AP PHYSICS C: MECHANICS ROTATION REVIEW

Transcription

1 AP PHYSICS C: MECHANICS ROTATION REVIEW 1. Two points, A and B, are on a disk that rotates about an axis. Point A is closer to the axis than point B. Which of the following is not true? A) Point B has the greater speed. B) Point A has the lesser centripetal acceleration. C) Points A and B have the same angular acceleration. D) Point B has the greater angular speed. E) Point A has the lesser tangential acceleration. 2. Two points, A and B, are on a disk that rotates about an axis. Point A is three times as far from the axis as point B. If the speed of point B is v, then what is the speed of point A? A) v B) 3v C) v/3 D) 9v E) v/9 3. Starting from rest, a disk rotates with constant angular acceleration. If it takes 10 rev to reach an angular velocity ω, then how many additional revolutions are required to reach an angular velocity 2ω? A) 10 rev B) 20 rev C) 30 rev D) 40 rev E) 50 rev 4. A record turntable rotates through 5.0 rad in 2.8 s as it is accelerated uniformly from rest. What is the angular velocity at the end of that time? A) 0.60 rad/s B) 0.90 rad/s C) 1.8 rad/s D) 3.6 rad/s E) 14 rad/s Page 1

2 5. You have a friend who lives in the southern part of the United States, and you live in the northern part. As the earth rotates, your linear velocity is hers, and your angular velocity is hers. A) greater than; equal to B) equal to; greater than C) greater than; less than D) less than; greater than E) less than; equal to 6. A wheel rotates through 6.0 rad in 2.0 s as it is uniformly brought to rest. The initial angular velocity of the wheel before braking began was A) 0.60 rad/s B) 0.90 rad/s C) 1.8 rad/s D) 6.0 rad/s E) 7.2 rad/s 7. You are whirling a stone on the end of a string in a horizontal circle of radius R = 0.65 m with a frequency of 4 rev/s when the string breaks. Just after the string breaks, the velocity of the stone is A) straight down. B) 32 m/s along a tangent to the circle. C) 16 m/s along the radius away from the center. D) 1.0 m/s along the radius toward the center. E) none of these. 8. You are pedaling a bicycle at 9.8 m/s. The radius of the wheels of the bicycle is 51.9 cm. The angular velocity of rotation of the wheels is A) 19 rad/s B) 2.5 rad/s C) 4.5 rad/s D) 3.0 rad/s E) 6.3 rad/s Page 2

3 9. The Empire's space station is a long way from any star. It is circular and has a radius of 5.10 km. The angular velocity that is needed to give the station an artificial gravity of 9.80 m/s 2 at its circumference is A) rad/s B) rad/s C) 0.28 rad/s D) 0.22 rad/s E) rad/s 10. A wheel is rotating at 30 rev/min. The angular velocity of the wheel is A) 2π 2 rad/s B) 2π rad/s C) 2 rad/s D) π /2 rad/s E) π rad/s 11. A particle moves uniformly around the circumference of a circle whose radius is 8.0 cm with a period of π/20 s. The angular velocity ω of the particle is A) 2.5 rad/s B) rad/s C) 40 rad/s D) 7.9 rad/s E) 0.96 rad/s 12. A particle is moving uniformly in a circle of radius 50 cm. Its angular velocity is 96 rad/s. The linear speed of the particle is A) 1.0 m/s B) 96 cm/s C) 48 m/s D) zero E) 15 m/s Page 3

4 13. A point P is at a distance R from the axis of rotation of a rigid body. The linear speed, centripetal acceleration, and tangential acceleration of the point can be expressed as A) B) C) D) Linear Centripetal Tangential speed acceleration acceleration Rω Rω 2 Rα Rω Rα Rω 2 Rω 2 Rα Rω Rω Rω 2 Rω E) Rω 2 Rα Rω A body that moves with a constant speed in a circle A) experiences no acceleration. B) undergoes no change in velocity. C) has no resultant force acting on it. D) has no work done on it. E) is described by all of these. 15. When an object is moving in a circle at constant speed, its acceleration is A) constantly increasing. B) constant in direction. C) zero. D) constant in magnitude. E) constant in both magnitude and direction. 16. A wheel rotates with a constant nonzero angular acceleration. Which of the following quantities remains constant in magnitude? A) v, tangential velocity B) ar, radial acceleration C) at, tangential acceleration D) ω, angular velocity E) All of these are correct. Page 4

5 17. A turntable rotating at 8.0 rad/s slows to a stop in 10 s. If the acceleration is constant, the angle through which the turntable rotates in the 10 s is A) 0.80 rad B) 0.40 rad C) 40 rad D) 80 rad E) 16 rad 18. What physical quantity is represented by the slope of the curve shown on the graph? A) displacement B) angular acceleration C) tangential acceleration D) velocity E) None of these is correct. 19. You give an orbiting satellite a command to rotate through an angle given by θ = at + bt 2 ct 4 where a, b, and c are constants and θ is in radians if t is in seconds. What is the angular acceleration of this satellite at time t? A) at B) a + b c C) 12 D) 2b 12ct 2 E) zero Page 5

