Q: A playground merry-go-round rotates through an angular displacement $$\Delta\theta in $$time $$\Delta t. $$What is the average angular velocity?

$$\omega_{av} = \frac{\Delta\theta}{\Delta t}$$

Q: A disk rotates with constant angular acceleration $$\alpha. If $$its initial angular velocity is $$\$$omega_0$$, what is the angular displacement $$\theta - \$$theta_0$$ after time $$t$$?

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

Q: A rotating disk slows down with constant angular acceleration. If its initial angular velocity is $$\$$omega_0$$ and it comes to rest after turning through angle $$\theta - \$$theta_0$$, what is the angular acceleration $$\alpha$$?

$$\alpha = -\frac{\$$omega_0$$^2}{2(\theta - \$$theta_0$$)}$$

Q: A rotating disk starts from rest and rotates with constant angular acceleration $$\alpha. $$What is the angular displacement after time $$t$$?

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

Question 1

A wind turbine blade rotates counterclockwise as viewed from above. If the angular velocity is decreasing, what are the signs of angular velocity $$\omega$$ and angular acceleration $$\alpha$$?

Accepted Answer

$$\omega is $$positive and $$\alpha is $$negative.

Question 2

A compact disc (CD) with radius $$r = 6.0\ 	ext{cm}$$ spins at $$7200\ 	ext{rev/min}. $$What is its angular velocity $$\omega in $$radians per second?

Accepted Answer

$$\omega = 754\ 	ext{rad/s}$$

Question 3

A merry-go-round rotates at a constant angular velocity. A boy is at the outer edge, and a girl is near the axis of rotation. How do their angular velocities compare?

Accepted Answer

Both children have the same angular velocity.

Question 4

A tractor wheel starts from rest and accelerates with constant angular acceleration. If it turns through $$25\ 	ext{rad} in $$time $$t$$, through what angle will it have turned after time $$2t$$?

Accepted Answer

$$100\ 	ext{rad}$$

Question 5

A wheel starts from rest and spins up to $$25\ 	ext{rpm} in $$time $$t$$ with constant angular acceleration. What will its angular velocity be after time $$2t$$?

Accepted Answer

$$50\ 	ext{rpm}$$

Question 6

A gyroscope is spinning clockwise when viewed from above. If it is slowing down, what are the signs of $$\omega$$ and $$\alpha$$?

Accepted Answer

$$\omega is $$negative and $$\alpha is $$positive.

Question 7

A playground merry-go-round rotates through an angular displacement $$\Delta	heta in $$time $$\Delta t. $$What is the average angular velocity?

Accepted Answer

$$\omega_{av} = \frac{\Delta	heta}{\Delta t}$$

Question 8

A rotating disk starts from rest and reaches an angular velocity $$\omega in $$time $$t$$ with constant angular acceleration $$\alpha. $$Which equation correctly relates these quantities?

Accepted Answer

$$\omega = \alpha t$$

Question 9

A wind turbine blade rotates through an arc length $$s$$ with radius $$r. $$What is the angular displacement $$	heta in $$radians?

Accepted Answer

$$	heta = \frac{s}{r}$$

Question 10

A rotating wheel has an initial angular velocity $$\$$omega_0$$ and angular acceleration $$\alpha$$. What is its angular velocity $$\omega$$ after time t$?

Accepted Answer

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

Question 11

A disk rotates with constant angular acceleration $$\alpha. If $$its initial angular velocity is $$\$$omega_0$$, what is the angular displacement $$	heta - \$$theta_0$$ after time $$t$$?

Accepted Answer

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

Question 12

A rotating engine pulley increases its angular velocity from $$\$$omega_1$$ to $$\$$omega_2 in $$time $$t_2$$ - $$t_1. $$What is the average angular acceleration?

Accepted Answer

$$\alpha_{av} = \frac{\$$omega_2$$ - \$$omega_1$$}{t_2$ - t_1$}$$

Question 13

A wind turbine blade rotates at $$1\ 	ext{rev/s}. $$What is its angular velocity in rad/s?

Accepted Answer

$$2\pi\ 	ext{rad/s}$$

Question 14

A rotating disk slows down with constant angular acceleration. If its initial angular velocity is $$\$$omega_0$$ and it comes to rest after turning through angle $$	heta - \$$theta_0$$, what is the angular acceleration $$\alpha$$?

Accepted Answer

$$\alpha = -\frac{\$$omega_0$$^2}{2(	heta - \$$theta_0$$)}$$

Question 15

A merry-go-round rotates counterclockwise. If $$\Delta	heta \u003e 0$$, what is the sign of angular velocity $$\omega$$?

Accepted Answer

$$\omega \u003e 0$$

Question 16

A rotating disk has angular velocity $$\omega$$ and angular acceleration $$\alpha. If$$ $$\alpha is $$negative and $$\omega is $$positive, what is happening to the disk?

Accepted Answer

The disk is slowing down.

Question 17

A child pushes a merry-go-round, causing it to accelerate from rest. If the angular acceleration is $$\alpha$$ and the time is $$t$$, what is the angular velocity after time $$t$$?

Accepted Answer

$$\omega = \alpha t$$

Question 18

A rotating disk starts from rest and rotates with constant angular acceleration $$\alpha. $$What is the angular displacement after time $$t$$?

Accepted Answer

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

Question 19

A rotating engine pulley increases its angular velocity from $$\$$omega_1$$ to $$\$$omega_2 in $$time $$t_2$$ - $$t_1. $$What is the average angular acceleration?

Accepted Answer

$$\alpha_{av} = \frac{\$$omega_2$$ - \$$omega_1$$}{t_2$ - t_1$}$$

Rotational Motion: Concepts, Equations, and Applications

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