Q: A wheel of radius 5 m accelerates from 60 RPM to 180 RPM in 2 s. What is its angular acceleration in $$\text{rad/s}^2$$?

$$\alpha = 10.47 \, \text{rad/s}^2$$

Q: A 20 kg, 4 m long rod is fixed at one end and points 37° above the horizontal. An 80 kg person stands at the other end. What is the net torque due to both weights about the axis? (Assume $$g = 9.8 \text{ m/s}^2$$ and uniform rod.)

$$\tau_{\text{net}} = (20 \times 9.8 \times 2) \sin(37^\circ) + (80 \times 9.8 \times 4) \sin(37^\circ)$$

Q: A 1,000 kg disc (radius 5 m) is mounted on a vertical axle (80 kg, radius 1 m). If a 200 N friction force is applied to the axle at 70 rad/s, how many revolutions until the system stops? (Disc is not a solid cylinder.)

$$N = \frac{I_{\text{total}} \$$omega_0$$}{2 \pi \tau}$$

Q: A 3 kg and 4 kg mass are attached to the ends of a 2-m rod. What is the moment of inertia about the center?

$$I = 3 \times 1^2$ + 4 \times 1^2$ = 7 \text{ kg m}^2$$

Question 1

A car tire of radius 0.40 m rotates 10,000 times while driving. How far did the car travel in meters? (Assume no slipping.)

Accepted Answer

$$2,512 \, 	ext{m}$$

Question 2

A wheel of radius 5 m accelerates from 60 RPM to 180 RPM in 2 s. What is its angular acceleration in $$	ext{rad/s}^2$$?

Accepted Answer

$$\alpha = 10.47 \, 	ext{rad/s}^2$$

Question 3

A solid disc of mass $$M = 100 	ext{ kg}$$ and radius $$R = 2 	ext{ m} is $$free to rotate about a fixed axis. You push tangentially with a force $$F = 50 	ext{ N}. $$What is the angular acceleration $$\alpha of $$the disc?

Accepted Answer

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

Question 4

A 20 kg, 4 m long rod is fixed at one end and points 37° above the horizontal. An 80 kg person stands at the other end. What is the net torque due to both weights about the axis? (Assume $$g = 9.8 	ext{ m/s}^2$$ and uniform rod.)

Accepted Answer

$$	au_{	ext{net}} = (20 	imes 9.8 	imes 2) \sin(37^\circ) + (80 	imes 9.8 	imes 4) \sin(37^\circ)$$

Question 5

A flywheel is a solid disc of mass $$8 	imes 10^4$ 	ext{ kg}$$ and diameter $$5.0 	ext{ m}. $$What is the maximum rotational kinetic energy it can store at $$120 	ext{ RPM}$$?

Accepted Answer

$$K = \frac{1}{2} I \$$omega^2$$, \quad I = \frac{1}{2} M R^2$, \quad \omega = \frac{2\pi 	imes 120}{60}$

Question 6

A solid sphere of mass $$2 	ext{ kg}$$ and radius $$0.3 	ext{ m}$$ rolls without slipping at $$10 	ext{ m/s}. $$What is its total kinetic energy?

Accepted Answer

$$K = \frac{1}{2} M v^2$ + \frac{1}{2} I \$$omega^2$$, \quad I = \frac{2}{5} M R^2$, \quad \omega = \frac{v}{R}$

Question 7

A 1,000 kg disc (radius 5 m) is mounted on a vertical axle (80 kg, radius 1 m). If a 200 N friction force is applied to the axle at 70 rad/s, how many revolutions until the system stops? (Disc is not a solid cylinder.)

Accepted Answer

$$N = \frac{I_{	ext{total}} \$$omega_0$$}{2 \pi 	au}$$

Question 8

A solid disc (mass $$M$$, radius $$R) is $$spinning at $$\$$omega_0$$. A second disc (mass m$, radius r$) is dropped on top and they spin together. What is the final angular speed if the second disc was initially at rest?

Accepted Answer

$$\omega_f = \frac{I_1$ \$$omega_0$$}{I_1$ + I_2$}$$

Question 9

A 4 m long blade starts from rest and spins with $$3 	ext{ rad/s}^2$$ for 10 s. What is the tangential speed at the tip after 10 s?

Accepted Answer

$$v = r \alpha t$$

Question 10

A 2-kg object spins in a circle of radius 4 m at $$3 	ext{ rad/s}. $$What is its kinetic energy using rotational variables?

Accepted Answer

$$K = \frac{1}{2} I \$$omega^2$$, \quad I = m r^2$

Question 11

A bicycle's back wheel (radius 0.7 m) spins at 15 m/s. What is its angular speed?

Accepted Answer

$$\omega = \frac{v}{r}$$

Question 12

A torque of $$100 	ext{ Nm} is $$needed to tighten a bolt with a 20-cm wrench. What is the minimum force required if you pull perpendicular to the wrench?

Accepted Answer

$$F = \frac{	au}{r}$$

Question 13

A solid sphere (mass $$M$$, radius $$R) $$rolls without slipping down an incline of height $$h. $$What is its speed at the bottom?

Accepted Answer

$$v = \sqrt{\frac{10}{7} g h}$$

Question 14

A 3 kg and 4 kg mass are attached to the ends of a 2-m rod. What is the moment of inertia about the center?

Accepted Answer

$$I = 3 	imes 1^2$ + 4 	imes 1^2$ = 7 	ext{ kg m}^2$$

Question 15

A 150-g baseball (radius 3.85 cm) leaves the pitcher’s hand at 30 m/s and spins at 20 rad/s. What is its total kinetic energy?

Accepted Answer

$$K = \frac{1}{2} m v^2$ + \frac{1}{2} I \$$omega^2$$, \quad I = \frac{2}{5} m r^2$

Question 16

A solid disc (mass $$M$$, radius $$R) is $$pulled by a rope wrapped around it. If you pull with force $$F$$ and the rope unwinds $$d$$ meters, what is the disc’s angular speed after pulling?

Accepted Answer

$$\omega = \sqrt{\frac{2 F d}{\frac{1}{2} M R^2$}}$$

Rotational Motion: Principles, Equations, and Applications

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