Q: Which of the following is NOT a correct equation for angular displacement $$\theta$$ with constant angular acceleration?

$$\theta - \$$theta_0$$ = \$$omega_0$$ t + \alpha t^3$

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 these quantities?

Accepted Answer

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

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

Accepted Answer

$$a = r\alpha$$

Question 5

A hollow cylinder and a solid cylinder, both with the same 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 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, assuming it unwinds without slipping?

Accepted Answer

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

Question 7

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

Accepted Answer

$$2\pi R$$

Question 8

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

Accepted Answer

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

Question 9

A winch drum (solid cylinder) of mass $$50\,	ext{kg}$$ and diameter $$0.12\,	ext{m} is $$rotated by a cable pulled 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

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

Question 10

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

Accepted Answer

Translational kinetic energy and rotational kinetic energy

Question 11

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 12

If a rigid body rotates about a moving axis, its total kinetic energy can be expressed as:

Accepted Answer

The sum of kinetic energy of the center of mass and rotational kinetic energy about the center of mass

Question 13

A sphere of mass $$M$$ and radius $$R$$ rolls down a slope without slipping. Which equation gives the speed $$v_{cm} of $$its center of mass at the bottom, starting from height $$h$$?

Accepted Answer

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

Question 14

Which of the following is NOT a correct equation for angular displacement $$	heta$$ with constant angular acceleration?

Accepted Answer

$$	heta - \$$theta_0$$ = \$$omega_0$$ t + \alpha t^3$

Question 15

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

Accepted Answer

$$s = r	heta$$

Question 16

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

Accepted Answer

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

Question 17

If a wheel rolls forward without slipping, what is the relationship between its linear speed $$v$$ and angular speed $$\omega$$?

Accepted Answer

$$v = \omega R$$

Question 18

Which physical quantity describes how mass is distributed relative to an axis of rotation?

Accepted Answer

Moment of inertia

Question 19

A solid sphere of mass $$M$$ and radius $$R$$ rolls down an incline. What is its moment of inertia?

Accepted Answer

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

Rotational Motion: Dynamics, Energy, and Applications

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