Deck 13: Dynamics of a Rigid Body

Question 1

All of the following quantities have the same dimensions except

A) kinetic energy.
B) torque.
C) work.
D) potential energy.
E) angular momentum.

Answer 1

angular momentum.

Question 2

All of the following angular quantities are vectors except

A) angular displacement.
B) angular velocity.
C) angular acceleration.
D) angular momentum.
E) torque.

Answer 2

angular displacement.

Question 3

The Earth moves about the Sun in an elliptical orbit. As the Earth moves closer to the Sun, the angular speed of the Earth about the Sun

A) increases.
B) stays the same.
C) decreases.
D) is zero.

Answer 3

increases.

Question 4

A baseball pitcher rotates his arm around his shoulder joint at 80.0 rad/s. Assuming the length of his arm is 0.60 m, the speed of the ball when it is still in his hand is

A) 302 m/s.
B) 133 m/s.
C) 48.0 m/s.
D) 21.2 m/s.
E) 7.64 m/s.

Question 5

Some old-fashioned bicycles had a front wheel with a large diameter compared to the diameter of the back wheel. When rolling without slipping on a horizontal surface, the two wheels were subject to the same

A) linear velocity at the rim.
B) angular velocity.
C) angular acceleration.
D) tangential acceleration at the rim.
E) centripetal acceleration at the rim.

Question 6

A centrifuge in a medical laboratory rotates at a rotational speed of 3600 rev/min. When turned off, it rotates 20.0 times at a constant angular acceleration before coming to rest. The angular acceleration of the centrifuge is

A) 1.50 rad/s².
B) 9.42 rad/s².
C) 18.8 rad/s².
D) 180 rad/s².
E) 565 rad/s².

Question 7

A centrifuge in a medical laboratory rotates at a rotational speed of 3600 rev/min. When turned off, it rotates 20.0 times at a constant angular acceleration before coming to rest. The angle through which the centrifuge rotates before coming to rest is

A) 3.18 rad.
B) 20.0 rad.
C) 80.0 rad.
D) 126 rad.
E) 180 rad.

Question 8

A centrifuge in a medical laboratory rotates at a rotational speed of 3600 rev/min. When turned off, it rotates 20.0 times at a constant angular acceleration before coming to rest. The time necessary for the centrifuge to come to rest is

A) 0.66 s.
B) 6.6 s.
C) 66 s.
D) 2.5 * 10² s.
E) 5.7 * 10² s.

Question 9

A wheel rotating at 7.5 rad/s slows down at 0.50 rad/s². The number of revolutions made by the wheel in the first 10 s is

A) 50 revolutions.
B) 38 revolutions.
C) 20 revolutions.
D) 7.9 revolutions.
E) 0.38 revolutions.

Question 10

You are trying to open a door that is stuck by pulling on the doorknob in a direction perpendicular to the door. If you instead tie a rope to the doorknob and then pull with the same force, the torque you exert on the door

A) increases.
B) stays the same.
C) decreases.
D) is zero.

Question 11

Consider an Atwood machine with m₂ > m₁ and a pulley of mass M. The tension in the string holding mass m₂ is

A) larger than the tension in the string holding mass m₁.
B) smaller than the tension in the string holding mass m₁.
C) equal to the tension in the string holding mass m₁.
D) unknown; more information is needed to work out the answer.

Question 12

Consider an Atwood machine with m₂ > m₁ and a pulley of mass M. The acceleration of the system, compared to the case when the pulley's mass is negligible, is

A) larger than the acceleration of the same system when the pulley's mass is negligible.
B) smaller than the acceleration of the same system when the pulley's mass is negligible.
C) equal to the acceleration of the same system when the pulley's mass is negligible.
D) unknown; more information is needed to work out the answer.

Question 13

A clamp is fastened at the 70-cm mark on a meterstick. Of the two arrangements pictured here, it is easier to balance the system vertically on your hand in

A) the arrangement with the clamp up.
B) the arrangement with the clamp down.
C) neither; both arrangements are equally difficult to balance.

Question 14

A 10-N force is applied tangent to a wheel of radius R, while a 15-N force is applied tangent to a wheel of radius R/3 mounted on top of the previous wheel, as pictured here. The torque acting on the system is

A) 5R in the negative x direction.
B) 5R in the positive x direction.
C) 5R in the negative y direction.
D) 5R in the positive y direction.
E) 5R in the xz plane.

Question 15

From a torque applied versus angular acceleration graph for the rotation of a rigid body about a fixed axis, the body's moment of inertia can be calculated as

A) the value of the torque divided by the value of the angular acceleration at a certain instant.
B) the value of the torque multiplied by the value of the angular acceleration at a certain instant.
C) the area bounded by the curve and the torque axis.
D) the area bounded by the curve and the angular acceleration axis.
E) the slope of the curve.