6 20. The angular acceleration of the flywheel of a generator is given by α(t) = 6bt 12ct 2 where b and c are constants and α is in rad/s 2 provided t is in seconds. If the initial angular velocity is taken to be ω0, the angular velocity at time t is given by A) ω0 + 6bt 2 12ct 3 B) 6b 24ct C) 3bt 2 4ct 3 + ω0 D) 3bt 2 4ct 3 E) 6b 24ct + ω0 21. The data used to construct the graph were taken from the tachometer of an airplane. The angular acceleration during the 10 s interval was A) 3.0 rad/s 2 B) 6.0 rad/s 2 C) 8.0 rad/s 2 D) 20 rad/s 2 E) 38 rad/s Which of the following statements about the motion of the second hand of a clock is true? A) The tangential velocity of the tip is constant. B) The angular velocity is zero. C) The angular acceleration is zero. D) The radial acceleration is zero. E) The tangential acceleration is nonzero. Page 6

7 23. A turntable has an angular velocity of 1.4 rad/s. The coefficient of static friction between the turntable and a block placed on it is The maximum distance from the center of the turntable that the block can be placed without sliding is approximately A) 0.50 m B) 1.0 m C) 1.4 m D) 2.0 m E) 4.4 m 24. A penny is placed 0.10 m from the center of a turntable. If the coefficient of static friction between the penny and the turntable is 0.50, the maximum linear speed at which the penny can travel without slipping is approximately A) 0.49 m/s B) 0.70 m/s C) 1.3 m/s D) 1.4 m/s E) 0.20 km/s 25. A 2.0-kg mass is attached to the end of a 5.0-m rope. The mass moves in a circular path on a horizontal frictionless surface. If the breaking strength of the rope is 40 N, the maximum translational speed with which you can swing the mass without breaking the rope is approximately A) 3.2 m/s B) 4.0 m/s C) 10 m/s D) 20 m/s E) 0.20 km/s Page 7

8 26. A 2-kg sphere attached to an axle by a spring is displaced from its rest position to a radius of 20 cm from the axle centerline by a standard mass of 20 kg, as in Figure 1. The same 2-kg sphere is also displaced 20 cm from the axle centerline, as in Figure 2, when the sphere is rotated at a speed of approximately A) 4.4 m/s B) 9.8 m/s C) 14 m/s D) 98 m/s E) 0.44 km/s 27. The ball shown in the figure will loop-the-loop if it starts from a point high enough on the incline. When the ball is at point A, the centripetal force on it is best represented by which of the following vectors? A) 1 B) 2 C) 3 D) 4 E) 5 Page 8

9 28. A kg dot of paint on the side of a rotating cylinder flies off when the angular speed of the cylinder reaches rad/s. The spin axis of the cylinder is vertical and its radius is 0.04 m. The force of adhesion between the paint and the surface is approximately A) 1 N B) 1 mn C) 5 mn D) 5 kn E) 5 N 29. A disk with a radius of 1.5 m whose moment of inertia is 34 kg m 2 is caused to rotate by a force of 160 N tangent to the circumference. The angular acceleration of the disk is approximately A) 0.14 rad/s 2 B) 0.23 rad/s 2 C) 4.4 rad/s 2 D) 7.1 rad/s 2 E) 23 rad/s Two small masses, ma = kg and mb = kg, are connected by a 1.0-m rod of negligible mass. The angular acceleration about B produced by a force of N applied at A is approximately A) 4.0 rad/s 2 B) 2.7 rad/s 2 C) 11 rad/s 2 D) 12 rad/s 2 E) rad/s 2 Page 9

10 31. A disk is free to rotate about an axis. A force applied at a distance d from the axis causes an angular acceleration α. What angular acceleration is produced if the same force is applied a distance 2d from the axis? A) α B) 2α C) α/2 D) 4α E) α/4 32. A bicycle wheel, a hollow sphere, and a solid sphere each have the same mass and radius. They each rotate about an axis through their centers. Which has the greatest moment of inertia and which has the least? A) The wheel has the greatest; the solid sphere has the least. B) The wheel has the greatest; the hollow sphere has the least. C) The hollow sphere has the greatest; the solid sphere has the least. D) The hollow sphere has the greatest; the wheel has the least. E) The solid sphere has the greatest; the hollow sphere has the least. 33. Water is drawn from a well in a bucket tied to the end of a rope whose other end wraps around a cylinder of mass 50 kg and diameter 25 cm. As you turn this cylinder with a crank, the rope raises the bucket. If the mass of a bucket of water is 20 kg, what torque must you apply to the crank to raise the bucket of water at a constant speed? A) 24 N m B) 2.5 N m C) 80 N m D) N m E) 49 N m 34. Water is drawn from a well in a bucket tied to the end of a rope whose other end wraps around a solid cylinder of mass 50 kg and diameter 25 cm. As this cylinder is turned with a crank, the rope raises the bucket. The mass of a bucket of water is 20 kg. Someone cranks the bucket up and then lets go of the crank, and the bucket of water falls down to the bottom of the well. Without friction or air resistance, what is the angular acceleration of the 50-kg cylinder? A) rad/s 2 B) 3.6 rad/s 2 C) 35 rad/s 2 D) 63 rad/s 2 E) 17 rad/s 2 Page 10

11 35. A disk-shaped grindstone of mass 3.0 kg and radius 8.0 cm is spinning at 600 rev/min. After the power is shut off, a man continues to sharpen his axe by holding it against the grindstone until it stops 10 s later. What is the average torque exerted by the axe on the grindstone? A) 9.6 mn m B) 0.12 N m C) 0.75 N m D) 0.60 kn m E) N m 36. A uniform stick 1 m long is placed horizontally on the ground along an east west axis. A force of 1.0 N is applied to the center of the stick in a direction 30º west of north. The torque exerted by the force relative to the east end of the stick is A) zero. B) 0.25 m, clockwise. C) 0.43 m, clockwise. D) 0.25 m, counterclockwise. E) 0.43 m, counterclockwise. 37. What constant torque, in the absence of friction, must be applied to a wheel to give it an angular velocity of 50 rad/s if it starts from rest and is accelerated for 10 s? The moment of inertia of the wheel about its axle is 9.0 kg m 2. A) 4.5 N m B) 9.0 N m C) 45 N m D) 30 N m E) 60 N m 38. A wheel slows from 20 rad/s to 12 rad/s in 5 s under the influence of a constant frictional torque. In these 5 s, the wheel turns through an angle of A) 2.4 rad B) 43 rad C) 60 rad D) 80 rad E) 100 rad Page 11

