Question 1

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

Accepted Answer

The tangential speed of a point on the rim with respect to the axle is equal to the speed of the axle with respect to the ground, which is 30 m/s, because the wheel is rolling without slipping.

Question 2

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

Accepted Answer

For a rolling object, the rotational kinetic energy is (1/2)Iω² and the translational kinetic energy is (1/2)mv². For a solid cylinder, I = (1/2)mr² and v = rω, so the ratio of rotational to translational kinetic energy is (1/2)(1/2)mr²ω² / (1/2)mv² = 1/2.

Question 3

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

Accepted Answer

The point on the rim directly above the axle moves at twice the speed of the axle, so its speed is 60 m/s.

Question 4

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

Accepted Answer

The rotational kinetic energy is associated with the wheels, while the translational kinetic energy is associated with the entire bicycle. The moment of inertia (I) for a uniform disk is given by $I = \frac{1}{2}MR^2$, where M is the mass of the disk and R is its radius. Since there are two wheels, the total moment of inertia for both wheels is $I_{total} = 2 \times \frac{1}{2}M_{wheel}R^2 = M_{wheel}R^2$. The rotational kinetic energy (K.E._rot) is given by $\frac{1}{2}I\omega^2$, and the translational kinetic energy (K.E._trans) for the entire bicycle (including the wheels) moving at a velocity v is $\frac{1}{2}M_{total}v^2$, where $M_{total}$ is the total mass of the bicycle and the wheels. The angular velocity $\omega$ and the linear velocity $v$ are related by $\omega = \frac{v}{R}$. Therefore, the ratio of rotational to total kinetic energy is:$$\frac{K.E._rot}{K.E._total} = \frac{\frac{1}{2}M_{wheel}R^2\left(\frac{v}{R}\right)^2}{\frac{1}{2}M_{total}v^2 + \frac{1}{2}M_{wheel}R^2\left(\frac{v}{R}\right)^2} = \frac{M_{wheel}R^2v^2}{R^2(M_{total}v^2 + M_{wheel}v^2)} = \frac{M_{wheel}}{M_{total} + M_{wheel}}$$Given that $M_{wheel} = 5.0 \, \text{kg}$ (for both wheels together) and $M_{total} = 25 \, \text{kg}$, the ratio becomes:$$\frac{5}{25 + 5} = \frac{5}{30} = \frac{1}{6}$$Converting this fraction to a percentage gives approximately 16.67%, which rounds to 17%, making option B the correct answer.

Question 5

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

Accepted Answer

The frictional force is tangent to the surface and points in the same direction as the velocity of the center of mass because it is the force that prevents the wheel from slipping.

Question 6

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

Accepted Answer

For a wheel rolling without slipping, the frictional force at the point of contact with the surface acts in the direction opposite to the direction of the velocity of the center of mass. This frictional force is necessary to prevent slipping and is tangent to the surface.

Question 7

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

Accepted Answer

The solid sphere has the shortest travel time because it has the smallest moment of inertia relative to its mass, which means it accelerates faster down the incline under the same conditions compared to the other shapes.

Question 8

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

Accepted Answer

The total kinetic energy of the bowling ball consists of both translational kinetic energy and rotational kinetic energy. The translational kinetic energy ($KE_{trans}$) is given by $\frac{1}{2}mv^2$, where $m$ is the mass and $v$ is the velocity. The rotational kinetic energy ($KE_{rot}$) is given by $\frac{1}{2}I\omega^2$, where $I$ is the moment of inertia and $\omega$ is the angular velocity. For a solid sphere, $I = \frac{2}{5}mr^2$, and $\omega = \frac{v}{r}$, where $r$ is the radius.First, calculate the translational kinetic energy:$$KE_{trans} = \frac{1}{2}mv^2 = \frac{1}{2} \times 6.0 \, \text{kg} \times (4.0 \, \text{m/s})^2 = 48 \, \text{J}$$Then, calculate the rotational kinetic energy:$$KE_{rot} = \frac{1}{2}I\omega^2 = \frac{1}{2} \times \frac{2}{5}mr^2 \times \left(\frac{v}{r}\right)^2 = \frac{1}{2} \times \frac{2}{5} \times 6.0 \, \text{kg} \times (0.09 \, \text{m})^2 \times \left(\frac{4.0 \, \text{m/s}}{0.09 \, \text{m}}\right)^2 = 19.2 \, \text{J}$$Finally, add the translational and rotational kinetic energies to get the total kinetic energy:$$KE_{total} = KE_{trans} + KE_{rot} = 48 \, \text{J} + 19.2 \, \text{J} = 67.2 \, \text{J}$$Rounding to the nearest whole number gives 67 J.