Question 16

A wheel of moment of inertia of 300 kg.m² initially at rest is accelerated by a constant torque of 5.00 N.m over 9.00 s. The wheel's rotational kinetic energy at the end of the 9.00 s is

A) 0.150 J.
B) 0.296 J.
C) 1.00 J.
D) 3.38 J.
E) 6.67 J.

Question 17

A solid cylinder of mass 10 kg and radius 1.0 m rotates about its own symmetry axis. A rope wrapped around it exerts a 5.0-N force tangent to the cylinder, causing the cylinder to rotate clockwise. A second rope wrapped around another section of the same cylinder exerts a 6.0-N force that is tangent to the cylinder, causing the cylinder to rotate counterclockwise. The magnitude of the cylinder's angular acceleration is

A) 0.20 rad/s².
B) 0.91 rad/s².
C) 1.1 rad/s².
D) 10 rad/s².
E) 11 rad/s².

Question 18

A disk of moment of inertia 80 kg.m² is free to rotate about its axis (through the center, perpendicular to the plane of the disk, which is set vertical). To increase its angular velocity uniformly from rest to 5.0 rad/s in 20.0 s, the work done by the applied torque is

A) 1.0 * 10³ J.
B) 4.0 F*10² J.
C) 2.0 * 10² J.
D) 3.2 J.
E) 1.6 J.

Question 19

A disk of moment of inertia 80 kg.m² is free to rotate about its axis (through the center, perpendicular to the plane of the disk, which is set vertical). To increase its angular velocity uniformly from rest to 5.0 rad/s in 20.0 s, the torque applied is

A) 8.0 * 10³ N.m.
B) 3.2 *10² N.m.
C) 20 N.m.
D) 32 N.m.
E) 0.80 N.m.

Question 20

A baseball player increases the speed of a bat from rest to 20.0 rad/s in 0.25 s at a constant acceleration. Assuming the bat is a 2.0-kg rod of uniform mass distribution and of length 1.0 m, the torque applied by the player to one end of the bat is

A) 10 N.m.
B) 13 N.m.
C) 40 N.m.
D) 53 N.m.
E) 1.6*10² N.m.

Question 21

A wheel is rolling (without slipping) on the ground. Its axle travels at 30 m/s with respect to the ground. The tangential speed of a point on the rim with respect to the axle is

A) 15 m/s.
B) 30 m/s.
C) 45 m/s.
D) 60 m/s.
E) 90 m/s.

Question 22

A solid cylinder (the moment of inertia about its axis is mr²/2) rolls without slipping. The ratio of its rotational to translational kinetic energy is

A) 1/4.
B) 1/2.
C) 1.
D) 2.
E) 4.

Question 23

A wheel is rolling (without slipping) on the ground. Its axle travels at 30 m/s with respect to the ground. The instantaneous speed (with respect to the ground) of a point on the rim directly above the axle is

A) 15 m/s.
B) 30 m/s.
C) 45 m/s.
D) 60 m/s.
E) 90 m/s.

Question 24

A bicycle of mass 25 kg has wheels of mass 5.0 kg and radii of 30 cm each (assume the wheels are uniform disks). The part of the kinetic energy that is rotational is

A) 10%.
B) 17%.
C) 20%.
D) 34%.
E) 40%.

Question 25

A wheel of radius R is rolling without slipping on a horizontal surface. The frictional force exerted by the surface on the wheel at the point where the rim is in contact with the surface is

A) tangent to the surface, pointing in the same direction as the velocity of the center of mass.
B) tangent to the surface, pointing in the direction opposite that of the velocity of the center of mass.
C) zero.
D) normal to the surface, pointing toward the wheel's center of mass.
E) normal to the surface, pointing away from the wheel's center of mass.

Question 26

A wheel of radius R is rolling without slipping down an inclined surface. The frictional force exerted by the surface on the wheel at the point where the rim is in contact with the surface is

A) tangent to the surface, pointing in the same direction as the velocity of the center of mass.
B) tangent to the surface, pointing in the direction opposite that of the velocity of the center of mass.
C) zero.
D) normal to the surface, pointing toward the wheel's center of mass.
E) normal to the surface, pointing away from the wheel's center of mass.

Question 27

A hollow sphere, a solid sphere, a hollow right cylinder (or hoop), and a solid right cylinder having the same total mass and identical maximum radius are rolled (without slipping) the same distance down an inclined plane. The object with the shortest travel time is the

A) hollow sphere.
B) solid sphere.
C) hollow right cylinder (or hoop).
D) solid right cylinder.

