Download AP PHYSICS C: MECHANICS ROTATION REVIEW 1. Two

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
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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
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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/s2 at its circumference is
A) 4.4 × 10–2 rad/s
B) 7.0 × 10–3 rad/s
C) 0.28 rad/s
D) –0.22 rad/s
E) 1.3 × 103 rad/s
10. A wheel is rotating at 30 rev/min. The angular velocity of the wheel is
A) 2π2 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) 3.2 × 102 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
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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
Linear
speed
A)
Centripetal
acceleration
Rω
Rω2
B)
Rω
Rα
Rω2
C)
Rω2
Rα
Rω
D)
Rω
Rω2
Rω
E)
Rω2
Rα
Rω2
Tangential
acceleration
Rα
14. 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.
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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 + bt2 – ct4
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 – 12ct2
E) zero
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20. The angular acceleration of the flywheel of a generator is given by
α(t) = 6bt – 12ct2
where b and c are constants and α is in rad/s2 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 + 6bt2 – 12ct3
B) 6b – 24ct
C) 3bt2 – 4ct3 + ω0
D) 3bt2 – 4ct3
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/s2
B) 6.0 rad/s2
C) 8.0 rad/s2
D) 20 rad/s2
E) 38 rad/s2
22. 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.
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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 0.20. 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
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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
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28. A 5 × 10–6-kg dot of paint on the side of a rotating cylinder flies off when the angular
speed of the cylinder reaches 5 × 103 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 · m2 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/s2
B) 0.23 rad/s2
C) 4.4 rad/s2
D) 7.1 rad/s2
E) 23 rad/s2
30.
Two small masses, mA = 4.0 × 10–3 kg and mB = 2.0 × 10–3 kg, are connected by a 1.0-m
rod of negligible mass. The angular acceleration about B produced by a force of 0.016
N applied at A is approximately
A) 4.0 rad/s2
B) 2.7 rad/s2
C) 11 rad/s2
D) 12 rad/s2
E) 4.0 × 102 rad/s2
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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) 2.4 × 103 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) 1.1 × 102 rad/s2
B) 3.6 rad/s2
C) 35 rad/s2
D) 63 rad/s2
E) 17 rad/s2
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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) 0.060 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 · m2.
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
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39. A solid cylinder has a moment of inertia of 2 kg · m2. It is at rest at time zero when a
net torque given by
τ = 6t2 + 6 (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 = ½MR2) 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/s2 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
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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 · m2
B) 2.0 kg · m2
C) 3.0 kg · m2
D) 4.0 kg · m2
E) 5.0 kg · m2
44. 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) 1.0 × 10–3 kg · m2
B) 4.0 × 10–3 kg · m2
C) 2.0 × 10–3 kg · m2
D) 8.0 × 10–3 kg · m2
E) 2.8 × 10–3 kg · m2
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45. The moment of inertia of a slim rod of mass m and length L about a transverse axis
through one end is mL2/3. 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) mL2/36
B) 7mL2/36
C) mL2/9
D) 2mL2/9
E) 4mL2/9
46. The moment of inertia of a slim rod about a transverse axis through one end is mL2/3,
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 · m2
B) 20 kg · m2
C) 0.020 kg · m2
D) 80 kg · m2
E) 4.5 kg · m2
47.
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) mL2
B) ½mL2
C) 2mL2
D) ¼ mL2
E) 4mL2
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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 ½mR2, the moment of inertia about the axis through P is
A) 0.4mR2
B) ½ mR2
C) 2/3 mR2
D) mR2
E) 1.5mR2
49. 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.
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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
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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 · m2
B) 47.0 kg · m2
C) 99.0 kg · m2
D) 148 kg · m2
E) 211 kg · m2
53. 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.
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54.
A stone of mass 10 kg has a rotational inertia of 2.4 kg · m2 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 · m2
B) 2.0 kg · m2
C) 2.4 kg · m2
D) 2.8 kg · m2
E) 4.4 kg · m2
55. A uniform disk (Io = ½ mR2) 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 mR2
B) mR2
C) 1.5mR2
D) 2mR2
E) 2mR2/3
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.
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57. A cord attached to a 3.63-kg mass is wrapped around a wheel of radius 0.610 m and
released. The moment of inertia of the wheel is 2.71 kg · m2. If the wheel rotates on
frictionless bearings, the acceleration of the falling weight is
A) 3.26 m/s2
B) 1.04 m/s2
C) 2.44 m/s2
D) 1.95 m/s2
E) 4.27 m/s2
58.
Two masses M and m (M > m) are hung over a disk (Idisk = ½ M'R2) 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
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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) 0.010 kg · m2
B) 0.011 kg · m2
C) 0.0125 kg · m2
D) 0.0138 kg · m2
E) 0.0225 kg · m2
60.
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
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61.
The moment of inertia of the wheel in the figure is 0.50 kg · m2, and the bearing is
frictionless. The acceleration of the 15-kg mass is approximately
A) 9.8 m/s2
B) 8.7 m/s2
C) 74 m/s2
D) 16 m/s2
E) 0.53 m/s2
62. 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.