12 39. A solid cylinder has a moment of inertia of 2 kg m 2. It is at rest at time zero when a net torque given by τ = 6t (SI units) is applied. After 2 s, the angular velocity of the cylinder will be A) 3.0 rad/s B) 12 rad/s C) 14 rad/s D) 24 rad/s E) 28 rad/s 40. A 7.00-kg mass and a 4.00-kg mass are mounted on a spindle that is free to turn about the x axis as shown. Assume the mass of the arms and the spindle to be negligible. The magnitude of the resultant torque is approximately A) 82.2 N m B) 157 N m C) 225 N m D) 392 N m E) 461 N m 41. A solid disk (I = ½MR 2 ) that is 10 cm in diameter has a mass of 4 kg. The force applied at the outer surface required to produce an angular acceleration of 6 rad/s 2 about an axis through the center of the disk is A) 0.24 kn B) 0.12 kn C) 0.30 N D) 0.60 N E) 1.2 N Page 12

13 42. Torque is defined as A) a force tending to cause rotation. B) the cross product of force and displacement. C) the product of the force and the angular displacement. D) the product of the force and the angular velocity. E) the rotational work done. 43. A thin, massless string is wrapped around a 0.25-m radius grindstone supported by bearings that produce negligible frictional torque. A steady tension of 20 N in the string causes the grindstone to move from rest to a speed of 60 rad/s in 12 s. The moment of inertia of the grindstone is A) 1.0 kg m 2 B) 2.0 kg m 2 C) 3.0 kg m 2 D) 4.0 kg m 2 E) 5.0 kg m Four 50-g point masses are at the corners of a square with 20-cm sides. What is the moment of inertia of this system about an axis perpendicular to the plane of the square and passing through its center? A) kg m 2 B) kg m 2 C) kg m 2 D) kg m 2 E) kg m 2 Page 13

14 45. The moment of inertia of a slim rod of mass m and length L about a transverse axis through one end is ml 2 /3. The moment of inertia of such a rod about a transverse axis through the rod at a distance L/3 from one end is A) ml 2 /36 B) 7mL 2 /36 C) ml 2 /9 D) 2mL 2 /9 E) 4mL 2 /9 46. The moment of inertia of a slim rod about a transverse axis through one end is ml 2 /3, where m is the mass of the rod and L is its length. The moment of inertia of a 0.24-kg meterstick about a transverse axis through its center is A) 0.14 kg m 2 B) 20 kg m 2 C) kg m 2 D) 80 kg m 2 E) 4.5 kg m The moment of inertia of a set of dumbbells, considered as two mass points m separated by a distance 2L about the axis AA, is A) ml 2 B) ½mL 2 C) 2mL 2 D) ¼ ml 2 E) 4mL 2 Page 14

15 48. A homogeneous solid cylinder of mass m, length L, and radius R rotates about an axis through point P, which is parallel to the cylinder axis. If the moment of inertia about the cylinder axis is ½mR 2, the moment of inertia about the axis through P is A) 0.4mR 2 B) ½ mr 2 C) 2/3 mr 2 D) mr 2 E) 1.5mR To increase the moment of inertia of a body about an axis, you must A) increase the angular acceleration. B) increase the angular velocity. C) decrease the angular velocity. D) make the body occupy less space. E) place part of the body farther from the axis. Page 15

16 50. If all of the objects illustrated in the figure have equal masses, the moment of inertia about the indicated axis is largest for the A) ring B) cross C) sphere D) cube E) rod 51. In the figure, R1 = R2 and cm is the center of mass. The rotational inertia about an axis through point P1 is I1, the rotational inertia about an axis through point P2 is I2, and the rotational inertia about an axis through the cm is Icm. The relationship among the moments is A) I1 = I2 > Icm B) I1 = I2 < Icm C) I1 > I2 > Icm D) I1 < Icm > I2 E) I1 = I2 = Icm Page 16

17 52. A 7.00-kg mass and a 4.00-kg mass are mounted on a spindle free to turn about the x axis as shown. Assume the mass of the arms and the spindle to be negligible. The rotational inertia of this system is approximately A) 44.0 kg m 2 B) 47.0 kg m 2 C) 99.0 kg m 2 D) 148 kg m 2 E) 211 kg m The rotational inertia of an object about an axis depends on the A) angular velocity about the axis. B) angular acceleration about the axis. C) mass distribution about the axis. D) torque about the axis. E) linear acceleration about the axis. Page 17

18 54. A stone of mass 10 kg has a rotational inertia of 2.4 kg m 2 about an axis A parallel to an axis through the center of mass. If axis A is 0.20 m from the center of mass axis, the rotational inertia about the center of mass axis is A) 0.40 kg m 2 B) 2.0 kg m 2 C) 2.4 kg m 2 D) 2.8 kg m 2 E) 4.4 kg m A uniform disk (Io = ½ mr 2 ) of mass m and radius R is suspended from a point on its rim. The moment of inertia of the disk about an axis perpendicular to the disk through the pivot point is A) ½ mr 2 B) mr 2 C) 1.5mR 2 D) 2mR 2 E) 2mR 2 /3 56. The torque exerted on a perfectly spherical satellite by the gravitational pull of the sun is A) zero. B) directed along the earth's axis to the north pole. C) directed along the earth's axis to the south pole. D) in the direction of the earth's orbit. E) directed toward the sun. Page 18

