Question 1

A small uniform disk and a small uniform sphere are released simultaneously at the top of a high inclined plane,and they roll down without slipping.Which one will reach the bottom first?
A)the one of smallest diameter
B)the one of greatest mass
C)the disk
D)the sphere
E)They will reach the bottom at the same time.

Accepted Answer

The sphere will reach the bottom first. This is because as the objects roll down the inclined plane, their translational kinetic energy is converted into rotational kinetic energy. The sphere has less rotational inertia (I) compared to the disk (I = 2/5MR^2 for a sphere and I = 1/2MR^2 for a disk). Therefore, the sphere will be able to convert more of its initial potential energy into rotational kinetic energy, allowing it to roll down the inclined plane faster than the disk.

Question 2

The figure shows scale drawings of four objects,each of the same mass and uniform thickness,with the mass distributed uniformly.Which one has the greatest moment of inertia when rotated about an axis perpendicular to the plane of the drawing at point P? 
A.

B.

C.

D.

A)A
B)B
C)C
D)D
E)The moment of inertia is the same for all of these objects.

Accepted Answer

B

Question 3

Consider a solid uniform sphere of radius R and mass M rolling without slipping.Which form of its kinetic energy is larger,translational or rotational?
A)Translational kinetic energy is larger.
B)Rotational kinetic energy is larger.
C)Both are equal.
D)You need to know the speed of the sphere to tell.

Accepted Answer

A

Question 4

A spinning ice skater is able to control the rate at which she rotates by pulling in her arms.Which of the following statements are true about the skater during this process? (There could be more than one correct choice.)
A)Her angular momentum remains almost constant.
B)Her moment of inertia remains almost constant.
C)Her kinetic energy remains almost constant.
D)She is subject to a constant non-zero torque.

Accepted Answer

When the skater pulls in her arms, her moment of inertia decreases, but her angular momentum remains constant due to the conservation of angular momentum, assuming no external torques act on her.

Question 5

Suppose a uniform solid sphere of mass M and radius R rolls without slipping down an inclined plane starting from rest.The linear velocity of the sphere at the bottom of the incline depends on
A)the mass of the sphere.
B)the radius of the sphere.
C)both the mass and the radius of the sphere.
D)neither the mass nor the radius of the sphere.

Accepted Answer

The linear velocity of a sphere rolling without slipping down an incline depends on the height of the incline and the acceleration due to gravity, not on the mass or radius of the sphere. This is because both the translational and rotational kinetic energies depend on these factors in such a way that they cancel out the mass and radius in the final expression for velocity.

Question 6

Two children,Ahmed and Jacques,ride on a merry-go-round.Ahmed is at a greater distance from the axis of rotation than Jacques.Which of the following are true statements? (There could be more than one correct choice.)
A)Jacques has a greater angular speed than Ahmed.
B)Jacques and Ahmed have the same angular speed.
C)Jacques has a smaller angular speed than Ahmed.
D)Ahmed has a greater tangential speed than Jacques.
E)Jacques and Ahmed have the same tangential speed.

Accepted Answer

B) Since both Ahmed and Jacques are on the same merry-go-round, they share the same angular speed because angular speed is uniform for all points on a rotating body.D) Ahmed, being at a greater distance from the axis of rotation, has a greater tangential speed. Tangential speed increases with distance from the axis of rotation.

Question 7

When is the angular momentum of a system constant?
A)Only when its total kinetic energy is constant.
B)Only when the linear momentum and the energy are constant.
C)Only when no net external torque is exerted on the system.
D)Only when the moment of inertia is constant.

Accepted Answer

Angular momentum of a system is conserved only when no net external torque is exerted on the system. This is known as the principle of conservation of angular momentum.

Question 8

A car is traveling along a freeway at 65 mph.What is the linear speed,relative to the highway,of each of the following points on one of its tires?
(a)the highest point on the tire
(b)the lowest point on a tire
(c)the center of the tire

Accepted Answer

The answer of A car is traveling along a freeway...

Question 9

A disk and a hoop of the same mass and radius are released at the same time at the top of an inclined plane.If both are uniform,which one reaches the bottom of the incline first if there is no slipping?
A)The hoop
B)The disk
C)Both reach the bottom at the same time.

Accepted Answer

The disk reaches the bottom first because it has a smaller moment of inertia compared to the hoop, allowing it to accelerate faster down the incline under the same conditions.

Question 10

Consider a uniform hoop of radius R and mass M rolling without slipping.Which is larger,its translational kinetic energy or its rotational kinetic energy?
A)Translational kinetic energy is larger.
B)Rotational kinetic energy is larger.
C)Both are equal.
D)You need to know the speed of the hoop to tell.

