College Physics EOC Questions Chapter 8

344 Chapter 8 Rotational Motion

Substitute this into the Newton’s second law equation for the sphere’s motion along the x axis:

Mg sin u - f = MaCM, x

Mg sin u -25

MaCM, x = MaCM, x

Mg sin u = MaCM, x +25

MaCM, x =75

MaCM, x

aCM, x =57

g sin u

Since 5>7 (about 0.714) is greater than 2>3 (about 0.667), this is in fact greater than the disk’s acceleration aCM, x = 12>32g sin u.

8-12 (c) The angular speed of the ball increases. Once the ball has been hit and set in motion it has angular momentum around the pole, and because we neglect resistive forces there is no external torque on it. (Yes, there is a force on the ball due to tension in the rope. This force is straight inward toward the pole, however, so the angle f between rs and Fs in Equation 8-20, the definition of torque, is 180°. Hence the magnitude of the torque is zero: t = rF sin f = rF sin 180° = 0.) Angular momentum Lz is therefore conserved. As the rope loops around the pole, the ball gets closer to the rotational axis and its

moment of inertia I decreases. Hence the angular speed (which is the magnitude of the angular velocity vz) must increase in order for the magnitude of Lz = Ivz to remain constant.

8-13 (d) The angular momentum L of a system is conserved only if no net torque acts on the system. This isn’t the case for either the pulley or the disk: A net torque due to the tension force acts on the pulley as it rotates, and a net torque due to the force of friction acts on the disk as it rolls downhill. For both objects the moment of inertia I remains unchanged (the objects don’t change shape) and the angular velocity vz increases, so Lz = Ivz increases. Conservation laws are great when they apply, but remember that they do not apply in all situations!

8-14 By bending in the middle, the cat divides its body into two segments, each of which can rotate around a different axis. The cat uses its muscles to rotate its front section into an upright position while simultaneously rotating its hind quarters slightly in the same direction. This small twist cancels the angu-lar momentum generated by the rotation of the front part of his body. Next, the opposite occurs: As the rear of the animal rotates into alignment with the front, it slightly twists its forequarters. Because the net angular momentum of the cat must remain zero, the cat’s body must twist in order to cancel the angular momen-tum associated with the front and rear of its body separately.

Questions and ProblemsIn a few problems, you are given more data than you actually need; in a few other problems, you are required to supply data from your general knowledge, outside sources, or informed estimate.Interpret as significant all digits in numerical values that have trailing zeros and no decimal points.For all problems, use g = 9.80 m>s2 for the free-fall accelera-tion due to gravity.• Basic, single-concept problem•• Intermediate-level problem, may require synthesis of con-cepts and multiple steps••• Challenging problemSSM Solution is in Student Solutions Manual

Conceptual Questions1. •Describe any inconsistencies in the following statement: The units of torque are N # m, but that’s not the same as the units of energy.

2. •What are the units of angular velocity (vs)? Why are factors of 2p present in many equations describing rotational motion?

3. •Why is it critical to define the axis of rotation when you set out to find the moment of inertia of an object? SSM

4. •Explain how an object moving in a straight line can have a nonzero angular momentum.

5. •What are the units of the following quantities: (a) rotational kinetic energy, (b) moment of inertia, and (c) angular momentum?

6. •Explain which physical quantities change when an ice skater moves her arms in and out as she rotates in a pirouette. What causes her angular velocity to change, if it changes at all?

7. •While watching two people on a seesaw, you notice that the person at the top always leans backward, while the person at the bottom always leans forward. (a) Why do the riders do this? (b) Assuming they are sitting equidistant from the pivot point

of the seesaw, what, if anything, can you say about the relative masses of the two riders? SSM

8. •The four solids shown in Figure 8-33 have equal heights, widths, and masses. The axes of rotation are located at the cen-ter of each object and are perpendicular to the plane of the paper. Rank the moments of inertia from greatest to least.

Figure 8-33 Problem8

Hoop Solid cylinder Solid sphere Hollow sphere

9. •Referring to the time-lapse photograph of a falling cat in Figure 8-32, do you think that a cat will fall on her feet if she does not have a tail? Explain your answer using the concepts of this chapter.

10. •What is the ratio of rotational kinetic energy for two balls, each tied to a light string and spinning in a circle with a radius equal to the length of the string? The first ball has a mass m and a string of length L, and rotates at a rate of v. The second ball has a mass 2m and a string of length 2L, and rotates at a rate of 2v.

11. •A student cannot open a door at her school. She pushes with ever-greater force, and still the door will not budge! Know-ing that the door does push open, is not locked, and a minimum torque is required to open the door, give a few reasons why this might be occurring.

