Q: A gyroscope spins with an angular velocity $$\omega = 10\ \mathrm{rad/s}$$ and has a moment of inertia $$I = 0.2\ \mathrm{kg\cdot m^2$}. $$What is its rotational kinetic energy?

$$K = \frac{1}{2} I \$$omega^2$$ = \frac{1}{2} \times 0.2 \times 100 = 10\ \mathrm{J}$$

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

$$\theta = \$$omega_0$$ t + \frac{1}{2} \alpha t^2$ = 0 + 0.5 \times 2 \times 9 = 9\ \mathrm{rad}$$

Q: A point on the rim of a rotating disk of radius $$0.5\ \mathrm{m}$$ has an angular velocity $$\omega = 6\ \mathrm{rad/s}. $$What is its linear speed?

$$v = r \omega = 0.5 \times 6 = 3\ \mathrm{m/s}$$

Q: A propeller is designed to spin at $$1500\ \mathrm{rev/min}$$, but the tip speed must not exceed $$300\ \mathrm{m/s}. $$What is the maximum radius allowed?

$$r = \frac{v}{\omega}$$, $$\omega = 1500 \times \frac{2\pi}{60} = 157.08\ \mathrm{rad/s}$$, $$r = 300 / 157.08 \approx 1.91\ \mathrm{m}$$

Question 1

A wind turbine blade rotates with a constant angular acceleration of $$\alpha = 0.5\ \mathrm{$$rad/s^2$$}$$ from rest. What is its angular velocity after $$8\ \mathrm{s}$$?

Accepted Answer

$$\omega = 0 + (0.5)(8) = 4\ \mathrm{rad/s}$$

Question 2

A gyroscope spins with an angular velocity $$\omega = 10\ \mathrm{rad/s}$$ and has a moment of inertia $$I = 0.2\ \mathrm{kg\cdot m^2$}. $$What is its rotational kinetic energy?

Accepted Answer

$$K = \frac{1}{2} I \$$omega^2$$ = \frac{1}{2} 	imes 0.2 	imes 100 = 10\ \mathrm{J}$$

Question 3

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

Accepted Answer

The solid cylinder, $$I = \frac{1}{2} mR^2$

Question 4

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

Accepted Answer

$$	heta = \$$omega_0$$ t + \frac{1}{2} \alpha t^2$ = 0 + 0.5 	imes 2 	imes 9 = 9\ \mathrm{rad}$$

Question 5

A point on the rim of a rotating disk of radius $$0.5\ \mathrm{m}$$ has an angular velocity $$\omega = 6\ \mathrm{rad/s}. $$What is its linear speed?

Accepted Answer

$$v = r \omega = 0.5 	imes 6 = 3\ \mathrm{m/s}$$

Question 6

A discus thrower spins with $$\alpha = 50\ \mathrm{$$rad/s^2$$}$$ and $$\omega = 10\ \mathrm{rad/s} at a $$radius $$r = 0.80\ \mathrm{m}. $$What is the magnitude of the total acceleration of the discus?

Accepted Answer

$$a = \sqrt{(r\$$alpha)^2$$ + (r\$$omega^2$$)^2$$} = \sqrt{$$(40)^2$$ + $$(64)^2$$} = \sqrt{1600 + 4096} = \sqrt{5696} \approx 75.5\ \mathrm{$$m/s^2$$}$

Question 7

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

Accepted Answer

$$r = \frac{v}{\omega}$$, $$\omega = 1500 	imes \frac{2\pi}{60} = 157.08\ \mathrm{rad/s}$$, $$r = 300 / 157.08 \approx 1.91\ \mathrm{m}$$

Question 8

If a merry-go-round spins at constant $$\omega$$, and Sofia is twice as far from the axis as Rasheed, what is Sofia's speed compared to Rasheed's?

Accepted Answer

Twice

Question 9

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 10

A rotating rigid body has masses $$m_A$$, $$m_B$$, $$m_C at $$distances $$r_A$$, $$r_B$$, $$r_C$$ from the axis. What is the total moment of inertia?

Accepted Answer

$$I = m_A r_A^2$ + m_B r_B^2$ + m_C r_C^2$

Question 11

Which of the following best describes the sign convention for angular velocity?

Accepted Answer

Counterclockwise is positive, clockwise is negative.

Question 12

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} M R^2$

Question 13

A point on a rotating object has a tangential acceleration $$a_{tan}$$ and a radial acceleration $$a_{rad}. $$Which equation gives the magnitude of the total acceleration?

Accepted Answer

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

Question 14

A rotating object has a constant angular velocity $$\omega. $$What is true about its radial (centripetal) acceleration?

Accepted Answer

It is always present and given by $$a_{rad} = \$$omega^2$$ r$$

Question 15

A rotating object increases its angular velocity from $$2\ \mathrm{rad/s} to$$ $$10\ \mathrm{rad/s} in$$ $$4\ \mathrm{s}. $$What is its angular acceleration?

Accepted Answer

$$\alpha = \frac{\Delta \omega}{\Delta t} = \frac{10 - 2}{4} = 2\ \mathrm{$$rad/s^2$$}$$

Question 16

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} M R^2$

Question 17

A rotating object has a tangential acceleration $$a_{tan} = 5\ \mathrm{m/s^2$}$$ and a radial acceleration $$a_{rad} = 12\ \mathrm{m/s^2$}. $$What is the magnitude of its total acceleration?

Accepted Answer

$$a = \sqrt{5^2$ + 12^2$} = 13\ \mathrm{m/s^2$}$$

Question 18

A rotating rigid body has a kinetic energy $$K = \frac{1}{2} I \$$omega^2$$. If its angular velocity doubles, what happens to its kinetic energy?

Accepted Answer

It increases by a factor of 4.

Rotational Motion: Concepts, Equations, and Applications

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