19 57. A cord attached to a 3.63-kg mass is wrapped around a wheel of radius m and released. The moment of inertia of the wheel is 2.71 kg m 2. If the wheel rotates on frictionless bearings, the acceleration of the falling weight is A) 3.26 m/s 2 B) 1.04 m/s 2 C) 2.44 m/s 2 D) 1.95 m/s 2 E) 4.27 m/s Two masses M and m (M > m) are hung over a disk (Idisk = ½ M'R 2 ) and are released so that they accelerate. If T1 is the tension in the cord on the left and T2 is the tension in the cord on the right, then A) T1 = T2 B) T2 > T1 C) T2 < T1 D) T2 = Mg E) T2 = Mg/m Page 19

20 59. In a laboratory experiment, various torques are applied to a rotor and the angular acceleration is measured. The results are plotted on the accompanying graph. From the graph, the moment of inertia of the rotor is A) kg m 2 B) kg m 2 C) kg m 2 D) kg m 2 E) kg m A wheel of radius R1 has an axle of radius R2 = ¼R1. If a force F1 is applied tangent to the wheel, a force F2, applied tangent to the axle that will keep the wheel from turning, is equal to A) F1/4 B) F1 C) 4F1 D) 16F1 E) F1/16 Page 20

21 61. The moment of inertia of the wheel in the figure is 0.50 kg m 2, and the bearing is frictionless. The acceleration of the 15-kg mass is approximately A) 9.8 m/s 2 B) 8.7 m/s 2 C) 74 m/s 2 D) 16 m/s 2 E) 0.53 m/s The torque exerted on a perfectly spherical orbiting communications satellite by the gravitational pull of the earth is A) directed toward the earth. B) directed parallel to the earth's axis and toward the north pole. C) directed parallel to the earth's axis and toward the south pole. D) directed toward the satellite. E) zero. Page 21

22 63. In the figure, the rotational inertia of the wheel and axle about the center is 12.0 kg m 2, the constant force F is 39.2 N, and the radius r is m. The wheel starts from rest. When the force has acted through 2.00 m, the rotational velocity ω acquired by the wheel due to this force will be A) 1.26 rad/s B) 3.33 rad/s C) 3.61 rad/s D) 6.24 rad/s E) 10.3 rad/s 64. A solid sphere (I = 0.4MR 2 ) of radius 0.06 m and mass 0.50 kg rolls without slipping 14 m down a 30º inclined plane. At the bottom of the plane, the linear velocity of the center of mass of the sphere is approximately A) 3.5 m/s B) 3.9 m/s C) 8.7 m/s D) 18 m/s E) 9.9 m/s 65. Power can be expressed as the product of A) force and displacement. B) torque and angular displacement. C) torque and angular acceleration. D) force and acceleration. E) torque and angular velocity. Page 22

23 66. A cylinder (I = ½mR 2 ) rolls along a level floor with a speed v. The work required to stop this cylinder is A) ¼ mv 2 B) ½ mv 2 C) ¾ mv 2 D) mv 2 E) 1.25mv The moment of inertia of a certain cylinder, whose mass is not distributed uniformly, is 0.6mR 2 about its geometric axis. The translational speed of the center of mass after it starts from rest and rolls 14 m down a 30º incline is approximately A) 9.3 m/s B) 86 m/s C) 13 m/s D) 3.1 m/s E) 41 m/s 68. The amount of work done on a rotating body can be expressed in terms of the product of A) force and lever arm. B) torque and angular velocity. C) torque and angular acceleration. D) force and time of application of the force. E) torque and angular displacement. 69. A body of mass m is whirled at a constant angular velocity on the end of a string of length R. To double the kinetic energy of the body as it whirls while maintaining the angular velocity, the length of the string must be changed to A) 2R B) C) R/2 D) 4R E) Page 23

24 70. A ball of mass m1, connected to another mass m2 by a string, is whirled at a constant speed in a horizontal circle of radius R equal to m. If the mass m2 = 5.00 kg, the kinetic energy of the ball is A) J B) 2.45 J C) 4.90 J D) 19.6 J E) 39.2 J 71. A hoop of mass 50 kg rolls without slipping. If the center of mass of the hoop has a translational speed of 4.0 m/s, the total kinetic energy of the hoop is A) 0.20 kj B) 0.40 kj C) 1.1 kj D) 3.9 kj E) None of these is correct. 72. Two solid balls (one large, the other small) and a cylinder roll down a hill. Which has the greatest speed at the bottom and which the least? A) The large ball has the greatest; the small ball has the least. B) The small ball has the greatest; the large ball has the least. C) The cylinder has the greatest; the small ball has the least. D) The cylinder has the greatest; both balls have the same lesser speed. E) Both balls have the same greater speed; the cylinder has the least. Page 24