Question 9

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

Accepted Answer

The kinetic energy of the hollow sphere is smaller than that of the solid sphere because the hollow sphere has a larger moment of inertia than the solid sphere, meaning it requires more energy to rotate. This results in less translational kinetic energy at the bottom of the incline compared to the solid sphere.

Question 10

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

Accepted Answer

The answer of A hollow sphere and a solid sphere...

Question 11

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

Accepted Answer

The moment of inertia of a hollow sphere is larger than that of a solid sphere, which means it will accelerate more slowly down the incline due to its distribution of mass. This results in a larger travel time for the hollow sphere compared to the solid sphere when rolling down an inclined plane without slipping.

Question 12

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

Accepted Answer

When a hoop rolls without slipping, the point on the rim in contact with the ground is momentarily at rest, and the topmost point moves at twice the speed of the axle. The points with the same speed as the axle are symmetrically located on either side of the vertical diameter, separated by 120 degrees.

Question 13

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

Accepted Answer

When the boy walks tangentially in a clockwise direction, by the principle of conservation of angular momentum, the platform will start rotating in the opposite direction, which is counterclockwise, to conserve the total angular momentum of the system.

Question 14

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

Accepted Answer

The platform's angular velocity decreases due to the conservation of angular momentum. As the boy moves radially outward, his moment of inertia with respect to the rotation axis increases, leading to a decrease in angular velocity to conserve angular momentum.

Question 15

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

Accepted Answer

The angular momentum of the Earth-meteor system is conserved, so the change in the Earth's angular momentum is equal to the angular momentum of the meteor. The angular momentum of a point mass is $L = mvr$, where $m$ is the mass, $v$ is the speed, and $r$ is the distance from the axis of rotation. In this case, the distance from the axis of rotation is the radius of the Earth, $r = 6.37 \times 10^6 \mathrm{~m}$. The speed of the meteor is $v = 80,000 \mathrm{~km/h} = 22.2 \mathrm{~km/s}$. The change in the Earth's angular momentum is $\Delta L = 80.0 \times 10^{12} \mathrm{~kg} \cdot \mathrm{m}^2 / \mathrm{s}$. So, the mass of the meteor is:$$m = \frac{\Delta L}{vr} = \frac{80.0 \times 10^{12} \mathrm{~kg} \cdot \mathrm{m}^2 / \mathrm{s}}{(22.2 \mathrm{~km/s})(6.37 \times 10^6 \mathrm{~m})} = 565 \mathrm{~kg}$$

Question 16

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

Accepted Answer

The answer of The work done in rotating a rigid...

Question 17

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

Accepted Answer

The translational kinetic energy (TKE) is (1/2)mv^2 and the rotational kinetic energy (RKE) for a solid sphere is (1/2)(2/5)mv^2 = (1/5)mv^2. The ratio of TKE to RKE is (1/2)mv^2 / (1/5)mv^2 = 5/2.

Question 18

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

Accepted Answer

The answer of A pencil of mass M and length...

Question 19

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

Accepted Answer

The speed of the fly as it hits the desk can be found using energy conservation. Initially, the potential energy is $mgh$ where $h = L/2$, and just before hitting the desk, all this potential energy converts into kinetic energy, $1/2 mv^2$. Solving for $v$ gives $v = \sqrt{2gL}$.

Question 20

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

Accepted Answer

The angular momentum of the skater remains constant when she extends her arms away from her body if frictional forces are neglected, due to the conservation of angular momentum.

Quiz 13: Dynamics of a Rigid Body

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 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 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 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 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 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 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 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 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 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 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 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 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 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

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

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 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

Quiz 13: Dynamics of a Rigid Body

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 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 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 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 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 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 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 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 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 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 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 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 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 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

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

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 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 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