12. •Analyze the following statement and determine if there are any physical inconsistencies: While rotating a ball on the end of a string of length L, the rotational kinetic energy remains constant as long as the length and angular speed are fixed. When the ball is pulled inward and the length of the string is

Freed_c08_290-352_st_hr1.indd 344 4/9/13 12:45 PM

Questions and Problems 345

shortened, the rotational kinetic energy will remain constant due to conservation of energy, but the angular momentum will not because there is an external force acting on the ball to pull it inward. The moment of inertia and angular speed will, of course, remain the same throughout the process because the ball is rotating in the same plane throughout the motion.

13. •A freely rotating turntable moves at a steady angu-lar velocity. A glob of cookie dough falls straight down and attaches to the very edge of the turntable. Describe which quan-tities (angular velocity, angular acceleration, torque, rotational kinetic energy, moment of inertia, or angular momentum) are conserved during the process and describe qualitatively what happens to the motion of the turntable. SSM

14. •Describe what a “torque wrench” is and discuss any dif-ficulties that a Canadian auto or bicycle mechanic might have working with an American mechanic’s tools (and vice versa).

15. •In describing rotational motion, it is often useful to develop an analogy with translational motion. First, write a set of equa-tions describing translational motion. Then write the rotational analogs (for example, u = u0 . . .) of the translational equations (for example, x = x0 + v0 t +

12at2) using the following legend:

x 3 u v 3 v a 3 a F 3 t m 3 I

p 3 L Ktranslational 3 Krotational

16. •In Chapter 4, you learned that the mass of an object deter-mines how that object responds to an applied force. Write a rota-tional analog to that idea based on the concepts of this chapter.

17. •Using the rotational concepts of this chapter, explain why a uniform solid sphere beats a uniform solid cylinder which beats a ring when the three objects “race” down an inclined plane while rolling without slipping. SSM

18. •Which quantity is larger: the angular momentum of Earth rotating on its axis each day or the angular momentum of Earth revolving about the Sun each year? Try to determine the answer without using a calculator.

19. •Define the SI unit radian. The unit appears in some physi-cal quantities (for example, the angular velocity of a turntable is 3.5 rad>s) and it is omitted in others (for example, the transla-tional velocity at the rim of a turntable is 0.35 m>s). Because the formula relating rotational and translational quantities involves multiplying by a radian (v = rv), discuss when it is appropriate to include radians and when the unit should be dropped.

20. •A hollow cylinder rolls without slipping up an incline, stops, and then rolls back down. Which of the following graphs in Figure 8-34 shows the (a) angular acceleration and (b) angular velocity for the motion? Assume that up the ramp is the positive direction.

+

(a)

–

+

(b)

–

+

(c)

–

+

(d)

–

+

(e)

–

+

(f)

–


21. •Consider a situation in which a merry-go-round, starting from rest, speeds up in the counterclockwise direction. It eventu-ally reaches and maintains a maximum angular velocity. After a short time the merry-go-round then starts to slow down and eventually stops. Assume the accelerations experienced by the merry-go-round have constant magnitudes. (a) Which graph in Figure 8-35 describes the angular velocity as the merry-go-round speeds up? (b) Which graph describes the angular position as the merry-go-round speeds up? (c) Which graph describes the angular velocity as the merry-go-round travels at its maximum velocity? (d) Which graph describes the angular position as the merry-go-round travels at its maximum velocity? (e) Which graph describes the angular velocity as the merry-go-round slows down? (f) Which graph describes the angular posi-tion as the merry-go-round slows down? (g) Draw a graph of the torque experienced by the merry-go-round as a function of time during the scenario described in the problem. SSM


(a)

t

w

(e)

t

w

(b)

t

w

(c)

t

w

(d)

t

w

(i)

t

q

(v)

t

q

(ii)

t

q

(iii)

t

q

(iv)

t

q

22. •Rank the torques exerted on the bolts in A–D (Figure 8-36) from least to greatest. Note that the forces in B and D make an angle of 45° with the wrench. Assume the wrenches and the magnitude of the force F are identical.

F

FFF

(a) (c)(b) (d)


Multiple-Choice Questions23. •A solid sphere of radius R, a solid cylinder of radius R, and a rod of length R all have the same mass, and all three are rotating with the same angular velocity. The sphere is rotat-ing around an axis through its center. The cylinder is rotating around its long axis, and the rod is rotating around an axis through its center but perpendicular to the rod. Which one has the greatest rotational kinetic energy?

A. the sphereB. the cylinderC. the rodD. the rod and cylinder have the same rotational kinetic

energyE. they all have the same kinetic energy



24. •How would a flywheel’s (spinning disk’s) kinetic energy change if its moment of inertia were five times larger but its angular speed were five times smaller?

A. 0.1 times as large as beforeB. 0.2 times as large as beforeC. same as beforeD. 5 times as large as beforeE. 10 times as large as before

25. •You have two steel spheres; sphere 2 has twice the radius of sphere 1. What is the ratio of the moment of inertia I2:I1?

A. 2B. 4C. 8D. 16E. 32

26. •A solid ball, a solid disk, and a hoop, all with the same mass and the same radius, are set rolling without slipping up an incline, all with the same initial energy. Which goes farthest up the incline?