25 73. Assume that all of the mass of a bicycle wheel is concentrated at its rim. Such a wheel of mass 1.2 kg and radius 30 cm starts from rest at the top of a hill 100 m long and inclined at 20º to the horizontal. What will be the speed of the wheel at the bottom of the hill if it rolls without slipping? A) 21 m/s B) 26 m/s C) 15 m/s D) 33 m/s E) 37 m/s 74. Starting from rest at the same time, a coin and a ring roll down an incline without slipping. Which reaches the bottom first? A) The ring reaches the bottom first. B) The coin reaches the bottom first. C) They arrive at the bottom simultaneously. D) The winner depends on the relative masses of the two. E) The winner depends on the relative diameters of the two. 75. For a hoop (ring) of mass M and radius R that is rolling without slipping, which is greater, its translational or its rotational kinetic energy? A) Its translational kinetic energy is greater. B) Its rotational kinetic energy is greater. C) They are equal. D) The answer depends on the radius. E) The answer depends on the mass. 76. For a disk of mass M and radius R that is rolling without slipping, which is greater, its translational or its rotational kinetic energy? A) Its translational kinetic energy is greater. B) Its rotational kinetic energy is greater. C) They are equal. D) The answer depends on the radius. E) The answer depends on the mass. 77. A wheel on a car is rolling without slipping along level ground. The speed of the car is 36 m/s. The wheel has an outer diameter of 50 cm. The speed of the top of the wheel is A) 36 m/s B) 3.6 m/s C) 72 m/s D) 18 m/s E) 98 m/s Page 25

26 78. A bicycle is moving at a speed v = 12.6 m/s. A small stone is stuck to one of the tires. At the instant the stone is at point A in the figure, it comes free. The velocity of the stone (magnitude and direction) relative to the earth just after release is A) 17.8 m/s at 45º above the horizontal, toward the front of the bicycle. B) 12.6 m/s at 45º above the horizontal, away from the bicycle. C) 12.6 m/s at 37º below the horizontal. D) 12.6 m/s straight up. E) 17.8 m/s at 45º above the horizontal, toward the back of the bicycle. 79. A wheel of radius R is rolling without slipping. The velocity of the point on the rim that is in contact with the surface, relative to the surface, is A) equal to Rω in the direction of motion of the center of mass. B) equal to Rω opposite the direction of motion of the center of mass. C) zero. D) equal to the velocity of the center of mass and in the same direction. E) equal to the velocity of the center of mass but in the opposite direction. 80. A solid cylinder, a hollow cylinder, and a square block of equal masses are released at the top of an inclined plane. The cylinders roll down and the block slides down, all with negligible frictional losses. Which of the following will happen? A) The hollow cylinder arrives at the bottom first. B) The solid cylinder arrives at the bottom first. C) The block arrives at the bottom first. D) They all arrive at the bottom simultaneously. E) None of these will happen. 81. A solid cylinder, a hollow cylinder, and a square block of equal masses are released at the top of an inclined plane. The cylinders roll down and the block slides down, all with negligible frictional losses. In what order will they arrive at the bottom? A) solid cylinder, hollow cylinder, block B) hollow cylinder, solid cylinder, block C) block, hollow cylinder, solid cylinder D) block, solid cylinder, hollow cylinder E) all at the same instant Page 26

27 82. Three solid, homogeneous spheres are on identical inclined planes. If there are no frictional losses, which of the following statements correctly relates the translational speeds at the bottoms of the inclined planes? A) va = vb = vc B) va > vb; va > vc C) va < vc; vb < vc D) va < vb; vb < vc E) va = vb; vb < vc 83. You are given two hoops (I = ½mR 2 ), which are (1) brass and (2) wood, and two cylinders (I = mr 2 ), which are (3) brass and (4) wood; each has radius R. If all are released from the same starting line at the same time, the one(s) that reach the bottom first are A) 1 and 2 B) 3 and 4 C) 1, 2, 3, and 4 D) 1 E) 3 Page 27

28 84. The moment of inertia of a certain wheel about its axle is ¾ mr 2. The translational speed of its axle after it starts from rest and rolls without slipping down an inclined plane 2.13 m high is A) 9.75 m/s B) 8.53 m/s C) 7.31 m/s D) 6.10 m/s E) 4.88 m/s 85. A uniform cylinder (I = ½ mr 2 ) of diameter 0.20 m and mass 12 kg rolls without slipping down a 37º inclined plane. The acceleration of the cylinder down the plane is approximately A) 2.0 m/s 2 B) 3.9 m/s 2 C) 4.9 m/s 2 D) 5.8 m/s 2 E) 9.8 m/s A uniform cylinder (I = ½ mr 2 ) of diameter 0.20 m and mass 12 kg rolls without slipping down a 37º inclined plane. The gain in translational kinetic energy of the cylinder when it has rolled 5 m down the incline of the plane is approximately A) 24 J B) 0.12 kj C) 0.24 kj D) 0.35 kj E) 0.59 kj Page 28

29 87. A solid disk (Icm = ½ mr 2 ) rolls without slipping up a plane a distance s. The plane is inclined at an angle θ with the horizontal. The disk has mass m, radius R, and an initial translational speed v. The distance s the disk rolls is A) ¾ v 2 /(g sin θ) B) ½ v 2 /(g sin θ) C) ½ Rv/(g sin θ) D) ½ mg(sin θ cos θ)(rv) 2 E) v 2 /(g sin θ) 88. In the laboratory, a solid cylinder is permitted to roll down a plane inclined at an angle with the horizontal. If no slipping occurs, which of the following is true? A) No slipping implies no frictional force to consider. B) Because there is no slipping, the motion of the cylinder can be considered to be a pure rotation about the center of mass. C) The change in potential energy is equal to ½ I0ω 2. D) The acceleration of the center of mass can be expressed in terms of the acceleration due to gravity and θ. E) The only force acting on the cylinder is that of gravity. Page 29

