Q: A wind turbine blade rotates about a fixed axis. If the blade completes 5 revolutions in 10 seconds, what is its average angular velocity?

$$\omega = \frac{5 \times 2\pi}{10} = 1\pi$$ rad/s

Q: A winch drum (solid cylinder, $$M = 50 kg$$, $$R = 0.06 m) is $$rotated by pulling a cable with a force of $$9.0 N $$over $$2.0 m. $$What is the work done by the force?

$$W = F \times d = 9.0 \times 2.0 = 18 J$$

Q: Which formula gives the angular displacement $$\theta$$ for constant angular acceleration $$\alpha$$ over time $$t$$?

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

Question 1

A wind turbine blade rotates about a fixed axis. If the blade completes 5 revolutions in 10 seconds, what is its average angular velocity?

Accepted Answer

$$\omega = \frac{5 	imes 2\pi}{10} = 1\pi$$ rad/s

Question 2

A gyroscope is spinning with an angular velocity $$\$$omega_0$$. If a constant angular acceleration $$\alpha$$ is applied for time t$, what is the final angular velocity?

Accepted Answer

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

Question 3

A solid disk of mass $$M$$ and radius $$R$$ rotates about its center. What is its moment of inertia?

Accepted Answer

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

Question 4

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

Accepted Answer

$$v = r\omega$$

Question 5

A hollow cylinder and a solid cylinder of equal mass and radius are released from rest at the top of an incline. Which reaches the bottom first?

Accepted Answer

The solid cylinder

Question 6

A winch drum (solid cylinder, $$M = 50 kg$$, $$R = 0.06 m) is $$rotated by pulling a cable with a force of $$9.0 N $$over $$2.0 m. $$What is the work done by the force?

Accepted Answer

$$W = F 	imes d = 9.0 	imes 2.0 = 18 J$$

Question 7

For a rolling object without slipping, which relationship holds between the linear speed $$v of $$the center of mass and the angular speed $$\omega$$?

Accepted Answer

$$v = \omega R$$

Question 8

A yo-yo (solid disk) of mass $$M$$ and radius $$R$$ drops a height $$h$$ from rest. What is the speed $$v_{cm} of $$its center of mass when it reaches the bottom?

Accepted Answer

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

Question 9

Which formula gives the angular displacement $$	heta$$ for constant angular acceleration $$\alpha$$ over time $$t$$?

Accepted Answer

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

Question 10

A solid sphere, a solid cylinder, and a hollow cylinder are released from rest at the top of an incline. Which object reaches the bottom first?

Accepted Answer

The solid sphere

Question 11

The kinetic energy of a rotating rigid body is given by which formula?

Accepted Answer

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

Question 12

If a wheel rolls forward without slipping for one revolution, what is the distance traveled?

Accepted Answer

$$\Delta x = 2\pi R$$

Question 13

A rectangular plate of mass $$M$$ and length $$L$$ rotates about its center. What is its moment of inertia?

Accepted Answer

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

Question 14

Which of the following best describes the total kinetic energy of a rolling object?

Accepted Answer

It is the sum of translational kinetic energy and rotational kinetic energy about the center of mass.

Question 15

A thin-walled hollow cylinder of mass $$M$$ and radius $$R$$ rotates about its axis. What is its moment of inertia?

Accepted Answer

$$I = MR^2$

Question 16

If the angular acceleration $$\alpha is $$constant, which equation gives the angular velocity $$\omega$$ after time $$t$$?

Accepted Answer

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

Question 17

A wheel of radius $$R$$ rolls without slipping. What is the angular velocity $$\omega if $$the linear speed of the center is $$v$$?

Accepted Answer

$$\omega = \frac{v}{R}$$

Question 18

A yo-yo is released from rest and drops a distance $$h. $$Which energy principle is used to find its speed at the bottom?

Accepted Answer

Conservation of mechanical energy, including rotational kinetic energy

Question 19

Which of the following is NOT a correct equation for angular motion with constant acceleration?

Accepted Answer

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

Rotational Motion: Concepts, Equations, and Applications

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