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

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

Q: For a particle moving in a circle of radius $$r$$ with angular velocity $$\omega$$, what is the radial (centripetal) acceleration?

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

Q: Which formula gives the angular displacement $$\theta$$ for an object with constant angular acceleration $$\alpha$$ starting from angular velocity $$\$$omega_0$$?

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

Q: A rotating object has a total acceleration $$\vec{a}$$ composed of tangential and radial components. Which equation correctly expresses the magnitude of $$\vec{a}$$?

$$a = \sqrt{a_{\text{tan}}^2 + a_{\text{rad}}^2}$$

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 $$\$$omega_0$$ = 10\ 	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

$$t = 5\ 	ext{s}$$

Question 3

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

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 spins with $$\alpha = 50\ 	ext{rad/s}^2$$ and $$\omega = 10\ 	ext{rad/s}$$, moving the discus in a circle of radius $$r = 0.80\ 	ext{m}. $$What is the magnitude of the discus's acceleration?

Accepted Answer

$$a = 88\ 	ext{m/s}^2$$

Question 6

If a propeller must not exceed a tip speed of $$v_{	ext{tip}} = 300\ 	ext{m/s}$$ and rotates at $$\omega = 1500\ 	ext{rev/min}$$, what is the maximum radius allowed? (1 rev = $$2\pi$$ rad)

Accepted Answer

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

Question 7

Which equation correctly relates linear and angular velocity for a point at distance $$r$$ from the axis?

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

The moment of inertia of a solid sphere of mass $$M$$ and radius $$R$$ about its symmetry axis is:

Accepted Answer

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

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

A rotating rigid body has mass distributed at distances $$r_A$$, $$r_B$$, and $$r_C$$ from the axis. Its kinetic energy is:

Accepted Answer

$$K = \frac{1}{2}(m_A r_A^2$ + m_B r_B^2$ + m_C r_C^2$)\$$omega^2$$

Question 12

For a particle moving in a circle of radius $$r$$ with angular velocity $$\omega$$, what is the radial (centripetal) acceleration?

Accepted Answer

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

Question 13

A solid cylinder of mass $$M$$ and radius $$R$$ rotates about its symmetry axis. What is its moment of inertia?

Accepted Answer

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

Question 14

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 kinetic energy?

Accepted Answer

$$K = 1.12\ 	ext{J}$$

Question 15

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 16

If an object rotates counterclockwise, what is the sign of its angular velocity?

Accepted Answer

Positive

Question 17

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

Accepted Answer

Thin-walled hollow cylinder

Question 18

Which formula gives the angular displacement $$	heta$$ for an object with constant angular acceleration $$\alpha$$ starting from angular velocity $$\$$omega_0$$?

Accepted Answer

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

Question 19

A rotating object has a total acceleration $$\vec{a}$$ composed of tangential and radial components. Which equation correctly expresses the magnitude of $$\vec{a}$$?

Accepted Answer

$$a = \sqrt{a_{	ext{tan}}^2 + a_{	ext{rad}}^2}$$

Rotational Motion: Concepts, Equations, and Applications

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