Q: A bicycle wheel increases its angular speed from 2 rad/s to 10 rad/s in 3 seconds at a constant angular acceleration. How many revolutions does the wheel make during this time?

$$\theta = \$$omega_0$$ t + \frac{1}{2} \alpha t^2$, where $$\alpha = \frac{\omega - \$$omega_0$$}{t}$$, then convert $$\theta$$ from radians to revolutions$

Q: A rigid body rotates with constant angular acceleration $$\alpha. If $$its initial angular velocity is $$\$$omega_0$$ and it rotates for time t$, what is its angular displacement?

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

Q: If a rigid body rotates about a fixed axis with constant angular acceleration, which equation relates the final angular velocity $$\omega to $$the initial angular velocity $$\$$omega_0$$, angular acceleration $$\alpha$$, and angular displacement $$\theta$$?

$$\$$omega^2$$ = \$$omega_0$$^2 + 2\alpha\theta$$

Question 1

A bicycle wheel increases its angular speed from 2 rad/s to 10 rad/s in 3 seconds at a constant angular acceleration. How many revolutions does the wheel make during this time?

Accepted Answer

$$	heta = \$$omega_0$$ t + \frac{1}{2} \alpha t^2$, where $$\alpha = \frac{\omega - \$$omega_0$$}{t}$$, then convert $$	heta$$ from radians to revolutions$

Question 2

Which of the following statements about the relationship between angular and linear quantities for a rotating rigid body is correct?

Accepted Answer

All points on the object have the same angular speed, but their linear speeds depend on their distance from the axis.

Question 3

A solid sphere and a hollow sphere, each of mass $$m$$ and radius $$r$$, are rotating about their central axes with the same angular speed $$\omega. $$Which has the greater rotational kinetic energy?

Accepted Answer

The hollow sphere, because $$I_{hollow} = \frac{2}{3}mr^2$ is greater than I$$_{solid} = \frac{2}{5}$$mr^2$$

Question 4

What is the moment of inertia of a thin rod of mass $$M$$ and length $$L$$ about an axis through its center and perpendicular to its length?

Accepted Answer

$$I = \frac{1}{12}ML^2$

Question 5

A point $$P on a $$rotating object is at a distance $$r$$ from the axis of rotation. If the angular acceleration is $$\alpha$$, what is the tangential acceleration of point $$P$$?

Accepted Answer

$$a_{tan} = r\alpha$$

Question 6

A thin, solid disk of mass $$m$$ and radius $$r$$ rotates about its central axis. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{2}mr^2$

Question 7

A rigid body rotates with constant angular acceleration $$\alpha. If $$its initial angular velocity is $$\$$omega_0$$ and it rotates for time t$, what is its angular displacement?

Accepted Answer

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

Question 8

Which of the following best describes the physical meaning of the moment of inertia?

Accepted Answer

It is a measure of an object's resistance to changes in its rotational motion.

Question 9

A system consists of four point masses at the ends of two rods as shown in the images. If $$m_1$$ = $$m_3 = 0.20$ kg, m_2$ = m_4$ = 0.30 kg$$, and the distance from the axis to each mass is $$0.25 m$$, what is the moment of inertia about the center?

Accepted Answer

$$I = 2m_1 r^2$ + 2m_2 r^2$ = 2(0.20)(0.25)^2$ + 2(0.30)(0.25)^2$ = 0.025 kg m$^2$$

Question 10

A rotating object has a moment of inertia $$I$$ and angular speed $$\omega. $$What is its rotational kinetic energy?

Accepted Answer

$$K_R = \frac{1}{2}I\$$omega^2$$

Question 11

A thin circular hoop of mass $$m$$ and radius $$r$$ rotates about its central axis. What is its moment of inertia?

Accepted Answer

$$I = mr^2$

Question 12

If a rigid body rotates about a fixed axis with constant angular acceleration, which equation relates the final angular velocity $$\omega to $$the initial angular velocity $$\$$omega_0$$, angular acceleration $$\alpha$$, and angular displacement $$	heta$$?

Accepted Answer

$$\$$omega^2$$ = \$$omega_0$$^2 + 2\alpha	heta$$

Question 13

A solid disk and a thin hoop, both with mass $$m$$ and radius $$r$$, are released from rest at the top of an incline. Which will reach the bottom first and why?

Accepted Answer

The solid disk, because it has a smaller moment of inertia relative to its mass and radius.

Question 14

Which of the following is NOT a correct formula for the moment of inertia of the given object?

Accepted Answer

$$I = \frac{2}{3}mr^2$ for a solid sphere

Question 15

A point on a rotating object has a radial (centripetal) acceleration. Which equation gives this acceleration?

Accepted Answer

$$a_{rad} = \$$omega^2$$ r$$

Question 16

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

Accepted Answer

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

Rotational Motion: Equations, Linear-Angular Relationships, and Moment of Inertia

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