A. the ballB. the diskC. the hoopD. the hoop and the disk roll to the same height, farther

than the ballE. they all roll to the same height

27. •A solid ball, a solid disk, and a hoop, all with the same mass and the same radius, are set rolling without slipping up an incline, all with the same initial linear speed. Which goes farthest up the incline?

A. the ballB. the diskC. the hoopD. the hoop and the disk roll to the same height, farther

than the ballE. they all roll to the same height SSM

28. •Bob and Lily are riding on a merry-go-round. Bob rides on a horse toward the outside of the circular platform, and Lily rides on a horse toward the center of the circular platform. When the merry-go-round is rotating at a constant angular speed, Bob’s angular speed is

A. exactly half as much as Lily’s.B. larger than Lily’s.C. smaller than Lily’s.D. the same as Lily’s.E. exactly twice as much as Lily’s.

29. •Bob and Lily are riding on a merry-go-round. Bob rides on a horse toward the outer edge of a circular platform. and Lily rides on a horse toward the center of the circular platform. When the merry-go-round is rotating at a constant angular speed v, Bob’s speed v is

A. exactly half as much as Lily’s.B. larger than Lily’s.C. smaller than Lily’s.D. the same as Lily’s.E. exactly twice as much as Lily’s.

30. •While a gymnast is in the air during a leap, which of the following quantities must remain constant for her?

A. positionB. velocityC. momentum

D. angular velocityE. angular momentum about her center of mass

31. •The moment of inertia of a thin ring about its symmetry axis is ICM = MR2. What is the moment of inertia if you twirl a large ring around your finger, so that in essence it rotates about a point on the ring, about an axis parallel to the symmetry axis?

A. 5MR2

B. 2MR2

C. MR2

D. 1.5MR2

E. 0.5MR2 SSM

32. •You give a quick push to a ball at the end of a massless, rigid rod, causing the ball to rotate clockwise in a horizontal circle (Fig-ure 8-37). The rod’s pivot is frictionless. After the push has ended, the ball’s angular velocity

A. steadily increases.B. increases for a while, then remains constant.C. decreases for a while, then remains constant.D. remains constant.E. steadily decreases.

Estimation/Numerical Analysis33. •Estimate the angular speed of a car moving around a clo-verleaf on-ramp of a typical freeway. Cloverleaf ramps extend through approximately three-quarters of a circle to connect two orthogonal freeways. SSM

34. •A fan is designed to last for a certain time before it will have to be replaced (planned obsolescence). The fan only has one speed (at a maximum of 750 rpm), and it reaches the speed in 2 s (starting from rest). It takes the fan 10 s for the blade to stop once it is turned off. If the manufacturer specifies that the fan will operate up to 1 billion rotations, estimate how many days you will be able to use the fan.

35. •Estimate the torque you apply when you open a door in your house. Be sure to specify the axis to which your estimate refers.

36. •Make a rough estimate of the moment of inertia of a pencil that is spun about its center by a nervous student during an exam.

37. •Estimate the moment of inertia of a figure skater as she rotates about the longitudinal axis that passes straight down through the center of her body into the ice. Make this estima-tion for the extreme parts of a pirouette (arms fully extended and arms drawn in tightly).

38. •Estimate the angular displacement (in radians and degrees) of Earth in one day of its orbit around the Sun.

39. •Estimate the angular speed of the apparent passage of the Sun across the sky of Earth (from dawn until dusk).

40. •Estimate the angular acceleration of a lone sock that is inside a washing machine that starts from rest and reaches the maximum speed of its spin cycle in typical fashion.

41. •Estimate the angular momentum about the center of rota-tion for a “skip-it ball” that is spun around on the ankle of a small child (the child hops over the ball as it swings around and around her feet). SSM

Push

Pivot

Top view




42. •Estimate the moment of inertia of Earth about its central axis as it rotates once in a day. Try to recall (or guess) the mass and radius of Earth before you look up the data.

43. •Using a spreadsheet and the data below, calculate the average angular speed of the rotating object over the first 10 s. Calculate the average angular acceleration from 15 to 25 s. If the object has a moment of inertia of 0.25 kg # m2 about the axis of rotation, calculate the average torque during the following time intervals: 0 < t < 10 s, 10 s < t < 15 s, and 15 s < t < 25 s.

t (s) u (rad) t (s) u (rad) t (s) u (rad)

0 0 9 3.14 18 6.481 0.349 10 3.50 19 8.532 0.700 11 3.50 20 11.03 1.05 12 3.49 21 14.14 1.40 13 3.50 22 17.65 1.75 14 3.51 23 21.66 2.10 15 3.51 24 26.27 2.44 16 3.98 25 31.08 2.80 17 5.01

Problems

8-2 An object’s rotational kinetic energy is related to its angular velocity and how its mass is distributed

44. •What is the angular speed of an object that completes 2.00 rev every 12.0 s? Give your answer in rad>s.45. •A car rounds a curve with a translational speed of 12.0 m>s.If the radius of the curve is 7.00 m, calculate the angular speed in rad>s.46. •Convert the following:

45.0 rev>min = ______rad>s331

3 rpm = ______rad>s2p rev>s = ______rad>s

47. •Calculate the angular speed of the Moon as it orbits Earth (recall that the Moon completes one orbit about Earth in 27.4 days and the Earth–Moon distance is 3.84 * 108 m). SSM

48. •If a 0.250-kg point object rotates at 3.00 rev>s about an axis that is 0.500 m away, what is the kinetic energy of the object?