Question 28

A bowling ball of mass 6.0 kg and radius 9.0 cm rolls without slipping down a lane at 4.0 m/s. The total kinetic energy of the bowling ball is

A) 19 J.
B) 24 J.
C) 48 J.
D) 67 J.
E) 96 J.

Question 29

A hollow sphere and a solid sphere each made from the same material and having equal maximum radius are rolled from rest (without slipping) the same distance down an inclined plane. At the bottom of the incline, the kinetic energy of the hollow sphere is

A) smaller than the kinetic energy of the solid sphere.
B) larger than the kinetic energy of the solid sphere.
C) the same as the kinetic energy of the solid sphere.
D) unknown; more information is needed to work out the answer.

Question 30

A hollow sphere and a solid sphere each made from the same material and having equal maximum radius are rolled from rest (without slipping) the same distance down an inclined plane. The torque experienced by the hollow sphere is

A) smaller than the torque experienced by the solid sphere.
B) larger than the torque experienced by the solid sphere.
C) the same as the torque experienced by the solid sphere.
D) unknown; more information is needed to work out the answer.

Question 31

A hollow sphere and a solid sphere each made from the same material and having equal maximum radius are rolled from rest (without slipping) the same distance down an inclined plane. The travel time of the hollow sphere is

A) smaller than the travel time of the solid sphere.
B) larger than the travel time of the solid sphere.
C) the same as the travel time of the solid sphere.
D) unknown; more information is needed to work out the answer.

Question 32

A hoop is rolling (without slipping) on the ground. Its axle travels at 20 m/s with respect to the ground. There are two angular positions on the rim where the instantaneous speed (with respect to the ground) is 20 m/s. The two angular positions are separated by an angle of

A) 30^o.
B) 60^o.
C) 90^o.
D) 120^o.
E) 180^o.

Question 33

A boy stands on a merry-go-round platform at rest. When the boy starts walking tangentially along the edge of the platform in a clockwise direction, the platform will

A) start rotating in a clockwise direction.
B) stay at rest until an external force acts on it.
C) stay at rest until an external torque acts on it.
D) start rotating in a counterclockwise direction.

Question 34

A boy stands at the center of a merry-go-round platform that rotates at constant angular velocity. When the boy starts walking radially toward the edge, the platform's angular velocity

A) increases.
B) does not change.
C) decreases.
D) becomes zero.

Question 35

Large meteors can strike the Earth with speeds of $80,000 \mathrm {~km} / \mathrm { h }$ . Such a meteor hits the Earth (radius $6.37 \times 10 ^ { 6 } \mathrm {~m}$ ) at a point on the plane of the Earth's equator at $80,000 \mathrm {~km} / \mathrm { h }$ , resulting in a change of the Earth's angular momentum of $80.0 \times 10 ^ { 12 } \mathrm {~kg} \cdot \mathrm { m } ^ { 2 } / \mathrm { s }$ . The meteor's mass is

A) 13.0 kg.
B) 56.0 kg.
C) 126 kg.
D) 565 kg.
E) 637 kg.

Question 36

The work done in rotating a rigid body can be expressed as

A) the scalar product between the torque applied and the angular displacement.
B) the vector product between the torque applied and the angular displacement.
C) the product between the magnitude of the torque applied and the angular displacement.
D) the scalar product between the force applied and the lever arm.
E) the vector product between the force applied and the lever arm.

Question 37

A solid sphere of radius R and mass m starts from rest at a height h and rolls down an incline without slipping. The ratio of translational to rotational kinetic energy at the bottom of the incline is

A) R/h.
B) 2R/5h.
C) 5R/2h.
D) 2/5.
E) 5/2.

Question 38

A pencil of mass M and length L has a moment of inertia about its center of mass of (1/12) M L² (assume uniform mass distribution). It balances vertically on the eraser in a state of unstable equilibrium. A fly of mass m is perched on the other end. Assuming the fly hangs on and the eraser does not slip, the kinetic energy of the system just before it hits the desk is

A) (M + m)gL.
B) 1/2(2M + m)gL.
C) 1/2(M + m)gL.
D) 1/2(M + 2m)gL.
E) 2(M + m)gL.

Question 39

A pencil of mass M and length L has a moment of inertia about its center of mass of (1/12) M L² (assume uniform mass distribution). It balances vertically on the eraser in a state of unstable equilibrium, with a fly of mass m perched on the other end. Assuming the fly hangs on and the eraser does not slip, when M approaches zero, the speed of the fly as it hits the desk is

A) v = $\sqrt { g L }$ /2.
B) v = $\sqrt { g L }$
C) v = $\sqrt { 2 g L }$
D) v = 2 $\sqrt { g L }$
E) v = 2 $\sqrt { 2 g L }$

Question 40

A skater is spinning with folded arms, and later she extends her arms away from her body. If frictional forces are neglected, the quantity that does not change during this event is

A) the angular momentum.
B) the linear momentum.
C) the kinetic energy.
D) the linear velocity.
E) the angular velocity.