30 89. The curve that most nearly represents the acceleration of an object rolling down an inclined plane as a function of the angle of inclination is A) 1 B) 2 C) 3 D) 4 E) A 1.0-kg metal hoop with a radius of 0.5 m has a translational velocity of 2.0 m/s as it rolls without slipping. The angular momentum of this hoop about its center of mass is A) 1.0 kg m 2 /s B) 2.0 kg m 2 /s C) 8.0 kg m 2 /s D) 4.0 kg m 2 /s E) 0.50 kg m 2 /s Page 30

31 91. A hoop of radius 3.05 n has a mass of 145 kg. Its moment of inertia is mr 2. The hoop rolls without slipping along a horizontal plane. If the center of mass of the hoop has a speed of m/s, the work required to bring the hoop to rest is A) 6.78 J B) 13.5 J C) 682 J D) 217 J E) 4.34 kj 92. A force Fx in the negative x direction is applied to a particle in the xy plane. The arrow that best represents the torque produced by Fx on the particle with respect to the origin is A) 1 B) 2 C) 3 D) 4 E) 5 Page 31

32 93. A phonograph turntable in the xz plane is rotating clockwise as viewed from above. The vector that represents the torque with which the motor turns the table is A) 1 B) 2 C) 3 D) 4 E) A torque is applied to a bolt by hanging a weight w from the end of the wrench, as shown. The coordinate axis along which the torque vector is directed is A) y B) x C) y D) x E) z Page 32

33 95. As a particle with a velocity v in the negative x direction passes through the point (0, 0, 1), it has an angular velocity relative to the origin that is best represented by vector A) 1 B) 2 C) 3 D) 4 E) zero 96. The vector C represents A) A B B) B A C) A B D) B A E) None of these is correct. Page 33

34 97. Vectors I and II lie in the xy plane. The vector product II (I II) could be represented by vector A) A B) B C) C D) D E) E Page 34

35 98. A 7-kg mass and a 4-kg mass are mounted on a spindle that is free to turn about the x axis as shown. Assume the mass of the arms and the spindle to be negligible. If the system is free to rotate and is released from rest, there will initially be a resultant torque in which of the following directions? A) z B) z C) y D) x E) x Page 35

36 99. A wheel is rotating clockwise on a fixed axis perpendicular to the page (x). A torque that causes the wheel to slow down is best represented by the vector A) 1 B) 2 C) 3 D) 4 E) A particle of mass m is moving with a velocity v, in the yz plane as shown. The vector that most nearly represents the angular momentum about the x axis is A) 1 B) 2 C) 3 D) 4 E) 5 Page 36

37 101. A wheel is set spinning and is then hung by a rope placed at one end of the axle. If the wheel is spinning as shown, the angular momentum of the wheel could be represented by vector A) 1 B) 2 C) 3 D) 4 E) A disk rotates clockwise in the plane of the page. What is the direction of the angular momentum vector? A) clockwise B) counterclockwise C) into the page D) out of the page E) Angular momentum has no direction. Page 37

38 103. Particles 1, 2, and 3 have equal masses and equal speeds. The angular momentum with respect to the origin for these three masses is A) the same for each particle. B) greatest for particle 1. C) greatest for particle 2. D) greatest for particle 3. E) least for particle The angular momentum of a flywheel about its axis is 925 kg m 2 /s. If its moment of inertia about the same axis is 2.50 kg m 2, its angular velocity is A) 370 rev/min B) 62 rev/min C) 36 rev/min D) 2210 rad/s E) 370 rad/s 105. If the angular momentum of a system is constant, which of the following statements must be true? A) No torque acts on any part of the system. B) A constant torque acts on each part of the system. C) Zero net torque acts on each part of the system. D) A constant external torque acts on the system. E) Zero net torque acts on the system The angular momentum of a system is conserved only if A) the angular velocity is a function of time. B) the sum of the external torques equals the sum of the internal torques. C) the moment of inertia of the system is constant. D) the sum of the external torques is zero. E) the sum of the internal torques is zero. Page 38

39 107. A constant torque of 15 N m acts for 3.0 s on a system of mass 2.0 kg. The change in angular momentum of the system during this period of time is A) 5.0 kg m 2 /s B) 7.5 kg m 2 /s C) 10 kg m 2 /s D) 23 kg m 2 /s E) 45 kg m 2 /s 108. The angular momentum of a body about a particular axis as a function of time is shown in the graph. The external torque acting on the body along this axis at t = 2 s is A) 0 B) 5 N m C) 10 N m D) 20 N m E) 40 N m 109. If the sum of the external torques acting on an isolated system of particles is zero, it must be true that A) the system can have no kinetic energy. B) the angular momentum of the system does not change. C) the system can have no angular velocity. D) the system can have no linear velocity. E) the angular momentum of the system must be continually decreasing. Page 39

40 110. A disk-shaped grindstone of mass 3.0 kg and radius 8.0 cm is spinning at 600 rev/min. After the power is shut off, a man continues to sharpen his axe by holding it against the grindstone until it stops 10 s later. What was the stone's initial kinetic energy when the power was turned off? A) 19 J B) J C) J D) J E) J 111. A spinning bicycle wheel is supported as shown by a line fastened to one end of its axle. The resultant torque acting on the wheel lies along which of the following axes? A) x B) y C) y D) z E) z 112. The angular momentum vector for a spinning wheel lies along its axle and is pointed east. To make this vector point south, it is necessary to exert a force on the east end of the axle in which direction? A) up B) down C) north D) south E) east Page 40