Accepted Answer

For a uniform hoop rolling without slipping, the translational kinetic energy (Kt) and rotational kinetic energy (Kr) are related by the equation Kt = 1/2 Mv^2 and Kr = 1/2 Iω^2, where v is the velocity of the center of mass and ω is the angular velocity, and I is the moment of inertia of the hoop about its center of mass. For a uniform hoop, I = MR^2/2, where R is the radius of the hoop and M is the mass of the hoop. Since the hoop is rolling without slipping, the velocity of the center of mass is related to the angular velocity by the equation v = ωR. Substituting these equations into the expressions for Kt and Kr, we get Kt = Kr. Therefore, the translational and rotational kinetic energies are equal for a uniform hoop rolling without slipping.

Question 11

A disk,a hoop,and a solid sphere are released at the same time at the top of an inclined plane.They are all uniform and roll without slipping.In what order do they reach the bottom?
A)disk, hoop, sphere
B)hoop, sphere, disk
C)sphere, disk, hoop
D)sphere, hoop, disk
E)hoop, disk, sphere

Accepted Answer

The moment of inertia is lowest for the sphere, higher for the disk, and highest for the hoop. Lower moment of inertia means more of the potential energy is converted into translational kinetic energy rather than rotational, allowing the sphere to reach the bottom first, followed by the disk, then the hoop.

Question 12

Suppose a solid uniform sphere of mass M and radius R rolls without slipping down an inclined plane starting from rest.The angular velocity of the sphere at the bottom of the incline depends on
A)the mass of the sphere.
B)the radius of the sphere.
C)both the mass and the radius of the sphere.
D)neither the mass nor the radius of the sphere.

Accepted Answer

The angular velocity of the sphere at the bottom of the incline only depends on the geometry of the sphere, specifically its radius R, and not on its mass M. This is because the potential energy of the sphere is converted into kinetic energy as it rolls down the incline, and this conversion depends on the distribution of mass (i.e. moment of inertia) which is a function of the sphere's radius. Therefore, the angular velocity of the sphere at the bottom of the incline is determined solely by its radius, and not affected by the mass.

Question 13

A uniform ball is released from rest on a no-slip surface,as shown in the figure.After reaching its lowest point,the ball begins to rise again,this time on a frictionless surface.When the ball reaches its maximum height on the frictionless surface,it is 
A)higher than when it was released.
B)lower than when it was released.
C)at the same height from which it was released.
D)It is impossible to tell without knowing the mass of the ball.
E)It is impossible to tell without knowing the radius of the ball.

Accepted Answer

The answer of A uniform ball is released from rest...

Question 14

Two uniform solid balls,one of radius R and mass M,the other of radius 2R and mass 8M,roll down a high incline.They start together from rest at the top of the incline.Which one will reach the bottom of the incline first?
A)The small sphere arrives first.
B)Both reach the bottom at the same time.
C)The large sphere arrives first.

Accepted Answer

Both spheres will reach the bottom at the same time because the acceleration of a rolling object down an incline does not depend on its mass or size but only on the gravitational acceleration and the incline angle. Since both spheres are solid and uniform, they have the same distribution of mass relative to their axes of rotation, leading to the same rolling motion characteristics.

Question 15

A person sits on a freely spinning lab stool that has no friction in its axle.When this person extends her arms,
A)her moment of inertia decreases and her angular speed increases.
B)her moment of inertia decreases and her angular speed decreases.
C)her moment of inertia increases and her angular speed increases.
D)her moment of inertia increases and her angular speed decreases.
E)her moment of inertia increases and her angular speed remains the same.

Accepted Answer

When the person extends her arms, her moment of inertia increases because the mass is distributed further from the axis of rotation. According to the conservation of angular momentum, as the moment of inertia increases, the angular speed decreases. Therefore, the correct choice is D.

Question 16

When a rigid object rotates about a fixed axis,what is true about all the points in the object? (There could be more than one correct choice.)
A)They all have the same angular speed.
B)They all have the same tangential speed.
C)They all have the same angular acceleration.
D)They all have the same tangential acceleration.
E)They all have the same radial acceleration.

Accepted Answer

A rigid object rotates about a fixed axis with a constant angular velocity. This means that all points on the rigid object travel along a circular arc with the same radius, which means they have the same angular velocity. Also, since the angular velocity is not changing, the angular acceleration is zero, which means all points have the same angular acceleration. However, as the object rotates, the distance traveled by each point from the axis of rotation can change, which means the tangential speed and tangential acceleration can be different for different points. The radial acceleration is also not constant for all points since it depends on the distance from the axis of rotation. Therefore, choices B, D, and E are incorrect.

Question 17

A solid sphere,solid cylinder,and a hollow pipe all have equal masses and radii.If the three of them are released simultaneously at the top of an inclined plane and do not slip,which one will reach the bottom first?
A)sphere
B)pipe
C)cylinder
D)The pipe and cylinder arrive together before the sphere.
E)They all reach the bottom at the same time.

Accepted Answer

The solid sphere reaches the bottom first because it has the smallest moment of inertia among the three, which means it accelerates faster down the incline under the same conditions.