49. •What is the rotational kinetic energy of an object that has a moment of inertia of 0.280 kg # m2 about the axis of rotation when its angular speed is 4.00 rad>s?50. •What is the moment of inertia of an object that rotates at 13.0 rev>min about an axis and has a rotational kinetic energy of 18.0 J?

51. •What is the angular speed of a rotating wheel that has a moment of inertia of 0.330 kg # m2 and a rotational kinetic energy of 2.75 J? Give your answer in both rad>s and rev>min. SSM

8-3 An object’s moment of inertia depends on its mass distribution and the choice of rotation axis

52. •What is the combined moment of inertia for the three point objects about the axis O in Figure 8-38?

Figure 8-38Problem52

2.4 kg 1.8 kg

O

3.0 kg

42 cm28 cm

53. •What is the combined moment of inertia of three point objects (m1 = 1.00 kg, m2 = 1.50 kg, m3 = 2.00 kg) tied together with massless strings and rotating about the axis O as shown in Figure 8-39?


m1 m2 m3

O

1 m 2 m 3 m

54. ••A baton twirler in a marching band complains that her baton is defective (Figure 8-40). The manufacturer specifies that the baton should have an overall length of L = 60.0 cm and a total mass between 940 and 950 g (there is one 350-g objects on each end). Also according to the manufacturer, the moment of inertia about the central axis passing through the baton should fall between 0.0750 and 0.0800 kg # m2. The twirler (who has completed a class in physics) claims this is impossible. Who’s right? Explain your answer.

Figure 8-40Problem54L

55. ••What is the moment of inertia of a steering wheel about the axis that passes through its center? Assume the rim of the wheel has a radius R and a mass M. Assume that there are five radial spokes that connect in the center as shown in Fig-ure 8-41. The spokes are thin rods of length R and mass 1

2M, and are evenly spaced around the wheel. SSM


56. •Using the parallel-axis theorem, calculate the moment of inertia for a solid, uniform sphere about an axis that is tangent to its surface (Figure 8-42).


R

I = ?I = MR22–5



57. •Calculate the moment of inertia for a uniform, solid cylinder (mass M, radius R) if the axis of rota-tion is tangent to the side of the cylinder as shown in Figure 8-43.

58. •Calculate the moment of inertia for a thin rod that is 1.25 m long and has mass of 2.25 kg. The axis of rotation passes through the rod at a point one-third of the way from the left end (Figure 8-44).

Figure 8-44 Problem58L1—3 L2—

3

59. •Calculate the moment of inertia of a thin plate that is 5.00 cm * 7.00 cm in area and has a mass density of 1.50 g>cm2. The axis of rotation is located at the left side, as shown in Figure 8-45.

60. •Calculate the radius of a solid sphere of mass M that has the same moment of inertia about an axis through its center of mass as a second solid sphere of radius R and mass M which has the axis of rotation passing tangent to the surface and par-allel to the center of mass axis (Figure 8-46).


r = ?

M

R

M

61. ••Two uniform, solid spheres (one has a mass M and a radius R and the other has a mass M and a radius 2R) are connected by a thin, uniform rod of length 3R and mass M (Figure 8-47). Find the moment of inertia about the axis through the center of the rod. SSM

62. ••What is the moment of inertia of the sphere–rod system shown in Figure 8-48 where the sphere has a radius R and a mass M and the rod is thin and massless, and has a length L? The sphere–rod sys-tem is spun about an axis A.

8-4 Conservation of mechanical energy also applies to rotating objects

63. •Calculate the final speed of a uniform, solid cylinder of radius 5.00 cm and mass 3.00 kg that starts from rest at the top of an inclined plane that is 2.00 m long and tilted at an angle of 25.0° with the horizontal. Assume the cylinder rolls without slipping down the ramp.

64. •Calculate the final speed of a uniform, solid sphere of radius 5.00 cm and mass 3.00 kg that starts with a transla-tional speed of 2.00 m>s at the top of an inclined plane that is 2.00 m long and tilted at an angle of 25.0° with the hori-zontal. Assume the sphere rolls without slipping down the ramp.

65. •••A spherical marble that has a mass of 50.0 g and a radius of 0.500 cm rolls without slipping down a loop-the-loop track that has a radius of 20.0 cm. The marble starts from rest and just barely clears the loop to emerge on the other side of the track. What is the minimum height that the marble must start from to make it around the loop?

