Q: A 2 kg ball is moving at 5 m/s. What is its momentum?

$$p = mv = 2 \times 5 = 10\ \mathrm{kg\cdot m/s}$$

Q: A 0.5 kg hockey puck slides across ice at 8 m/s and is stopped by a stick in 0.2 s. What is the average force exerted by the stick?

$$F_{avg} = \frac{\Delta p}{\Delta t} = \frac{0.5 \times 8}{0.2} = 20\ \mathrm{N}$$

Q: A force of 15 N is applied perpendicular to a wrench 0.3 m from the bolt. What is the torque produced?

$$\tau = F r = 15 \times 0.3 = 4.5\ \mathrm{N\cdot m}$$

Q: A 5 kg object is lifted to a height of 3 m. What is its gravitational potential energy relative to the ground?

$$U = Mgh = 5 \times 9.8 \times 3 = 147\ \mathrm{J}$$

Q: A wheel accelerates from rest at a constant angular acceleration of $$2\ \mathrm{$$rad/s^2$$}$$ for 5 seconds. What is its final angular velocity?

$$\omega = \$$omega_0$$ + \alpha t = 0 + 2 \times 5 = 10\ \mathrm{rad/s}$$

Q: A 0.2 kg mass moves in a circle of radius 0.6 m at 3 m/s. What is its angular momentum about the center?

$$L = m v r = 0.2 \times 3 \times 0.6 = 0.36\ \mathrm{kg\cdot m^2$/s}$$

Q: A seesaw is balanced with a 30 kg child sitting 2 m from the pivot on one side. Where should a 20 kg child sit on the other side to balance the seesaw?

$$30 \times 2 = 20 \times x \implies x = 3\ \mathrm{m}$$

Q: A rotating platform (moment of inertia $$I = 5\ \mathrm{kg\cdot m^2$}) $$spins at $$4\ \mathrm{rad/s}. If a $$child sits on the edge and increases the moment of inertia to $$8\ \mathrm{kg\cdot m^2$}$$, what is the new angular speed (assuming no external torque)?

$$I_1$$ \$$omega_1$$ = $$I_2$$ \$$omega_2$$ \implies 5 \times 4 = 8 \times \$$omega_2$$ \implies \$$omega_2 = 2.5$$\ \mathrm{rad/s}$$

Question 1

A 2 kg ball is moving at 5 m/s. What is its momentum?

Accepted Answer

$$p = mv = 2 	imes 5 = 10\ \mathrm{kg\cdot m/s}$$

Question 2

A 0.5 kg hockey puck slides across ice at 8 m/s and is stopped by a stick in 0.2 s. What is the average force exerted by the stick?

Accepted Answer

$$F_{avg} = \frac{\Delta p}{\Delta t} = \frac{0.5 	imes 8}{0.2} = 20\ \mathrm{N}$$

Question 3

A disk of mass 3 kg and radius 0.4 m rotates at 10 rad/s. What is its rotational kinetic energy?

Accepted Answer

$$K_{rot} = \frac{1}{2} I \$$omega^2$$ = \frac{1}{2} \left(\frac{1}{2} m r^2$ight) \$$omega^2$$ = \frac{1}{4} 	imes 3 	imes 0.16 	imes 100 = 12\ \mathrm{J}$$

Question 4

A force of 15 N is applied perpendicular to a wrench 0.3 m from the bolt. What is the torque produced?

Accepted Answer

$$	au = F r = 15 	imes 0.3 = 4.5\ \mathrm{N\cdot m}$$

Question 5

A solid sphere (mass 2 kg, radius 0.5 m) rolls without slipping at 4 m/s. What is its total kinetic energy?

Accepted Answer

$$K_{total} = \frac{1}{2} M v_{cm}^2 + \frac{1}{2} I \$$omega^2$$ = \frac{1}{2} 	imes 2 	imes 16 + \frac{1}{2} 	imes \frac{2}{5} 	imes 2 	imes 0.25 	imes (8)^2$ = 16 + 3.2 = 19.2\ \mathrm{J}$$

Question 6

A 5 kg object is lifted to a height of 3 m. What is its gravitational potential energy relative to the ground?

Accepted Answer

$$U = Mgh = 5 	imes 9.8 	imes 3 = 147\ \mathrm{J}$$

Question 7

A wheel accelerates from rest at a constant angular acceleration of $$2\ \mathrm{$$rad/s^2$$}$$ for 5 seconds. What is its final angular velocity?

