Q: A car's speedometer needle rotates about a fixed axis. If the needle moves from $$\$$theta_1$$ = 0.5$$ rad to $$\$$theta_2$$ = 2.0$$ rad in $$\Delta t = 3.0 s$$, what is the average angular velocity of the needle?

$$\frac{2.0 - 0.5}{3.0} = 0.5$$ \text{ rad/s}$

Q: The instantaneous angular velocity $$\omega_z is $$defined as:

$$\omega_z = \lim_{\Delta t \to 0} \frac{\Delta \theta}{\Delta t} = \frac{d\theta}{dt}$$

Q: A rotating disc slows down with a constant angular acceleration $$\alpha_z = -10.0$$ rad/s$$^2. If $$its initial angular velocity is $$\$$omega_0$$ = 27.5$$ rad/s, what is its angular velocity after $$0.300 s$$?

$$\omega = 27.5 + (-10.0)(0.300) = 24.5$$ \text{ rad/s}$

Q: An athlete whirls a discus of radius $$0.80 m at$$ $$10.0$$ rad/s, with angular acceleration $$50.0$$ rad/s$$^2. $$What is the tangential acceleration of the discus?

$$a_{\text{tan}} = r \alpha = 0.80 \times 50.0 = 40.0$$ \text{ m/s}^2$

Q: A point on a rotating wheel has a centripetal (radial) acceleration given by:

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

Q: A propeller is designed to rotate at $$2400$$ rpm. If the maximum tip speed is $$270 m/s$$, what is the maximum possible radius of the propeller? (Use $$\omega = 2\pi f$$ and $$v = r\omega)$$

$$r = \frac{v}{\omega} = \frac{270}{2\pi \times 40} \approx 1.07$$ \text{ m}$

Q: A rotating rigid body has an angular position given by $$\theta = (2.0 \text{ rad/s}^3)t^3$. What is the instantaneous angular velocity at t = 2.0$ s?

$$\omega = 3 \times 2.0 \times (2.0)^2$ = 24.0 \text{ rad/s}$$

Q: Which of the following is a correct analogy between linear and rotational kinematics for constant acceleration?

$$v_x = v_{0x} + a_x t$$ \text{ corresponds to } $$\omega_z = \omega_{0z} + \alpha_z t$$

Question 1

A car's speedometer needle rotates about a fixed axis. If the needle moves from $$\$$theta_1$$ = 0.5$$ rad to $$\$$theta_2$$ = 2.0$$ rad in $$\Delta t = 3.0 s$$, what is the average angular velocity of the needle?

Accepted Answer

$$\frac{2.0 - 0.5}{3.0} = 0.5$$ 	ext{ 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 Ferris wheel rotates counterclockwise. According to the sign convention, what is the sign of its angular velocity $$\omega_z$$?

Accepted Answer

Positive, because counterclockwise rotation is defined as positive.

Question 4

The instantaneous angular velocity $$\omega_z is $$defined as:

Accepted Answer

$$\omega_z = \lim_{\Delta t 	o 0} \frac{\Delta 	heta}{\Delta t} = \frac{d	heta}{dt}$$

Question 5

A rotating disc slows down with a constant angular acceleration $$\alpha_z = -10.0$$ rad/s$$^2. If $$its initial angular velocity is $$\$$omega_0$$ = 27.5$$ rad/s, what is its angular velocity after $$0.300 s$$?

Accepted Answer

$$\omega = 27.5 + (-10.0)(0.300) = 24.5$$ 	ext{ rad/s}$

Question 6

Which equation correctly relates the linear speed $$v of a $$point at distance $$r$$ from the axis of rotation to the angular speed $$\omega$$?

Accepted Answer

$$v = r \omega$$

Question 7

An athlete whirls a discus of radius $$0.80 m at$$ $$10.0$$ rad/s, with angular acceleration $$50.0$$ rad/s$$^2. $$What is the tangential acceleration of the discus?

Accepted Answer

$$a_{	ext{tan}} = r \alpha = 0.80 	imes 50.0 = 40.0$$ 	ext{ m/s}^2$

Question 8

A point on a rotating wheel has a centripetal (radial) acceleration given by:

Accepted Answer

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

Question 9

A propeller is designed to rotate at $$2400$$ rpm. If the maximum tip speed is $$270 m/s$$, what is the maximum possible radius of the propeller? (Use $$\omega = 2\pi f$$ and $$v = r\omega)$$

Accepted Answer

$$r = \frac{v}{\omega} = \frac{270}{2\pi 	imes 40} \approx 1.07$$ 	ext{ m}$

Question 10

Which of the following statements about angular velocity $$\omega$$ and angular acceleration $$\alpha is $$correct?

Accepted Answer

If $$\omega$$ and $$\alpha$$ are in the same direction, the object is speeding up.

Question 11

A rotating rigid body has an angular position given by $$	heta = (2.0 	ext{ rad/s}^3)t^3$. What is the instantaneous angular velocity at t = 2.0$ s?

Accepted Answer

$$\omega = 3 	imes 2.0 	imes (2.0)^2$ = 24.0 	ext{ rad/s}$$

Question 12

Which of the following is a correct analogy between linear and rotational kinematics for constant acceleration?

Accepted Answer

$$v_x = v_{0x} + a_x t$$ 	ext{ corresponds to } $$\omega_z = \omega_{0z} + \alpha_z t$$

Question 13

A point on a rotating disk is $$0.20 m $$from the axis. If the disk rotates at $$5.0$$ rad/s, what is the linear speed of the point?

Accepted Answer

$$v = r \omega = 0.20 	imes 5.0 = 1.0$$ 	ext{ m/s}$

Question 14

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

Accepted Answer

Because only radians allow the equations $$s = r	heta$$ and $$v = r\omega to be $$dimensionally consistent.

Question 15

A wheel rotates with constant angular acceleration. If its initial angular velocity is $$2.0$$ rad/s and its angular acceleration is $$3.0$$ rad/s$$^2$$, what is its angular velocity after $$4.0 s$$?

Accepted Answer

$$\omega = 2.0 + 3.0 	imes 4.0 = 14.0$$ 	ext{ rad/s}$

Question 16

A rotating object has $$\omega = 0$$ and $$\alpha \u003e 0. $$What is happening to the object?

Accepted Answer

It is speeding up from rest.

Question 17

If a rotating rigid body has $$\omega = 10$$ rad/s and $$\alpha = -2$$ rad/s$$^2$$, what will its angular velocity be after $$3$$ seconds?

Accepted Answer

$$\omega = 10 + (-2) 	imes 3 = 4$$ 	ext{ rad/s}$

Rotation of Rigid Bodies: Angular Kinematics and Dynamics

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