66. •••A billiard ball of mass 160 g and radius 2.50 cm starts with a translational speed of 2.00 m>s at point A on the track as shown in Figure 8-49. If point B is at the top of a hill that has a radius of curvature of 60 cm, what is the normal force acting on the ball at point B? Assume the billiard ball rolls without slipping on the track.

60 cm

B

A

10 cm


67. •Sports A bowling ball that has a radius of 11.0 cm and a mass of 5.00 kg rolls without slipping on a level lane at 2.00 rad>s. Calculate the ratio of the translational kinetic energy to the rotational kinetic energy of the bowling ball. SSM

68. •Astronomy Earth is approximately a solid sphere, has a mass of 5.98 * 1024 kg and a radius of 6.38 * 106 m, and completes one rotation about its central axis each day. Cal-culate the rotational kinetic energy of Earth as it spins on its axis.69. •Astronomy Calculate the translational kinetic energy of Earth as it orbits the Sun once each year (the Earth–Sun distance

I = ?I = MR21–2


7.00 cm

5.00 cm


M

M

M2R

3R

R


AL R

M




is 1.50 * 1011 m). Calculate the ratio of the translational kinetic energy to the rotational kinetic energy calculated in the previ-ous problem.

70. •A potter’s flywheel is made of a 5.00-cm-thick, round slab of concrete that has a mass of 60.0 kg and a diameter of 35.0 cm. This disk rotates about an axis that passes through its center, perpendicular to its round area. Calculate the angu-lar speed of the slab about its center if the rotational kinetic energy is 15.0 J. Express your answer in both rad>s and rev>min.

71. ••Sports A flying disk (160 g, 25.0 cm in diameter) spins at a rate of 300 rpm with its center balanced on a fingertip. What is the rotational kinetic energy of the Frisbee if the disc has 70% of its mass on the outer edge (basically a thin ring 25.0-cm in diameter) and the remaining 30% is a nearly flat disk 25.0-cm in diameter? SSM

8-5 The equations for rotational kinematics are almost identical to those for linear motion

72. •Suppose a roulette wheel is spinning at 1 rev>s. (a) How long will it take for the wheel to come to rest if it experiences an angular acceleration of -0.02 rad>s2? (b) How many rota-tions will it complete in that time?

73. •A spinning top completes 6000 rotations before it starts to topple over. The average speed of the rotations is 800 rpm. Calculate how long the top spins before it begins to topple.

74. •A child pushes a merry-go-round that has a diameter of 4.00 m and goes from rest to an angular speed of 18.0 rpm in a time of 43.0 s. (a) Calculate the angular displacement and the average angular acceleration of the merry-go-round. (b) What is the maximum tangential speed of the child if she rides on the edge of the platform?

75. •Jerry twirls an umbrella around its central axis so that it completes 24.0 rotations in 30.0 s. (a) If the umbrella starts from rest, calculate the angular acceleration of a point on the outer edge. (b) What is the maximum tangential speed of a point on the edge if the umbrella has a radius of 55.0 cm?

76. •Prior to the music CD, stereo systems had a phono-graphic turntable on which vinyl disk recordings were played. A particular phonographic turntable starts from rest and achieves a final constant angular speed of 331

3 rpm in a time of 4.5 s. How many rotations did the turntable undergo during that time? The classic Beatles album Abbey Road is 47 min and 7 s in duration. If the turntable requires 8 s to come to rest once the album is over, calculate the total number of rota-tions for the complete start-up, playing, and slowdown of the album.

77. •A CD player varies its speed as it moves to another cir-cular track on the CD. A CD player is rotating at 300 rpm. To read another track, the angular speed is increased to 450 rpm in a time of 0.75 s. Calculate the average angular acceleration in rad>s2 for the change to occur. SSM

78. •Astronomy A communication satellite circles Earth in a geosynchronous orbit such that the satellite remains directly above the same point on the surface of Earth. (a) What angular displacement (in radians) does the satellite undergo in 1 h of its orbit? (b) Calculate the angular speed of the satellite in rev>min and rad>s.

8-6 Torque is to rotation as force is to translation

79. •What is the torque about your shoulder axis if you hold a 10-kg barbell in one hand straight out and at shoulder height? Assume your hand is 75 cm from your shoulder. SSM

80. •A driver applies a horizontal force of 20.0 N (to the right) to the top of a steering wheel, as shown in Figure 8-50. The steering wheel has a radius of 18.0 cm and a moment of iner-tia of 0.0970 kg # m2. Calculate the angular acceleration of the steering wheel about the central axis.


81. •Medical When the palmaris longus muscle in the forearm is flexed, the wrist moves back and forth (Figure 8-51). If the muscle generates a force of 45.0 N and it is acting with an effec-tive lever arm of 22.0 cm, what is the torque that the muscle pro-duces on the wrist? Curiously, over 15% of all Caucasians lack this muscle; a smaller percentage of Asians (around 5%) also lack it. Some studies correlate the absence of the muscle with carpal tunnel syndrome.

