Q: A bicycle wheel starts from rest and accelerates at $$\alpha = 2.00\ \text{rad/s}^2. $$What is its angular velocity after $$3.00\ \text{s}$$?

$$\omega = 6.00\ \text{rad/s}$$

Q: Which equation describes the angular displacement $$\theta$$ for a rotating object with constant angular acceleration $$\alpha$$?

$$\theta = \$$theta_0$$ + \$$omega_0$$ t + \frac{1}{2}\alpha t^2$

Q: A rotating object has angular velocity $$\omega$$ and radius $$r. $$What is the radial (centripetal) acceleration?

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

Question 1

A wind turbine blade rotates with a constant angular velocity of $$\omega = 2.5\ 	ext{rad/s}. If $$the tip of the blade is $$r = 40\ 	ext{m}$$ from the axis, what is the linear speed of the tip?

Accepted Answer

$$v = 100\ 	ext{m/s}$$

Question 2

A gyroscope is spinning with an angular velocity $$\$$omega_0$$ = 5\ 	ext{rad/s}$$ and an angular acceleration $$\alpha = -2\ 	ext{rad/s}^2. $$How long will it take for the gyroscope to come to rest?

Accepted Answer

$$t = 2.5\ 	ext{s}$$

Question 3

A bicycle wheel starts from rest and accelerates at $$\alpha = 2.00\ 	ext{rad/s}^2. $$What is its angular velocity after $$3.00\ 	ext{s}$$?

Accepted Answer

$$\omega = 6.00\ 	ext{rad/s}$$

Question 4

A merry-go-round spins at a constant angular velocity. If Sofia is twice as far from the axis as Rasheed, what is Sofia’s speed compared to Rasheed’s?

Accepted Answer

Twice Rasheed’s speed

Question 5

A discus thrower accelerates a discus in a circle of radius $$0.80\ 	ext{m}$$ with angular acceleration $$\alpha = 50\ 	ext{rad/s}^2$$ and angular velocity $$\omega = 10\ 	ext{rad/s}. $$What is the magnitude of the discus’s total acceleration?

Accepted Answer

$$a = 56\ 	ext{m/s}^2$$

Question 6

A propeller is designed to turn at $$1500\ 	ext{rev/min}$$, and the tip speed must not exceed $$300\ 	ext{m/s}. $$What is the maximum radius allowed for the propeller?

Accepted Answer

$$r = 1.91\ 	ext{m}$$

Question 7

If a rigid body rotates about an axis and a point $$P is at a $$distance $$r$$ from the axis, which equation correctly relates the linear speed $$v of$$ $$P to $$the angular velocity $$\omega$$?

Accepted Answer

$$v = r\omega$$

Question 8

A solid sphere and a solid cylinder are made from the same mass $$m$$ and radius $$R. $$Which has the larger moment of inertia about its symmetry axis?

Accepted Answer

Cylinder

Question 9

For a rotating rigid body, the kinetic energy is given by $$K = \frac{1}{2}I\$$omega^2$$. What does I$ represent?

Accepted Answer

Moment of inertia

Question 10

A bicycle’s rear sprocket has $$N_B$$ teeth and the front sprocket has $$N_A$$ teeth. If the chain moves without slipping, what is the relationship between their angular velocities?

Accepted Answer

$$\omega_A N_A = \omega_B N_B$$

Question 11

Which of the following best describes the difference between tangential and radial acceleration in rotational motion?

Accepted Answer

Tangential acceleration changes the speed, radial acceleration changes the direction

Question 12

A rotating object has angular acceleration $$\alpha$$ and radius $$r. $$What is the tangential acceleration at a point $$r$$ from the axis?

Accepted Answer

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

Question 13

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 14

A wheel rotates counterclockwise. What is the sign of its angular velocity?

Accepted Answer

Positive

Question 15

A sculpture consists of three point masses: $$m_A = 0.30\ 	ext{kg} at$$ $$r_A = 0.40\ 	ext{m}$$, $$m_B = 0.10\ 	ext{kg} at$$ $$r_B = 0.50\ 	ext{m}$$, and $$m_C = 0.20\ 	ext{kg} at$$ $$r_C = 0.30\ 	ext{m}. If it $$rotates at $$\omega = 4\ 	ext{rad/s}$$ about an axis through $$A$$, what is its rotational kinetic energy?

Accepted Answer

$$K = 1.2\ 	ext{J}$$

Question 16

Which equation describes the angular displacement $$	heta$$ for a rotating object with constant angular acceleration $$\alpha$$?

Accepted Answer

$$	heta = \$$theta_0$$ + \$$omega_0$$ t + \frac{1}{2}\alpha t^2$

Question 17

If a rotating object’s angular velocity increases, what can be said about its angular acceleration?

Accepted Answer

Angular acceleration is positive

Question 18

A hollow cylinder and a solid cylinder have the same mass $$M$$ and radius $$R. $$Which has the greater moment of inertia about its symmetry axis?

Accepted Answer

Hollow cylinder

Question 19

A rotating object has angular velocity $$\omega$$ and radius $$r. $$What is the radial (centripetal) acceleration?

Accepted Answer

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

Question 20

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

Accepted Answer

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

Rotational Motion: Concepts, Equations, and Applications

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