Q: A 2 kg block is released from rest at a height of 5 m above the ground. Neglecting air resistance, what is the speed of the block just before it hits the ground? (Use $$g = 9.8\ \mathrm{m/s^2$})$$

$$v = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 5} = 9.9\ \mathrm{m/s}$$

Q: A spring with spring constant $$k = 200\ \mathrm{N/m} is $$compressed by $$0.1\ \mathrm{m}. $$What is the elastic potential energy stored in the spring?

$$U_s = \frac{1}{2}k(\Delta s)^2$ = \frac{1}{2} \times 200 \times (0.1)^2$ = 1\ \mathrm{J}$$

Q: A 1 kg object is attached to a spring and oscillates horizontally on a frictionless surface. If the spring constant is $$k = 100\ \mathrm{N/m}$$ and the amplitude is $$0.2\ \mathrm{m}$$, what is the maximum kinetic energy of the object?

$$K_{max} = \frac{1}{2}kA^2$ = \frac{1}{2} \times 100 \times (0.2)^2$ = 2\ \mathrm{J}$$

Q: A disk of mass $$2\ \mathrm{kg}$$ and radius $$0.5\ \mathrm{m}$$ rotates about its center with angular velocity $$\omega = 4\ \mathrm{rad/s}. $$What is its rotational kinetic energy?

$$K_{rot} = \frac{1}{2}I\$$omega^2$$ = \frac{1}{2} \times (\frac{1}{2} \times 2 \times 0.5^2$) \times 16 = 2\ \mathrm{J}$$

Q: A 3 kg object moves in a circle of radius $$2\ \mathrm{m} at a $$constant speed of $$6\ \mathrm{m/s}. $$What is its angular momentum about the center?

$$L = mvr = 3 \times 6 \times 2 = 36\ \mathrm{kg\cdot m^2$/s}$$

Q: A 0.2 kg ball is dropped from a height of $$10\ \mathrm{m}$$ and rebounds to a height of $$8\ \mathrm{m}. $$What is the change in the ball's mechanical energy due to the collision with the ground?

$$\Delta E = m g (h_{final} - h_{initial}) = 0.2 \times 9.8 \times (8 - 10) = -3.92\ \mathrm{J}$$

Q: A force of $$10\ \mathrm{N} is $$applied tangentially to the rim of a wheel of radius $$0.3\ \mathrm{m}. $$What is the torque produced about the center?

$$\tau = rF = 0.3 \times 10 = 3\ \mathrm{N\cdot m}$$

Question 1

A 2 kg block is released from rest at a height of 5 m above the ground. Neglecting air resistance, what is the speed of the block just before it hits the ground? (Use $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$v = \sqrt{2gh} = \sqrt{2 	imes 9.8 	imes 5} = 9.9\ \mathrm{m/s}$$

Question 2

A spring with spring constant $$k = 200\ \mathrm{N/m} is $$compressed by $$0.1\ \mathrm{m}. $$What is the elastic potential energy stored in the spring?

Accepted Answer

$$U_s = \frac{1}{2}k(\Delta s)^2$ = \frac{1}{2} 	imes 200 	imes (0.1)^2$ = 1\ \mathrm{J}$$

Question 3

A 0.5 kg ball moving at $$4\ \mathrm{m/s}$$ collides elastically with a stationary 0.5 kg ball. What is the velocity of the first ball after the collision?

Accepted Answer

$$0\ \mathrm{m/s} ($$the first ball stops, the second moves at $$4\ \mathrm{m/s})$$

Question 4

A force $$F(x) = 3x^2$ (in N) acts on a particle as it moves from x = 0$ to x = 2$$\ \mathrm{m}$$. What is the work done by this force?

Accepted Answer

$$W = \$$int_0$$^2 3x^2$ dx = [x^3$]_0^2$ = 8\ \mathrm{J}$$

Question 5

A 1 kg object is attached to a spring and oscillates horizontally on a frictionless surface. If the spring constant is $$k = 100\ \mathrm{N/m}$$ and the amplitude is $$0.2\ \mathrm{m}$$, what is the maximum kinetic energy of the object?

Accepted Answer

$$K_{max} = \frac{1}{2}kA^2$ = \frac{1}{2} 	imes 100 	imes (0.2)^2$ = 2\ \mathrm{J}$$

Question 6

A disk of mass $$2\ \mathrm{kg}$$ and radius $$0.5\ \mathrm{m}$$ rotates about its center with angular velocity $$\omega = 4\ \mathrm{rad/s}. $$What is its rotational kinetic energy?

Accepted Answer

$$K_{rot} = \frac{1}{2}I\$$omega^2$$ = \frac{1}{2} 	imes (\frac{1}{2} 	imes 2 	imes 0.5^2$) 	imes 16 = 2\ \mathrm{J}$$

Question 7

A 3 kg object moves in a circle of radius $$2\ \mathrm{m} at a $$constant speed of $$6\ \mathrm{m/s}. $$What is its angular momentum about the center?

