Q: Which of the following is the correct formula for the instantaneous angular velocity $$\omega_z$$?

$$\omega_z = \frac{d\theta}{dt}$$

Q: Why must angles be expressed in radians when relating linear and angular quantities?

Because the formulas $$s = r\theta$$ and $$v = r\omega$$ are only valid if $$\theta$$ and $$\omega$$ are in radians.

Q: If a rigid body rotates with constant angular acceleration, which kinematic equation is analogous to $$x = x_0$ + v_0$ t + \frac{1}{2} a t^2$?

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

Question 1

A Ferris wheel rotates about a fixed axis. If it completes one full revolution in 40 seconds, what is its average angular velocity in radians per second?

Accepted Answer

$$0.079$$ rad/s

Question 2

Which of the following best describes the relationship between the arc length $$s$$, the radius $$r$$, and the angle $$	heta (in $$radians) for a rotating rigid body?

Accepted Answer

$$s = r 	heta$$

Question 3

A car’s speedometer needle rotates from $$\$$theta_1$$ = 0.2$$ rad to $$\$$theta_2$$ = 1.0$$ rad in $$4$$ seconds. What is the average angular velocity?

Accepted Answer

$$0.2$$ 	ext{ rad/s}$

Question 4

Which statement about angular velocity is correct?

Accepted Answer

Angular velocity is a vector whose direction is given by the right-hand rule.

Question 5

A rotating disk has an angular velocity $$\omega_z = 5$$ rad/s and angular acceleration $$\alpha_z = -2$$ rad/s$$^2. $$What is happening to the disk?

Accepted Answer

The disk is slowing down.

Question 6

If a point on a rotating wheel is $$0.5 m $$from the axis and the wheel rotates at $$4$$ rad/s, what is the linear speed of the point?

Accepted Answer

$$2 m/s$$

Question 7

Which of the following is the correct formula for the instantaneous angular velocity $$\omega_z$$?

Accepted Answer

$$\omega_z = \frac{d	heta}{dt}$$

Question 8

A ceiling fan starts from rest and reaches an angular velocity of $$6$$ rad/s in $$3$$ seconds with constant angular acceleration. What is the angular acceleration?

Accepted Answer

$$2$$ $$rad/s^2$

Question 9

A point on a rotating disk has a tangential acceleration $$a_{tan}$$ and a centripetal acceleration $$a_{rad}. $$Which of the following is true?

Accepted Answer

$$a_{tan} = r\alpha$$ and $$a_{rad} = r\$$omega^2$$

Question 10

Why must angles be expressed in radians when relating linear and angular quantities?

Accepted Answer

Because the formulas $$s = r	heta$$ and $$v = r\omega$$ are only valid if $$	heta$$ and $$\omega$$ are in radians.

Question 11

A rotating propeller speeds up from $$10$$ rad/s to $$40$$ rad/s in $$5$$ seconds. What is its average angular acceleration?

Accepted Answer

$$6$$ $$rad/s^2$

Question 12

If a rigid body rotates with constant angular acceleration, which kinematic equation is analogous to $$x = x_0$ + v_0$ t + \frac{1}{2} a t^2$?

Accepted Answer

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

Question 13

A disk rotates at $$\omega = 12$$ rad/s. What is the centripetal acceleration of a point $$0.2 m $$from the center?

Accepted Answer

$$28.8$$ $$m/s^2$

Question 14

A rotating object’s angular velocity is negative. What does this indicate about its rotation?

Accepted Answer

It is rotating clockwise.

Question 15

A discus thrower accelerates the discus from rest with $$\alpha = 500$$ rad/s$$^2$$ for $$0.2 s. $$What is the final angular velocity?

Accepted Answer

$$100$$ rad/s

Question 16

Which of the following is a correct application of the right-hand rule for angular velocity?

Accepted Answer

Curl the fingers of your right hand in the direction of rotation; your thumb points in the direction of $$\vec{\omega}.$$

Question 17

A wheel rotates with constant angular acceleration. If its angular velocity increases from $$2$$ rad/s to $$10$$ rad/s in $$4$$ seconds, what is the angular displacement during this time?

Accepted Answer

$$24$$ rad

Question 18

A point on a rotating disk is $$0.3 m $$from the axis. If the angular velocity is $$5$$ rad/s and the angular acceleration is $$2$$ rad/s$$^2$$, what is the magnitude of the total acceleration?

Accepted Answer

$$8.5$$ $$m/s^2$

Rotation of Rigid Bodies: Angular Kinematics and Dynamics

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