82. •A torque wrench is used to tighten a nut on a bolt. The wrench is 25 cm long, and a force of 120 N is applied at the end of the wrench as shown in Figure 8-52. Calculate the torque about the axis that passes through the bolt.

25 cm

20°F = 120 N


83. •An 85.0-cm-wide door is pushed open with a force of F = 75.0 N. Calculate the torque about an axis that passes through the hinges in each of the cases in Figure 8-53. SSM


Palmaris longus muscle




85 cm

Hingeaxis

F F

FF

(a) (b)

(c) (d)

25°

70°

84. •••A 50.0-g meter stick is balanced at its midpoint (50 cm). Then a 200-g weight is hung with a light string from the 70.0-cm point, and a 100-g weight is hung from the 10.0-cm point (Figure 8-54). Calculate the clockwise and counterclockwise torques acting on the board due to the four forces shown about the following axes: (a) the 0-cm point, (b) the 50-cm point, and (c) the 100-cm point.

200 g100 g

10 cm 70 cm

Fpivot

50 cm

mstickg


85. •A robotic arm lifts a barrel of radioactive waste (Figure 8-55). If the maximum torque delivered by the arm about the axis O is 3000 N # m and the dis-tance r in the diagram is 3.00 m, what is the maximum mass of the barrel?

86. •A typical adult can deliver about 10 N # m of torque when attempting to open a twist-off cap on a bottle. What is the max-imum force that the average person can exert with his fingers if most bottle caps are about 2 cm in diameter?

8-7 The techniques used for solving problems with Newton’s second law also apply to rotation problems

87. •A potter’s wheel is initially at rest. A constant external torque of 75.0 N # m is applied to the wheel for 15.0 s, giving the wheel an angular speed of 500 rev>min. (a) What is the

moment of inertia of the wheel? The external torque is then removed, and a brake is applied. If it takes the wheel 200 s to come to rest after the brake is applied, what is the magnitude of the torque exerted by the brake?

88. •A flywheel of mass 35.0 kg and diameter 60.0 cm spins at 400 rpm when it experiences a sudden power loss. The flywheel slows due to friction in its bearings during the 20.0 s the power is off. If the flywheel makes 200 complete revolutions during the power failure, (a) at what rate is the flywheel spinning when the power comes back on? (b) How long would it have taken for the flywheel to come to a complete stop?

89. ••A solid cylindrical pulley with a mass of 1.00 kg and a radius of 0.25 m is free to rotate about its axis. An object of mass 0.250 kg is attached to the pulley with a light string (Figure 8-56). Assuming the string does not stretch or slip, calcu-late the tension in the string and the angular acceleration of the pulley.

90. ••A block with mass m1 = 2.00 kg rests on a frictionless table. It is connected with a light string over a pulley to a hanging block of mass m2 = 4.00 kg. The pulley is a uniform disk with a radius of 4.00 cm and a mass of 0.500 kg (Figure 8-57). (a) Calculate the accel-eration of each block and the tension in each segment of the string. (b) How long does it take the blocks to move a distance of 2.25 m? (c) What is the angular speed of the pulley at this time?

91. ••A yo-yo with a rolling radius of r = 2.50 cm rolls down a string with a linear acceleration of 6.50 m>s2 (Fig-ure 8-58). (a) Calculate the tension in the string and the angular accel-eration of the yo-yo. (b) What is the moment of inertia of this yo-yo?

8-8 Angular momentum is conserved when there is zero net torque on a system

92. •What is the angular momentum about the central axis of a thin disk that is 18.0 cm in diameter, has a mass of 2.50 kg, and rotates at a constant 1.25 rad>s?93. •What is the angular momentum of a 300-g tetherball when it whirls around the central pole at 60.0 rpm and at a radius of 125 cm?

r

O



m1

m2


r

w




94. •Astronomy Calculate the angular momentum of Earth as it orbits the Sun. Recall that the mass of Earth is 5.98 * 1024 kg, the distance between Earth and the Sun is 1.50 * 1011 m, and the time for one orbit is 365.3 days.

95. •Astronomy Calculate the angular momentum of Earth as it spins on its central axis once each day. Assume Earth is approximately a uniform, solid sphere that has a mass of 5.98 * 1024 kg and a radius of 6.38 * 106 m.

96. •What is the speed of an electron in the lowest energy orbital of hydrogen, of radius equal to 5.29 * 10211 m? The mass of an electron is 9.11 * 10231 kg, and its angular momen-tum in this orbital is 1.055 * 10234 J # s.97. •What is the angular momentum of a 70.0-kg person rid-ing on a Ferris wheel that has a diameter of 35.0 m and rotates once every 25.0 s? SSM

98. •A professor sits on a rotating stool that spins at 10.0 rpm while she holds a 1.00-kg weight in each of her hands. Her out-stretched arms are 0.750 m from the axis of rotation, which passes through her head into the center of the stool. When she draws the weights in toward her body, her angular speed increases to 20.0 rpm. Neglecting the mass of her arms, how far are the weights from the rotational axis at the increased speed?

