Q: A wind turbine blade rotates with a constant angular acceleration. If its initial angular velocity is $$\$$omega_0$$ and after time t$ its angular velocity is $$\omega$$, which equation correctly relates angular displacement $$\theta$$ to these quantities?

$$\theta - \$$theta_0$$ = \frac{1}{2}(\omega + \$$omega_0$$)t$$

Question 1

A wind turbine blade rotates with a constant angular acceleration. If its initial angular velocity is $$\$$omega_0$$ and after time t$ its angular velocity is $$\omega$$, which equation correctly relates angular displacement $$	heta$$ to these quantities?

Accepted Answer

$$	heta - \$$theta_0$$ = \frac{1}{2}(\omega + \$$omega_0$$)t$$

Question 2

A gyroscope is spinning with angular velocity $$\omega. If $$the radius of the spinning disk is $$r$$, what is the linear velocity $$v of a $$point on the rim?

Accepted Answer

$$v = r\omega$$

Question 3

A 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 4

A hollow cylinder and a solid cylinder, both with mass $$M$$ and radius $$R$$, are released from rest at the top of an incline. Which reaches the bottom first and why?

Accepted Answer

The solid cylinder, because it has a smaller moment of inertia and more energy is converted to translational motion.

Question 5

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

Accepted Answer

$$v_{cm} = \sqrt{\frac{4gh}{3}}$$

Question 6

Which equation describes the angular velocity $$\omega$$ after time $$t$$ for an object starting at angular velocity $$\$$omega_0$$ with constant angular acceleration $$\alpha$$?

Accepted Answer

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

Question 7

A wheel rolls without slipping. If its radius is $$R$$ and its center moves with speed $$v_{cm}$$, what is its angular velocity $$\omega$$?

Accepted Answer

$$\omega = \frac{v_{cm}}{R}$$

Question 8

A winch drum (solid cylinder) of mass $$50\,	ext{kg}$$ and diameter $$0.12\,	ext{m} is $$rotated by a $$9.0\,	ext{N}$$ force pulling a cable for $$2.0\,	ext{m}. $$What is the final angular velocity $$\omega of $$the drum?

Accepted Answer

$$\omega = \sqrt{\frac{2Fd}{I}}$$

Question 9

For a rolling object, the total kinetic energy is the sum of translational and rotational kinetic energies. Which equation expresses this?

Accepted Answer

$$K_{total} = \frac{1}{2}Mv_{cm}^2 + \frac{1}{2}I\$$omega^2$$

Question 10

A thin-walled hollow cylinder of mass $$M$$ and radius $$R$$ rolls down an incline. What is its moment of inertia?

Accepted Answer

$$I = MR^2$

Question 11

A 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 sphere, because it has the smallest moment of inertia relative to its mass.

Question 12

If a point $$P on a $$rotating object moves in a circle of radius $$r$$ and angular displacement $$	heta$$, what is the arc length $$s$$ traveled by $$P$$?

Accepted Answer

$$s = r	heta$$

Question 13

A wheel of radius $$R$$ rolls one revolution without slipping. What is the distance $$\Delta x$$ traveled by its center?

Accepted Answer

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

Question 14

Which of the following best describes the relationship between linear acceleration $$a$$ and angular acceleration $$\alpha$$ for a point at distance $$r$$ from the axis?

Accepted Answer

$$a = r\alpha$$

Question 15

A rigid body rotates about a moving axis. Its total kinetic energy is:

Accepted Answer

$$K_{total} = \frac{1}{2}Mv_{cm}^2 + \frac{1}{2}I_{cm}\$$omega^2$$

Question 16

If a solid sphere of mass $$M$$ and radius $$R$$ rolls down a ramp of height $$h$$, what is its speed $$v_{cm} at $$the bottom?

Accepted Answer

$$v_{cm} = \sqrt{\frac{10gh}{7}}$$

Question 17

Which of the following is NOT a correct equation for fixed-axis rotation with constant angular acceleration?

Accepted Answer

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

Question 18

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

Accepted Answer

$$I = \frac{1}{12}M(L^2$ + b^2$)$$

Question 19

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

Accepted Answer

$$I = \frac{2}{3}MR^2$

Rotational Motion: Concepts, Equations, and Applications

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