41 113. If the sum of the external torques on a system is zero, there is A) a change in the system's moment of inertia. B) no change in the system's moment of inertia. C) a change in the system's angular momentum. D) no change in the system's angular momentum. E) a precessional angular velocity If the sum of the torques on a body about a fixed axis is not zero, the body most certainly A) experiences translational acceleration. B) experiences angular acceleration. C) experiences precession. D) experioences rotational inertia. E) remains in equilibrium A woman sits on a spinning piano stool with her arms folded. When she extends her arms, which of the following occurs? A) She increases her moment of inertia, thereby increasing her angular speed. B) She increases her moment of inertia, thereby decreasing her angular speed. C) She decreases her moment of inertia, thereby increasing her angular speed. D) She decreases her moment of inertia, thereby decreasing her angular speed. E) Both her moment of inertia and her angular speed remain constant A man turns with an angular velocity on a rotating table, holding two equal masses at arms' length. If he drops the two masses without moving his arms, his angular velocity A) decreases. B) remains the same. C) increases. D) increases as the angular velocity of the masses decreases. E) decreases as the angular velocity of the masses increases A man stands on the center of a platform that is rotating on frictionless bearings at a speed of 1.00 rad/s. Originally his arms are outstretched and he holds a 4.54-kg mass in each hand. He then pulls the weights in toward his body. Assume the moment of inertia of the man, including his arms, to remain constant at 5.42 kg m 2. If the original distance of the weights from the axis is m and their final distance is m, the final angular velocity is A) 1.14 rad/s B) 1.27 rad/s C) 1.58 rad/s D) 2.08 rad/s E) 7.70 rad/s Page 41

42 118. A woman sits on a stool that can turn friction-free about its vertical axis. She is handed a spinning bicycle wheel that has angular momentum L0 and she turns it over (that is, through 180º). She thereby acquires an angular momentum of magnitude A) 0 B) ½ L0 C) L0 D) 2L0 E) 4L Two identical cylindrical disks have a common axis. Initially one of the disks is spinning. When the two disks are brought into contact, they stick together. Which of the following is true? A) The total kinetic energy and the total angular momentum are unchanged from their initial values. B) Both the total kinetic energy and the total angular momentum are reduced to half of their original values. C) The total angular momentum is unchanged, but the total kinetic energy is reduced to half its original value. D) The total angular momentum is reduced to half its original value, but the total kinetic energy is unchanged. E) The total angular momentum is unchanged, and the total kinetic energy is reduced to one-quarter of its original value A wheel is rotating freely with an angular speed of 20 rad/s on a shaft whose moment of inertia is negligible. A second identical wheel, initially at rest, is suddenly coupled to the same shaft. The angular speed of the coupled wheels is A) 10 rad/s B) 14 rad/s C) 20 rad/s D) 28 rad/s E) 40 rad/s Page 42

43 121. A merry-go-round with a moment of inertia of kg m 2 is coasting at 2.20 rad/s. When a 72.6-kg man steps onto the rim, the angular velocity decreases to 2.0 rad/s. The radius of the merry-go-round is A) 3.06 m B) 3.66 m C) 4.27 m D) 4.88 m E) 5.49 m 122. In a playground there is a small merry-go-round of radius 1.25 m and mass 175 kg. Assume the merry-go-round to be a uniform disk. A child of mass 45 kg runs at a speed of 3.0 m/s tangent to the rim of the merry-go-round (initially at rest) and jumps on. If we neglect friction, what is the angular speed of the merry-go-round after the child has jumped on and is standing at its outer rim? A) 0.82 rad/s B) 2.4 rad/s C) 0.49 rad/s D) 1.2 rad/s E) 0.41 rad/s 123. A hoop rotates about an axis through its center with an angular velocity of 40.0 rad/s. If the rotational kinetic energy of the hoop is 400 J, its angular momentum is A) 800 kg m 2 /s B) 400 kg m 2 /s C) 200 kg m 2 /s D) 20 kg m 2 /s E) 5 kg m 2 /s Page 43

44 124. Which of the following are required for the total momentum (both angular and linear) of a system to be conserved? 1 The sum of the external torques acting on the system must be zero.. 2 The sum of the external forces acting on the system must be zero.. 3 The total kinetic energy must remain constant.. 4 There can be no external torques or forces acting on the system.. 5 There can be no internal torques or forces acting on the system.. A) 1 and 2 B) 1, 2, and 3 C) 1, 2, and 4 D) 1, 2, 3, and 4 E) 1, 2, 3, 4, and A man is walking north carrying a suitcase that contains a spinning gyroscope mounted on an axle attached to the front and back of the case. The angular velocity of the gyroscope points north. The man now begins to turn to walk east. As a result, the front end of the suitcase A) resists his attempt to turn and tries to remain pointed north. B) fights his attempt to turn and pulls to the west. C) rises upward. D) dips downward. E) does nothing whatever unusual. Page 44