General Problems99. •A baton is constructed by attaching two small objects that each have a mass M to the ends of a rod that has a length L and a uniform mass M. Find an expression for the moment of inertia of the baton when it is rotated around a point (3>8) L from one end.

100. •The outside diameter of the playing area of an optical Blu-ray disc is 11.75 cm, and the inside diameter is 4.50 cm. When viewing movies, the disc rotates so that a laser maintains a constant linear speed relative to the disc of 7.50 m>s as it tracks over the playing area. (a) What are the maximum and minimum angular speeds (in rad>s and rpm) of the disc? (b) At which location of the laser on the playing area do these speeds occur? (c) What is the average angular acceleration of a Blu-ray disc as it plays an 8.0-h set of movies?

101. •A table saw has a 25.0-cm-diameter blade that rotates at a rate of 7000 rpm. It is equipped with a safety mechanism that can stop the blade within 5.00 ms if something like a finger is accidentally placed in contact with the blade. (a) What average angular acceleration occurs if the saw starts at 7000 rpm and comes to rest in this time? (b) How many rotations does the blade complete during the stopping period?

102. •In 1932 Albert Dremel of Racine, Wisconsin, created his rotary tool that has come to be known as a dremel. (a) Sup-pose a dremel starts from rest and achieves an operating speed of 35,000 rev>min. If it requires 1.20 s for the tool to reach operating speed and it is held at that speed for 45.0 s, how many rotations has the bit made? Suppose it requires another 8.50 s for the tool to return to rest. (b) What are the average angular accel-erations for the start-up and the slow-down periods? (c) How many rotations does the tool complete from start to finish?

103. ••Medical, Sports On average, both arms and hands together account for 13% of a person’s mass, while the head is 7.0% and the trunk and legs account for 80%. We can model a spinning skater with his arms outstretched as a vertical cylinder (head + trunk + legs) with two solid uniform rods (arms + hands)

extended horizontally. Suppose a 62.0-kg skater is 1.80 m tall, has arms that are each 65.0 cm long (including the hands), and a trunk that can be modeled as being 35.0 cm in diameter. If the skater is initially spinning at 70.0 rpm with his arms out-stretched, what will his angular velocity be (in rpm) when he pulls in his arms until they are at his sides parallel to his trunk?

104. •Derive an expression for the moment of inertia of a spherical shell (for example, the peel of an orange) that has a mass M and a radius R, and rotates about an axis that is tan-gent to the surface.

105. •Because of your success in physics class you are selected for an internship at a prestigious bicycle company in its research and development division. Your first task involves designing a wheel made of a hoop that has a mass of 1.00 kg and a radius of 50.0 cm, and spokes with a mass of 10.0 g each. The wheel should have a total moment of inertia 0.280 kg # m2. (a) How many spokes are necessary to construct the wheel? (b) What is the mass of the wheel? SSM

106. ••Two beads that each have a mass M are attached to a thin rod that has a length 2L and a mass M>8. Each bead is initially a distance L>4 from the center of the rod. The whole system is set into uniform rotation about the center of the rod, with initial angular frequency v = 20p rad>s. If the beads are then allowed to slide to the ends of the rod, what will the angu-lar frequency become?

107. ••A uniform disk that has a mass M of 0.300 kg and a radius R of 0.270 m rolls up a ramp of angle u equal to 55.0° with initial speed v of 4.8 m>s. If the disk rolls without slip-ping, how far up the ramp does it go?

108. •In a new model of a machine, a spinning solid spherical part of radius R must be replaced by a ring of the same mass which is to have the same kinetic energy. Both parts need to spin at the same rate, the sphere about an axis through its cen-ter and the ring about an axis perpendicular to its plane at its center. (a) What should the radius of the ring be in terms of R? (b) Will both parts have the same angular momentum? If not, which one will have more?

109. •Many 2.5-in-diameter (6.35-cm) computer hard disks spin at a constant 7200 rpm operating speed. The disks have a mass of about 7.50 g and are essentially uniform throughout with a very small hole at the center. If they reach their operating speed 2.50 s after being turned on, what average torque does the disk drive supply to the disk during the acceleration?

110. ••Sports At the 1984 Olympics, the great diver Greg Louganis won one of his 10 gold medals for the reverse 31

2 somersault tuck dive. In the dive, Louganis began his 312

turns with his body tucked in at a maximum height of approx-imately 2.0 m above the platform, which itself was 10.0 m above the water. He spun uniformly 31

2 times and straight-ened out his body just as he reached the water. A reasonable approximation is to model the diver as a thin uniform rod 2.0 m long when he is stretched out and as a uniform solid cylinder of diameter 0.75 m when he is tucked in. (a) What was Louganis’s average angular speed as he fell toward the water with his body tucked in? Hint: How long did it take him to reach the water from his highest point? (b) What was his angular speed just as he stretched out? (c) How much did Louganis’s rotational kinetic energy change while extending his body if his mass was 75 kg?