Accepted Answer

$$\omega = \$$omega_0$$ + \alpha t = 0 + 2 	imes 5 = 10\ \mathrm{rad/s}$$

Question 8

A 0.2 kg mass moves in a circle of radius 0.6 m at 3 m/s. What is its angular momentum about the center?

Accepted Answer

$$L = m v r = 0.2 	imes 3 	imes 0.6 = 0.36\ \mathrm{kg\cdot m^2$/s}$$

Question 9

A seesaw is balanced with a 30 kg child sitting 2 m from the pivot on one side. Where should a 20 kg child sit on the other side to balance the seesaw?

Accepted Answer

$$30 	imes 2 = 20 	imes x \implies x = 3\ \mathrm{m}$$

Question 10

A rotating platform (moment of inertia $$I = 5\ \mathrm{kg\cdot m^2$}) $$spins at $$4\ \mathrm{rad/s}. If a $$child sits on the edge and increases the moment of inertia to $$8\ \mathrm{kg\cdot m^2$}$$, what is the new angular speed (assuming no external torque)?

Accepted Answer

$$I_1$$ \$$omega_1$$ = $$I_2$$ \$$omega_2$$ \implies 5 	imes 4 = 8 	imes \$$omega_2$$ \implies \$$omega_2 = 2.5$$\ \mathrm{rad/s}$$

Question 11

A car tire (radius 0.3 m) rotates at 600 rpm. What is the tangential speed at the edge of the tire?

Accepted Answer

$$\omega = 600 	imes \frac{2\pi}{60} = 62.83\ \mathrm{rad/s}$$, $$v = r \omega = 0.3 	imes 62.83 = 18.85\ \mathrm{m/s}$$

Question 12

A force of 10 N is applied at an angle of $$30^\circ to a $$lever arm of 0.5 m. What is the torque produced?

Accepted Answer

$$	au = F r \sin 	heta = 10 	imes 0.5 	imes \sin 30^\circ = 2.5\ \mathrm{N\cdot m}$$

Question 13

A uniform rod of length 2 m and mass 4 kg is pivoted at one end. What is its moment of inertia about the pivot?

Accepted Answer

$$I = \frac{1}{3} m L^2$ = \frac{1}{3} 	imes 4 	imes 4 = 5.33\ \mathrm{kg\cdot m^2$}$$

Question 14

A flywheel is spinning at $$20\ \mathrm{rad/s}$$ and slows to rest in 10 seconds due to a constant frictional torque. What is the angular acceleration?

Accepted Answer

$$\alpha = \frac{\omega_f - \omega_i}{t} = \frac{0 - 20}{10} = -2\ \mathrm{$$rad/s^2$$}$$

Question 15

A 1 kg mass is attached to a string and swung in a horizontal circle of radius 0.8 m at 6 m/s. What is the magnitude of its angular velocity?

Accepted Answer

$$\omega = \frac{v}{r} = \frac{6}{0.8} = 7.5\ \mathrm{rad/s}$$

Question 16

A gymnast spins with arms extended (moment of inertia $$I_1) at $$angular speed $$\$$omega_1$$. She pulls her arms in, reducing her moment of inertia to I_2$ = 0.5 I_1$. What happens to her angular speed?

Accepted Answer

$$I_1$$ \$$omega_1$$ = $$I_2$$ \$$omega_2$$ \implies \$$omega_2 = 2$$ \$$omega_1$$

Question 17

A uniform solid cylinder (mass 10 kg, radius 0.2 m) is rolling without slipping at 2 m/s. What is its rotational kinetic energy?

Accepted Answer

$$K_{rot} = \frac{1}{2} I \$$omega^2$$ = \frac{1}{2} 	imes \frac{1}{2} m r^2$ 	imes \left(\frac{v}{r}\$$right)^2$$ = \frac{1}{4} 	imes 10 	imes 0.04 	imes 100 = 10\ \mathrm{J}$$

Question 18

A 0.5 kg ball is dropped from a height of 2 m. What is its speed just before hitting the ground? (Ignore air resistance, $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$v = \sqrt{2gh} = \sqrt{2 	imes 9.8 	imes 2} = 6.26\ \mathrm{m/s}$$

Rotational Motion, Momentum, and Energy: Study Notes

Study Guide - Practice Questions

Study Guide - Flashcards