Accepted Answer

$$L = mvr = 3 	imes 6 	imes 2 = 36\ \mathrm{kg\cdot m^2$/s}$$

Question 8

A 0.2 kg ball is dropped from a height of $$10\ \mathrm{m}$$ and rebounds to a height of $$8\ \mathrm{m}. $$What is the change in the ball's mechanical energy due to the collision with the ground?

Accepted Answer

$$\Delta E = m g (h_{final} - h_{initial}) = 0.2 	imes 9.8 	imes (8 - 10) = -3.92\ \mathrm{J}$$

Question 9

A force of $$10\ \mathrm{N} is $$applied tangentially to the rim of a wheel of radius $$0.3\ \mathrm{m}. $$What is the torque produced about the center?

Accepted Answer

$$	au = rF = 0.3 	imes 10 = 3\ \mathrm{N\cdot m}$$

Question 10

A 4 kg object is moving at $$5\ \mathrm{m/s}$$ and collides inelastically with a stationary 2 kg object. What is the final velocity of the combined mass after the collision?

Accepted Answer

$$v_f = \frac{m_1$ v_1$ + m_2$ v_2$}{m_1$ + m_2$} = \frac{4 	imes 5 + 2 	imes 0}{4 + 2} = 3.33\ \mathrm{m/s}$$

Question 11

A uniform rod of length $$2\ \mathrm{m}$$ and mass $$3\ \mathrm{kg} is $$pivoted at one end. What is its moment of inertia about the pivot?

Accepted Answer

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

Question 12

A car of mass $$1000\ \mathrm{kg}$$ accelerates from rest to $$20\ \mathrm{m/s} in$$ $$10\ \mathrm{s}. $$What is the average power delivered to the car?

Accepted Answer

$$P = \frac{\Delta K}{\Delta t} = \frac{\frac{1}{2} m v^2$}{t} = \frac{0.5 	imes 1000 	imes 400}{10} = 20,000\ \mathrm{W}$$

Question 13

A 0.5 kg ball is attached to a string and swung in a horizontal circle of radius $$1.5\ \mathrm{m} at a $$speed of $$3\ \mathrm{m/s}. $$What is the tension in the string?

Accepted Answer

$$T = m \frac{v^2$}{r} = 0.5 	imes \frac{9}{1.5} = 3\ \mathrm{N}$$

Question 14

A 2 kg object is moving at $$10\ \mathrm{m/s}$$ and comes to rest due to friction after traveling $$25\ \mathrm{m}. $$What is the work done by friction?

Accepted Answer

$$W = \Delta K = 0 - \frac{1}{2} m v^2$ = -100\ \mathrm{J}$$

Question 15

A solid sphere of mass $$5\ \mathrm{kg}$$ and radius $$0.2\ \mathrm{m}$$ rolls without slipping down a $$2\ \mathrm{m}$$ high incline. What is its speed at the bottom? (Use $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$v = \sqrt{\frac{10}{7}gh} = \sqrt{\frac{10}{7} 	imes 9.8 	imes 2} \approx 5.3\ \mathrm{m/s}$$

Question 16

A 0.1 kg bullet is fired at $$500\ \mathrm{m/s}$$ into a stationary 2 kg wooden block resting on a frictionless surface. The bullet embeds in the block. What is the final velocity of the block-bullet system?

Accepted Answer

$$v_f = \frac{m_{bullet} v_{bullet}}{m_{bullet} + m_{block}} = \frac{0.1 	imes 500}{2.1} \approx 23.8\ \mathrm{m/s}$$

Question 17

A force $$F = 50\ \mathrm{N} is $$applied at an angle of $$30^\circ to $$the horizontal to push a box $$4\ \mathrm{m}$$ along a frictionless floor. What is the work done by the force?

Accepted Answer

$$W = Fd \cos	heta = 50 	imes 4 	imes \cos 30^\circ = 50 	imes 4 	imes 0.866 = 173.2\ \mathrm{J}$$

Question 18

A wheel starts from rest and accelerates at $$2\ \mathrm{$$rad/s^2$$}$$ for $$5\ \mathrm{s}. $$What is its angular displacement during this time?

Accepted Answer

$$	heta = \$$omega_0$$ t + \frac{1}{2}\alpha t^2$ = 0 + 0.5 	imes 2 	imes 25 = 25\ \mathrm{rad}$$

Question 19

A 1 kg mass is attached to a string and swung in a vertical circle of radius $$1\ \mathrm{m}. $$What is the minimum speed at the top of the circle to keep the string taut?

Accepted Answer

$$v_{min} = \sqrt{g r} = \sqrt{9.8 	imes 1} = 3.13\ \mathrm{m/s}$$

Study Guide: Energy, Work, Momentum, and Rotational Motion (Chapters 9–12)

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