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63.
In the figure, the rotational inertia of the wheel and axle about the center is 12.0 kg · m2,
the constant force F is 39.2 N, and the radius r is 0.800 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 = 0.4MR2) 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.
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66. A cylinder (I = ½mR2) rolls along a level floor with a speed v. The work required to
stop this cylinder is
A) ¼ mv2
B) ½ mv2
C) ¾ mv2
D) mv2
E) 1.25mv2
67. The moment of inertia of a certain cylinder, whose mass is not distributed uniformly, is
0.6mR2 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
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 0.800 m. If the mass m2 = 5.00 kg, the
kinetic energy of the ball is
A) 0.981 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.
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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
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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
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 = ½mR2), which are (1) brass and (2) wood, and two
cylinders (I = mR2), 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
84. The moment of inertia of a certain wheel about its axle is ¾ mR2. 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 mR2) 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/s2
B) 3.9 m/s2
C) 4.9 m/s2
D) 5.8 m/s2
E) 9.8 m/s2
86. A uniform cylinder (I = ½ mR2) 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
87.
A solid disk (Icm = ½ mR2) 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 v2/(g sin θ)
B) ½ v2/(g sin θ)
C) ½ Rv/(g sin θ)
D) ½ mg(sin θ – cos θ)(Rv)2
E) v2/(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 I0ω2.
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
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) 5
90.
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 · m2/s
B) 2.0 kg · m2/s
C) 8.0 kg · m2/s
D) 4.0 kg · m2/s
E) 0.50 kg · m2/s
Page 30
91. A hoop of radius 3.05 n has a mass of 145 kg. Its moment of inertia is mR2. The hoop
rolls without slipping along a horizontal plane. If the center of mass of the hoop has a
speed of 0.305 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
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) 5
94.
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
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
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
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
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) 5
100.
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
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) 5
102. 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
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 2.
104. The angular momentum of a flywheel about its axis is 925 kg · m2/s. If its moment of
inertia about the same axis is 2.50 kg · m2, 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.
106. 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
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 · m2/s
B) 7.5 kg · m2/s
C) 10 kg · m2/s
D) 23 kg · m2/s
E) 45 kg · m2/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
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) 3.8 × 10–3 J
C) 4.8 × 10–5 J
D) 1.9 × 10–3 J
E) 2.4 × 10–2 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
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.
114. 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.
115. 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.
116. 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.
117. 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 · m2. If the original
distance of the weights from the axis is 0.914 m and their final distance is 0.305 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
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) 4L0
119. 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.
120. 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
121. A merry-go-round with a moment of inertia of 6.78 × 103 kg · m2 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 · m2/s
B) 400 kg · m2/s
C) 200 kg · m2/s
D) 20 kg · m2/s
E) 5 kg · m2/s
Page 43
124. Which of the following are required for the total momentum (both angular and linear) of
a system to be conserved?
A)
B)
C)
D)
E)
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.
.
1 and 2
1, 2, and 3
1, 2, and 4
1, 2, 3, and 4
1, 2, 3, 4, and 5
125.
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
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.
127. A wheel of moment of inertia 0.136 kg · m2 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) 0.100 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
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.
130.
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
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.
132.
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
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
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?
τ
1
2
3
4
5
A)
B)
C)
D)
E)
ωp
L
–z
–x
–x
x
x
y
–y
–y
y
y
x
–z
z
z
–z
1
2
3
4
5
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 0.305 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) 0.0125 rad/s
B) 0.0318 rad/s
C) 0.200 rad/s
D) 0.100 rad/s
E) 0.625 rad/s
Page 49
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.
137. 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.
138. 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
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.
140. 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.
141.
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
Answer Key
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
D
B
C
D
E
D
E
D
B
E
C
C
A
D
D
C
C
B
D
C
B
C
B
B
C
A
C
E
D
A
B
A
A
C
E
C
C
D
C
B
D
B
A
B
Page 52
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
C
C
C
E
E
A
A
E
C
B
C
A
A
B
C
C
B
E
C
E
E
C
A
E
B
D
E
E
C
B
C
A
C
A
C
C
D
A
B
E
B
C
A
D
A
A
Page 53
91.
92.
93.
94.
95.
96.
97.
98.
99.
100.
101.
102.
103.
104.
105.
106.
107.
108.
109.
110.
111.
112.
113.
114.
115.
116.
117.
118.
119.
120.
121.
122.
123.
124.
125.
126.
127.
128.
129.
130.
131.
132.
133.
134.
135.
136.
B
A
B
D
A
D
A
E
A
D
E
C
C
E
E
D
E
B
B
A
E
A
D
B
B
B
D
D
E
A
A
A
D
A
D
C
B
C
A
C
A
D
E
E
C
B
Page 54
137.
138.
139.
140.
141.
E
E
E
D
A
Page 55