Question 18

A merry-go-round spins freely when Diego moves quickly to the center along a radius of the merry-go-round.As he does this,it is true to say that
A)the moment of inertia of the system decreases and the angular speed increases.
B)the moment of inertia of the system decreases and the angular speed decreases.
C)the moment of inertia of the system decreases and the angular speed remains the same.
D)the moment of inertia of the system increases and the angular speed increases.
E)the moment of inertia of the system increases and the angular speed decreases.

Accepted Answer

When Diego moves towards the center, the distribution of mass relative to the axis of rotation becomes more compact, decreasing the moment of inertia. According to the conservation of angular momentum, if the moment of inertia decreases, the angular speed must increase to keep the angular momentum constant.

Question 19

A solid sphere and a solid cylinder,both uniform and of the same mass and radius,roll without slipping at the same forward speed.It is correct to say that the total kinetic energy of the solid sphere is
A)more than the total kinetic energy of the cylinder.
B)less than the total kinetic energy of the cylinder.
C)equal to the total kinetic energy of the cylinder.

Accepted Answer

The answer of A solid sphere and a solid cylinder,both...

Question 20

The rotating systems shown in the figure differ only in that the two identical movable blocks are positioned a distance r from the axis of rotation (left),or a distance r/2 from the axis of rotation (right).If you release the hanging blocks simultaneously from rest, 
A)the block at the left lands first.
B)the block at the right lands first.
C)both blocks land at the same time.

Accepted Answer

The time period of oscillation for a pendulum is given by T = 2π√(l/g), where l is the length of the pendulum and g is the acceleration due to gravity. In the left case, the length of the pendulum is r, while in the right case, it is r/2. Since the length of the pendulum affects the time period, the right block will oscillate with a shorter time period than the left block. This means that the right block will reach the bottom of its swing before the left block, so answer choice B is correct.

Quiz 9: Rotational Motion

A small uniform disk and a small uniform sphere are released simultaneously at the top of a high inclined plane,and they roll down without slipping.Which one will reach the bottom first?

The figure shows scale drawings of four objects,each of the same mass and uniform thickness,with the mass distributed uniformly.Which one has the greatest moment of inertia when rotated about an axis perpendicular to the plane of the drawing at point P? A. B. C. D.

Consider a solid uniform sphere of radius R and mass M rolling without slipping.Which form of its kinetic energy is larger,translational or rotational?

A spinning ice skater is able to control the rate at which she rotates by pulling in her arms.Which of the following statements are true about the skater during this process? (There could be more than one correct choice.)

Suppose a uniform solid sphere of mass M and radius R rolls without slipping down an inclined plane starting from rest.The linear velocity of the sphere at the bottom of the incline depends on

Two children,Ahmed and Jacques,ride on a merry-go-round.Ahmed is at a greater distance from the axis of rotation than Jacques.Which of the following are true statements? (There could be more than one correct choice.)

When is the angular momentum of a system constant?

A car is traveling along a freeway at 65 mph.What is the linear speed,relative to the highway,of each of the following points on one of its tires? (a)the highest point on the tire (b)the lowest point on a tire (c)the center of the tire

A disk and a hoop of the same mass and radius are released at the same time at the top of an inclined plane.If both are uniform,which one reaches the bottom of the incline first if there is no slipping?

Consider a uniform hoop of radius R and mass M rolling without slipping.Which is larger,its translational kinetic energy or its rotational kinetic energy?

A disk,a hoop,and a solid sphere are released at the same time at the top of an inclined plane.They are all uniform and roll without slipping.In what order do they reach the bottom?

Suppose a solid uniform sphere of mass M and radius R rolls without slipping down an inclined plane starting from rest.The angular velocity of the sphere at the bottom of the incline depends on

A uniform ball is released from rest on a no-slip surface,as shown in the figure.After reaching its lowest point,the ball begins to rise again,this time on a frictionless surface.When the ball reaches its maximum height on the frictionless surface,it is

Two uniform solid balls,one of radius R and mass M,the other of radius 2R and mass 8M,roll down a high incline.They start together from rest at the top of the incline.Which one will reach the bottom of the incline first?

A person sits on a freely spinning lab stool that has no friction in its axle.When this person extends her arms,

When a rigid object rotates about a fixed axis,what is true about all the points in the object? (There could be more than one correct choice.)

A solid sphere,solid cylinder,and a hollow pipe all have equal masses and radii.If the three of them are released simultaneously at the top of an inclined plane and do not slip,which one will reach the bottom first?

A merry-go-round spins freely when Diego moves quickly to the center along a radius of the merry-go-round.As he does this,it is true to say that

A solid sphere and a solid cylinder,both uniform and of the same mass and radius,roll without slipping at the same forward speed.It is correct to say that the total kinetic energy of the solid sphere is

The rotating systems shown in the figure differ only in that the two identical movable blocks are positioned a distance r from the axis of rotation (left) ,or a distance r/2 from the axis of rotation (right) .If you release the hanging blocks simultaneously from rest,