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 cylinder (mass $$M$$, radius $$R) is $$rolling down an incline without slipping. What is its moment of inertia about its axis?

Accepted Answer

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

Question 4

Which of the following best describes the difference between linear and angular acceleration?

Accepted Answer

Linear acceleration is the rate of change of velocity, while angular acceleration is the rate of change of angular velocity.

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 yo-yo (solid disk, mass $$M$$, 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 7

A winch drum (solid cylinder, mass $$50\,	ext{kg}$$, radius $$0.06\,	ext{m}) is $$rotated by pulling a cable with a constant force of $$9.0\,	ext{N}$$ for $$2.0\,	ext{m}. $$What is the final angular velocity $$\omega of $$the drum?

Accepted Answer

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

Question 8

For a rolling object without slipping, which equation relates the linear velocity $$v of $$the center of mass to the angular velocity $$\omega$$?

Accepted Answer

$$v = \omega R$$

Question 9

The total kinetic energy of a rolling object is the sum of which two components?

Accepted Answer

Translational kinetic energy and rotational kinetic energy about the center of mass

Question 10

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

Question 11

If a point $$P on a $$rotating object is at a distance $$r$$ from the axis and rotates through an angle $$	heta$$, what is the arc length $$s$$ traveled by $$P$$?

Accepted Answer

$$s = r	heta$$

Question 12

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

Accepted Answer

$$I = MR^2$

Question 13

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

Accepted Answer

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

Question 14

A wheel rolls forward without slipping for one revolution. What is the distance $$\Delta x$$ traveled by the center of mass?

Accepted Answer

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

Question 15

Which of the following is true about the moment of inertia?

Accepted Answer

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

Question 16

A disk rotates about its center with angular velocity $$\omega. If $$its mass is $$M$$ and radius $$R$$, what is its rotational kinetic energy?

Accepted Answer

$$K = \frac{1}{2}I\$$omega^2$$ = \frac{1}{4}MR^2$\$$omega^2$$

Question 17

If the axis of rotation of a rigid body moves, how is the total kinetic energy expressed?

Accepted Answer

As the sum of translational kinetic energy of the center of mass and rotational kinetic energy about the center of mass

Question 18

A rectangular plate (mass $$M$$, length $$L$$, 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 solid sphere (mass $$M$$, radius $$R) $$rolls down an incline without slipping. What is its moment of inertia?

Accepted Answer

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

Rotational Motion: Dynamics, Energy, and Applications

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