111. ••Medical The bones of the forearm (radius and ulna) are hinged to the humerus at the elbow (Figure 8-59). The biceps muscle connects to the bones of the forearm about 2 cm beyond the joint. Assume the forearm has a mass of 2 kg and a length of 0.4 m. When the humerus and the biceps are nearly vertical and the forearm is horizontal, if a person wishes to hold an object of mass M so that her forearm remains motionless, what is the relationship between the force exerted by the biceps muscle and the mass of the object?

Muscle

Elbow

Radius

Ulna Hand

Humerus

M


112. ••Medical The femur of a human leg (mass 10 kg, length 0.9 m) is in traction (Figure 8-60). The center of gravity of the leg is one-third of the distance from the pelvis to the bottom of the foot. Two objects, with masses m1 and m2, are hung at the ends of the leg using pulleys to provide upward support. A third object of 8 kg is hung to provide tension along the leg. The body provides tension as well. (a) What is the mathemati-cal relationship between m1 and m2? Is this relationship unique in the sense that there is only one combination of m1 and m2 that maintains the leg in static equilibrium? (b) How does the relationship change if the tension force due to m1 is applied at the leg’s center of mass?

m1 m2

8 kgFbody


113. •Astronomy It is estimated that 60,000 tons of mete-ors and other space debris accumulates on Earth each year. Assume the debris is accumulated uniformly across the surface of Earth. (a) How much does Earth’s rotation rate change per year as a result of this accumulation? (That is, find the change

in angular velocity.) (b) How long would it take the accumula-tion of debris to change the rotation period by 1 s? SSM

114. •Astronomy Suppose we decided to use the rotation of Earth as a source of energy. (a) What is the maximum amount of energy we could obtain from this source? (b) By the year 2020 the projected rate at which the world uses energy is expected to be 6.4 * 1020 J>y. If energy use continues at that rate, for how many years would the spin of Earth supply our energy needs? Does this seem long enough to justify the effort and expense involved? (c) How long would it take before our day was extended to 48 h instead of 24 h? Assume that Earth is uni-form throughout.

115. •Astronomy In a little over 5 billion years, our Sun will collapse to a white dwarf approximately 16,000 km in diam-eter. (Ignore the fact that the Sun will lose mass as it ages.) (a) What will our Sun’s angular momentum and rotation rate be as a white dwarf? (Express your answers as multiples of its present-day values.) (b) Compared to its present value, will the Sun’s rotational kinetic energy increase, decrease, or stay the same when it becomes a white dwarf? If it does change, by what factor will it change? The radius of the Sun is presently 6.96 * 108 m.

116. •Astronomy (a) If all the people in the world (~7 billion) lined up along the equator, would Earth’s rotation rate increase or decrease? Justify your answer. (b) How would the rotation rate change if all people were no longer on Earth? Assume the average mass of a human is 70.0 kg.

117. •A 1000-kg merry-go-round (a flat, solid cylinder) sup-ports 10 children, each with a mass of 50.0 kg, located at the axis of rotation (thus you may assume the children have no angular momentum at that location). Describe a plan to move the children such that the angular velocity of the merry-go-round decreases to one-half its initial value.

118. •One way for pilots to train for the physical demands of flying at high speeds is with a device called the “human cen-trifuge.” It involves having the pilots travel in circles at high speeds so that they can experience forces greater than their own weight. The diameter of the NASA device is 17.8 m. (a) Suppose a pilot starts at rest and accelerates at a constant rate so that he undergoes 30 rev in 2 min. What is his angular acceleration (in rad>s2)? (b) What is his angular velocity (in rad>s) at the end of that time? (c) After the 2-min period, the centrifuge moves at a constant speed. The g-force experienced is the centripetal force keeping the pilot moving along a circular path. What is the g-force experienced by the pilot? (1 g = mass * 9.80 m>s2)(d) The pilot can tolerate 12 g’s in the horizontal direction. How long would it take the centrifuge to reach that state if it starts at the angular speed found in part (c) and accelerates at the rate found in part (a)?

119. •••The moment of inertia of a rolling marble is I =25MR2,

where M is the mass of the marble and R is the radius. The marble is placed in front of a spring that has a constant k and has been compressed a distance xc. The spring is released, and as the marble comes off the spring it begins to roll without slipping. Note: The static friction that causes rolling without slipping does not do work. (a) Derive an expression for the time it takes for the marble to travel a distance D along the surface after it has lost contact with the spring. (b) Show that your answer for part (a) has the correct units. SSM


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College Physics EOC Questions Chapter 8