Question 41

A potter's wheel (assuming a uniform disk of mass 5.0 kg and diameter 0.50 m) rotates around a vertical axis through its center at a frequency of 10 rad/s. The potter throws a 4.0-kg chunk of clay, shaped as a flat disk of radius 6.0 cm, onto the center of the rotating wheel. The frequency of rotation of the wheel after the clay sticks to it is

A) 8.4 rad/s.
B) 9.6 rad/s.
C) 10 rad/s.
D) 12 rad/s.
E) 19 rad/s.

Question 42

The protons and neutrons making up a nucleus are suddenly rearranging themselves so that the nucleus' moment of inertia decreases by a factor of 4. The change in the rotational kinetic energy during this process is

A) zero.
B) 2 times the initial kinetic energy of rotation.
C) 3 times the initial kinetic energy of rotation.
D) 4 times the initial kinetic energy of rotation.
E) 16 times the initial kinetic energy of rotation.

Question 43

A diver can reduce her moment of inertia by a factor of about 3.14 when changing from the straight position to the tuck position. If she makes 3.0 rotations in 2.0 s when in the tuck position, her angular speed when in the straight position is

A) 0.48 rad/s.
B) 0.52 rad/s.
C) 3.0 rad/s.
D) 4.7 rad/s.
E) 30 rad/s.

Question 44

The mean orbital speed of Mars is 0.80 times that of the Earth in their orbits around the Sun. Because the mean orbital radius of Mars is 1.52 times that of the Earth, and the mass of Mars is 0.11 times that of the Earth, the ratio of the angular momentum of Mars to that of the Earth in their orbits around the Sun is

A) 99.
B) 26.
C) 0.26.
D) 0.21.
E) 0.13.

Question 45

You are sitting on a stool that can rotate friction-free about its vertical axis, holding a mass in each outstretched hand. When you suddenly release and drop the masses, your angular velocity

A) increases.
B) stays the same.
C) decreases.
D) becomes zero.

Question 46

You are sitting on a stool that can rotate friction-free about its vertical axis, holding a spinning bicycle wheel that rotates in a horizontal plane. When you flip the bicycle wheel (turn it over through 180^o), you will

A) rotate in the same direction the bicycle wheel was originally rotating.
B) stay still.
C) rotate in the direction opposite to that of the bicycle wheel's original rotation.

Question 47

A uniform thin rod of length 0.40 m and mass 4.0 kg can rotate in the horizontal plane about a vertical axis through one end. The rod is at rest when a 3.0-g bullet traveling in the horizontal plane of the rod is fired into the free end of the rod. As viewed from above, the direction of the bullet's velocity makes an angle of 60^o with the rod. The bullet lodges in the rod, and the angular velocity of the rod is 10 rad/s immediately after collision. The magnitude of the bullet's velocity just before impact was

A) 2.1 $\times$ 10² m/s.
B) 1.8 $\times$ 10³ m/s.
C) 3.6 $\times$ 10³ m/s.
D) 1.8 $\times$ 10⁴ m/s.
E) 2.1 $\times$ 10⁴ m/s.

Question 48

A top is spinning rapidly on a table, its axis inclined to the vertical. As the spin of the top gradually decreases, the precession frequency of the top

A) increases.
B) decreases.
C) remains unchanged.
D) is unknown; more information is needed to work out the answer.

Question 49

A top is spinning rapidly on a table, its axis inclined to the vertical. If only the angle of inclination is changed, each of the following quantities changes except the

A) torque.
B) precession frequency.
C) angular momentum.
D) precessional angular velocity.

Question 50

A girl carries a box that contains a gyroscope (a flywheel suspended in gimbals) that spins around the east-west axis. The girl is walking in the same direction as the rotational axis of the gyroscope, then suddenly turns to the left. The front part of the box

A) moves upward.
B) moves downward.
C) does not change its position.
D) rotates north.
E) rotates south.

Question 51

A thin, 1.00-m-long rod of mass 1.00 kg has a solid sphere of radius 0.25 m and mass 0.50 kg attached to one end. The arrangement is originally vertical and stationary, with the sphere at the top. The system is free to pivot about the bottom end of the rod after being given a slight nudge. The speed of the sphere after the system rotates through 90^o is

A) 1.10 m/s.
B) 1.22 m/s.
C) 1.80 m/s.
D) 3.14 m/s.
E) 5.53 m/s.