45 126. Two wheels with identical moments of inertia are rotating about the same axle. The first is rotating clockwise at 2.0 rad/s, and the second is rotating counterclockwise at 6.0 rad/s. If the two wheels are brought into contact so that they rotate together, their final angular velocity will be A) 2.0 rad/s, counterclockwise. B) 3.0 rad/s, clockwise. C) 4.0 rad/s, counterclockwise. D) 5.0 rad/s, clockwise. E) 6.0 rad/s, clockwise A wheel of moment of inertia kg m 2 is spinning with an angular speed of 5000 rad/s. A torque is applied about an axis perpendicular to the spin axis. If the applied torque has a magnitude of 67.8 N m, the angular velocity of precession will be A) 1.00 rad/s B) rad/s C) 10.0 rad/s D) 100 rad/s E) 1000 rad/s 128. A certain airplane engine rotates counterclockwise when viewed from aft (that is, from the back of the airplane). When the plane turns to the left, A) the engine makes it turn faster than when it turns to the right. B) the engine makes it turn slower than when it turns to the right. C) it tends to dive. D) it tends to climb. E) the engine has no effect on the turn. Page 45

46 129. A solid cylinder is spinning counterclockwise about a longitudinal axis when a net torque τ is applied, as shown. The cylinder A) speeds up. B) slows down. C) precesses about a vertical axis. D) precesses about a horizontal axis. E) does none of these A wheel is set spinning and then is hung by a rope placed at one end of the axle. The precession vector of the spinning wheel points in the direction of A) z B) y C) z D) x E) y Page 46

47 131. The propeller of a motorboat turns clockwise relative to a water skier being towed by the boat. As the boat makes a sharp turn to the left, gyroscopic action tends to A) cause the front of the boat to rise. B) cause the front of the boat to dip. C) cause the boat to tip to the left. D) cause the boat to tip to the right. E) keep the boat headed in its original direction A wheel is rotating in the direction indicated. If you pull down on the end of the axle nearest you, that end of the axle tends to move A) up. B) to the right. C) down. D) to the left. E) directly inward. Page 47

48 133. A gyroscopic wheel spins clockwise as shown. The set of vectors that correctly describes the directions of the torque τ, angular momentum L, and angular velocity of precession ωp, is A) τ(+z); L( x); ω(+y) B) τ( z); L(+x); ω( y) C) τ(+y); L( x); ω(+z) D) τ( y); L( z); ω( x) E) τ(+y); L( x); ω( z) Page 48

49 134. A gyroscopic toy is spinning as shown. The torque τ, angular momentum of the wheel L, and angular precession velocity ωp are in which directions? A) 1 B) 2 C) 3 D) 4 E) 5 τ L ωp 1 z y x 2 x y z 3 x y z 4 x y z 5 x y z 135. A spinning bicycle wheel with a loaded rim (essentially a hoop) is supported by a line at one end of its axle. The radius of the wheel is m, and the wheel has a mass of 3.63 kg. It is spinning at 80.0 rad/s, and the center of mass is 15.2 cm from the point of support. The angular velocity of precession is A) rad/s B) rad/s C) rad/s D) rad/s E) rad/s Page 49

50 136. The figure shows vectors representing the angular velocity of precession ωp and the spin velocity ωs. The associated torque vector points along which of the axes? A) x B) y C) z D) z E) None of these is correct Spin-½ particles A) are called bosons. B) have spin angular momenta that can be changed by applying a net torque to them. C) can have angular momenta that change continuously from one value to another. D) can very accurately be thought of as spinning spheres. E) are described by none of the above Which of the following statements is true? A) The angular momentum of a particle due to its motion is its orbital angular momentum. B) Spin-½ particles are called fermions. C) The fundamental unit of angular momentum is U. D) The units of U are J s. E) All of these are correct. Page 50

51 139. Which of the following statements is true? A) Stable matter consists of electrons, protons, and neutrons. B) Electrons, protons, and neutrons have an intrinsic angular momentum that is called spin. C) Bosons have zero spin or integral spin. D) The spin angular momentum of a particle is a fundamental property of the particle and as such cannot be changed. E) All of these are correct Which of the following statements is not true? A) Stable matter consists of electrons, protons, and neutrons. B) Electrons, protons, and neutrons have an intrinsic angular momentum that is called spin. C) Bosons have zero spin or integral spin. D) An electron is well known to have a finite size. E) The spin angular momentum of a particle is a fundamental property of the particle and as such cannot be changed The energy-level diagram that most closely represents a rotating molecule with constant moment of inertia is A) 1 B) 2 C) 3 D) 4 E) 5 Page 51

52 Answer Key 1. D 2. B 3. C 4. D 5. E 6. D 7. E 8. D 9. B 10. E 11. C 12. C 13. A 14. D 15. D 16. C 17. C 18. B 19. D 20. C 21. B 22. C 23. B 24. B 25. C 26. A 27. C 28. E 29. D 30. A 31. B 32. A 33. A 34. C 35. E 36. C 37. C 38. D 39. C 40. B 41. D 42. B 43. A 44. B Page 52

53 45. C 46. C 47. C 48. E 49. E 50. A 51. A 52. E 53. C 54. B 55. C 56. A 57. A 58. B 59. C 60. C 61. B 62. E 63. C 64. E 65. E 66. C 67. A 68. E 69. B 70. D 71. E 72. E 73. C 74. B 75. C 76. A 77. C 78. A 79. C 80. C 81. D 82. A 83. B 84. E 85. B 86. C 87. A 88. D 89. A 90. A Page 53

54 91. B 92. A 93. B 94. D 95. A 96. D 97. A 98. E 99. A 100. D 101. E 102. C 103. C 104. E 105. E 106. D 107. E 108. B 109. B 110. A 111. E 112. A 113. D 114. B 115. B 116. B 117. D 118. D 119. E 120. A 121. A 122. A 123. D 124. A 125. D 126. C 127. B 128. C 129. A 130. C 131. A 132. D 133. E 134. E 135. C 136. B Page 54

55 137. E 138. E 139. E 140. D 141. A Page 55