedes’ principle. Thus, we have FB = WH

2

O

585

= Wice + Wbear , the expression with which

we will obtain Wbear. We can express each of the weights WH

2

and Wice as mass times the

O

magnitude of the acceleration due to gravity (Equation 4.5) and then relate the mass to the density and the displaced volume by using Equation 11.1. SOLUTION Since the ice with the bear on it is floating, the upward-acting buoyant force FB balances the downward-acting weight Wice of the ice and the weight Wbear of the bear. The buoyant force has a magnitude that equals the weight WH

2

O

of the displaced water, as

stated by Archimedes’ principle. Thus, we have

FB = WH

2

O

= Wice + Wbear

or

Wbear = WH

2

O

− Wice

In Equation (1), we can use Equation 4.5 to e