Q: A wheel of radius $$R$$ and mass $$M is $$acted on by a net torque $$\tau. $$What is the resulting angular acceleration $$\alpha if $$the wheel is a solid disk?

$$\alpha = \frac{2\tau}{M R^2$}$$

Q: A bug sits on the rim of a spinning disk of radius $$r = 0.2\,\mathrm{m}$$, rotating at a constant angular velocity $$\omega = 5\,\mathrm{rad/s}. $$What is the magnitude of the bug's acceleration?

$$a = r \$$omega^2$$ = 0.2 \times 25 = 5.0\,\mathrm{m/s^2$}$$

Q: A wheel initially spinning at $$\$$omega_0$$ = 40\,\mathrm{rad/s} is $$brought to rest with a constant angular acceleration $$\alpha = -8\,\mathrm{$$rad/s^2$$}. $$Through what angle $$\Delta \theta$$ does it rotate before stopping?

$$\Delta \theta = \frac{\$$omega_0$$^2}{2|\alpha|} = \frac{1600}{16} = 100\,\mathrm{rad}$$

Question 1

A solid disk of mass $$M$$ and radius $$R is $$rotating about its central axis with angular velocity $$\omega. $$What is the rotational kinetic energy of the disk?

Accepted Answer

$$\displaystyle \frac{1}{2} \left( \frac{1}{2} M R^2$ \right) \$$omega^2$$

Question 2

A door is 0.9 m wide. You push on the edge of the door with a force of 15 N at an angle of $$45^\circ to $$the plane of the door. What is the magnitude of the torque you apply about the hinge?

Accepted Answer

$$\displaystyle (0.9\,\mathrm{m})(15\,\mathrm{N}) \sin 45^\circ = 9.55\,\mathrm{N \cdot m}$$

Question 3

A hoop, a solid disk, and a solid sphere, all with the same mass $$M$$ and radius $$R$$, are released from rest at the top of an incline and roll without slipping to the bottom. Which reaches the bottom first and why?

Accepted Answer

The solid sphere, because it has the smallest moment of inertia relative to its mass, so more energy goes into translation.

Question 4

A vinyl record of radius $$r = 0.15\,\mathrm{m}$$ spins at $$45\,\mathrm{rpm}. $$What is the speed $$v of a $$point on the rim?

Accepted Answer

$$v = r \omega$$, where $$\omega = 2\pi f = 2\pi \left( \frac{45}{60} ight) = 4.71\,\mathrm{rad/s}$$, so $$v = 0.15 	imes 4.71 = 0.707\,\mathrm{m/s}$$

Question 5

A stick of length $$L$$ and mass $$M is $$pivoted at one end and released from rest at an angle $$	heta$$ above the horizontal. What is the speed $$v of $$the free end when the stick hits the ground?

Accepted Answer

$$v = \sqrt{3gL \sin 	heta}$$

Question 6

A force $$F is $$applied at a distance $$r$$ from the axis of rotation at an angle $$	heta to $$the lever arm. Which expression gives the magnitude of the torque produced?

Accepted Answer

$$	au = r F \sin 	heta$$

Question 7

A solid sphere (mass $$M$$, radius $$R) $$rolls without slipping at speed $$v. $$What is its total kinetic energy?

Accepted Answer

$$\displaystyle \frac{1}{2} M v^2$ + \frac{1}{2} I \$$omega^2$$ = \frac{1}{2} M v^2$ + \frac{1}{2} \left( \frac{2}{5} M R^2$ \right) \left( \frac{v}{R} \$$right)^2$$ = \frac{7}{10} M v^2$

Question 8

A wheel of radius $$R$$ and mass $$M is $$acted on by a net torque $$	au. $$What is the resulting angular acceleration $$\alpha if $$the wheel is a solid disk?

Accepted Answer

$$\alpha = \frac{2	au}{M R^2$}$$

Question 9

A thin rod of length $$L$$ and mass $$M$$ rotates about an axis through one end perpendicular to its length. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{3} M L^2$

Question 10

A bug sits on the rim of a spinning disk of radius $$r = 0.2\,\mathrm{m}$$, rotating at a constant angular velocity $$\omega = 5\,\mathrm{rad/s}. $$What is the magnitude of the bug's acceleration?

Accepted Answer

$$a = r \$$omega^2$$ = 0.2 	imes 25 = 5.0\,\mathrm{m/s^2$}$$

Question 11

A wheel initially spinning at $$\$$omega_0$$ = 40\,\mathrm{rad/s} is $$brought to rest with a constant angular acceleration $$\alpha = -8\,\mathrm{$$rad/s^2$$}. $$Through what angle $$\Delta 	heta$$ does it rotate before stopping?

Accepted Answer

$$\Delta 	heta = \frac{\$$omega_0$$^2}{2|\alpha|} = \frac{1600}{16} = 100\,\mathrm{rad}$$

Question 12

A force $$F is $$applied tangentially to the rim of a pulley (solid disk, mass $$M$$, radius $$R). $$What is the resulting angular acceleration $$\alpha$$?

Accepted Answer

$$\alpha = \frac{F R}{\frac{1}{2} M R^2$} = \frac{2F}{M R}$$

Question 13

A rolling object has both translational and rotational kinetic energy. Which equation correctly expresses the total kinetic energy for a rolling object of mass $$M$$, center-of-mass speed $$v$$, and moment of inertia $$I$$?

Accepted Answer

$$KE_{tot} = \frac{1}{2} M v^2$ + \frac{1}{2} I \$$omega^2$$

Question 14

A thin rod of length $$L$$ and mass $$M$$ rotates about its center. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{12} M L^2$

Question 15

A force is applied at a point on a lever arm at a distance $$r$$ from the axis, but the force is not perpendicular to the lever arm. Which component of the force produces torque?

Accepted Answer

The component of the force perpendicular to the lever arm

Question 16

A hoop and a disk, both with mass $$M$$ and radius $$R$$, are released from rest at the top of an incline. Which has the greater final speed at the bottom and why?

Accepted Answer

The disk, because it has a smaller moment of inertia and thus more energy goes into translation.

Question 17

A rotating object has angular velocity $$\omega$$ and angular acceleration $$\alpha. If$$ $$\alpha is $$constant, which equation gives the angular displacement $$\Delta 	heta$$ after time $$t$$?

Accepted Answer

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

Question 18

A solid disk and a thin spherical shell, both with mass $$M$$ and radius $$R$$, are spun up to the same angular velocity $$\omega. $$Which has more rotational kinetic energy?

Accepted Answer

The thin spherical shell, because $$I_{shell} = \frac{2}{3} M R^2$ is greater than I$$_{disk} = \frac{1}{2} M $$R^2.$$

Question 19

A force $$F is $$applied at a distance $$r$$ from the axis of rotation, but at an angle $$	heta = 0^\circ (i.e$$., along the lever arm). What is the torque produced?

Accepted Answer

$$	au = r F \sin 0^\circ = 0$$

Rotational Motion: Concepts, Equations, and Applications

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