Q: A CD spins up from rest with a constant angular acceleration $$\alpha. $$After time $$t$$, what is its angular displacement $$\theta$$?

$$\theta = \$$omega_0$$ t + \frac{1}{2}\alpha t^2$

Question 1

A solid disk and a thin ring both have the same mass $$M$$ and radius $$R. $$Which has the greater moment of inertia about an axis through their centers perpendicular to the plane?

Accepted Answer

The thin ring, because all its mass is at distance $$R$$ from the axis.

Question 2

A ceiling fan accelerates from rest to an angular velocity of $$6\pi$$ rad/s in 3 seconds with constant angular acceleration. What is the angular acceleration?

Accepted Answer

$$2\pi$$ rad/s$$^2$$

Question 3

Which equation correctly relates the linear speed $$v of a $$point at distance $$r$$ from the axis of rotation to the angular velocity $$\omega$$?

Accepted Answer

$$v = r \omega$$

Question 4

A Ferris wheel rotates at a constant angular velocity $$\omega. $$What is the centripetal acceleration of a passenger sitting at a distance $$R$$ from the center?

Accepted Answer

$$a_{cp} = R \$$omega^2$$

Question 5

A wind turbine blade of mass $$m is $$rotating about its axis at angular velocity $$\omega. $$What is its rotational kinetic energy?

Accepted Answer

$$K = \frac{1}{2} m \$$omega^2$$ r^2$

Question 6

A solid sphere of mass $$M$$ and radius $$R$$ has a moment of inertia $$I_{cm} = \frac{2}{5}MR^2$ about its center. What is its moment of inertia about a parallel axis tangent to its surface?

Accepted Answer

$$I = \frac{2}{5}MR^2$ + MR^2$ = \frac{7}{5}MR^2$

Question 7

A CD spins up from rest with a constant angular acceleration $$\alpha. $$After time $$t$$, what is its angular displacement $$	heta$$?

Accepted Answer

$$	heta = \$$omega_0$$ t + \frac{1}{2}\alpha t^2$

Question 8

A child sits on a merry-go-round at a distance $$r$$ from the center, rotating with angular velocity $$\omega. $$What is the child’s tangential acceleration if the angular acceleration is $$\alpha$$?

Accepted Answer

$$a_t = r \alpha$$

Question 9

A thin rectangular plate of mass $$M$$ and sides $$a$$ and $$b$$ rotates about an axis through its center and perpendicular to the plate. What is its moment of inertia?

Accepted Answer

$$I = M(a^2$ + b^2$)/12$$

Question 10

A rotating object has both tangential acceleration $$a_t$$ and centripetal acceleration $$a_{cp}. $$What is the magnitude of its total acceleration?

Accepted Answer

$$a = \sqrt{a_t^2$ + a_{cp}^2}$$

Question 11

A solid cylinder of mass $$M$$ and radius $$R$$ rolls down a hill of height $$h$$ without slipping. Using energy conservation, what is its final speed at the bottom?

Accepted Answer

$$v = \sqrt{\frac{2gh}{1 + \frac{I}{MR^2$}}}$$

Question 12

Which of the following statements about the moment of inertia is correct?

Accepted Answer

It depends on both the mass and the distribution of mass relative to the axis of rotation.

Question 13

A car’s speedometer needle rotates about a fixed axis. If the needle sweeps an angle of $$\frac{\pi}{2}$$ radians in 0.5 seconds, what is its average angular velocity?

Accepted Answer

$$\omega_{av} = \frac{\pi}{2} / 0.5 = \pi$$ rad/s

Question 14

A thin rod of length $$L$$ and mass $$M$$ rotates about an axis perpendicular to the rod and passing through one end. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{3}ML^2$

Question 15

If a rotating object’s angular velocity vector and angular acceleration vector point in opposite directions, what is happening to the object?

Accepted Answer

It is slowing down.

Question 16

A disk rotates with constant angular acceleration $$\alpha. If $$its initial angular velocity is $$\$$omega_0$$, what is its angular velocity after time t$?

Accepted Answer

$$\omega = \$$omega_0$$ + \alpha t$$

Question 17

A rigid body rotates about a fixed axis. Which of the following is true for all points on the body?

Accepted Answer

All points have the same angular velocity.

Question 18

The parallel axis theorem relates the moment of inertia $$I_{cm}$$ about the center of mass to the moment of inertia $$I_P$$ about a parallel axis a distance $$d$$ away. Which formula expresses this relationship?

Accepted Answer

$$I_P = I_{cm} + Md^2$

Question 19

A rotating rigid body has a total kinetic energy $$K = \frac{1}{2}I\$$omega^2$$. If the moment of inertia doubles and the angular velocity is halved, what happens to the kinetic energy?

Accepted Answer

It is reduced to one quarter of its original value.

Rotation of Rigid Bodies: Rotational Kinematics and Dynamics

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