Q: A 2 kg block slides down a frictionless incline of height 5 m. What is the speed of the block at the bottom of the incline? (Assume $$g = 9.8 \ \mathrm{m/s^2$})$$

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

Q: A car of mass 1200 kg is traveling at 20 m/s. What is its kinetic energy?

$$KE = \frac{1}{2}mv^2$ = \frac{1}{2} \times 1200 \times 20^2$ = 240,000 \ \mathrm{J}$$

Q: A 0.5 kg ball is dropped from a height of 10 m. Ignoring air resistance, what is the potential energy lost by the ball when it reaches the ground?

$$\Delta PE = mgh = 0.5 \times 9.8 \times 10 = 49 \ \mathrm{J}$$

Q: A 3 kg object moves in a circle of radius 2 m at a constant speed of 4 m/s. What is the magnitude of the centripetal force acting on the object?

$$F_c = \frac{mv^2$}{r} = \frac{3 \times 16}{2} = 24 \ \mathrm{N}$$

Q: A 0.2 kg ball is attached to a string and swung in a horizontal circle of radius 1.5 m at a speed of 3 m/s. What is the tension in the string?

$$T = \frac{mv^2$}{r} = \frac{0.2 \times 9}{1.5} = 1.2 \ \mathrm{N}$$

Q: A 5 kg object is moving in a circle of radius 4 m with a period of 2 s. What is the centripetal acceleration?

$$a_c = \frac{4\pi^2$ r}{T^2$} = \frac{4 \times \pi^2$ \times 4}{4} = \pi^2$ \approx 9.87 \ \mathrm{m/s^2$}$$

Q: A 2 kg object is rotating at 10 rad/s with a radius of 0.5 m. What is its angular momentum?

$$L = mvr = m(\omega r)r = m\omega r^2$ = 2 \times 10 \times 0.5^2$ = 5 \ \mathrm{kg \cdot m^2$/s}$$

Question 1

A 2 kg block slides down a frictionless incline of height 5 m. What is the speed of the block at the bottom of the incline? (Assume $$g = 9.8 \ \mathrm{m/s^2$})$$

Accepted Answer

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

Question 2

Which of the following best describes the work done by a constant force $$\vec{F}$$ acting over a displacement $$\vec{d}$$?

Accepted Answer

$$W = Fd \cos 	heta$$

Question 3

A car of mass 1200 kg is traveling at 20 m/s. What is its kinetic energy?

Accepted Answer

$$KE = \frac{1}{2}mv^2$ = \frac{1}{2} 	imes 1200 	imes 20^2$ = 240,000 \ \mathrm{J}$$

Question 4

A 0.5 kg ball is dropped from a height of 10 m. Ignoring air resistance, what is the potential energy lost by the ball when it reaches the ground?

Accepted Answer

$$\Delta PE = mgh = 0.5 	imes 9.8 	imes 10 = 49 \ \mathrm{J}$$

Question 5

A 3 kg object moves in a circle of radius 2 m at a constant speed of 4 m/s. What is the magnitude of the centripetal force acting on the object?

Accepted Answer

$$F_c = \frac{mv^2$}{r} = \frac{3 	imes 16}{2} = 24 \ \mathrm{N}$$

Question 6

A 0.2 kg ball is attached to a string and swung in a horizontal circle of radius 1.5 m at a speed of 3 m/s. What is the tension in the string?

Accepted Answer

$$T = \frac{mv^2$}{r} = \frac{0.2 	imes 9}{1.5} = 1.2 \ \mathrm{N}$$

Question 7

A 5 kg object is moving in a circle of radius 4 m with a period of 2 s. What is the centripetal acceleration?

Accepted Answer

$$a_c = \frac{4\pi^2$ r}{T^2$} = \frac{4 	imes \pi^2$ 	imes 4}{4} = \pi^2$ \approx 9.87 \ \mathrm{m/s^2$}$$

Question 8

A 0.1 kg bullet is fired horizontally at 300 m/s into a 2 kg wooden block initially at rest on a frictionless surface. The bullet embeds in the block. What is the final velocity of the block-bullet system?

Accepted Answer

$$v_f = \frac{m_1$ v_1$ + m_2$ v_2$}{m_1$ + m_2$} = \frac{0.1 	imes 300 + 2 	imes 0}{0.1 + 2} = \frac{30}{2.1} \approx 14.3 \ \mathrm{m/s}$$

Question 9

A 60 kg astronaut pushes off a 180 kg satellite in space. If the astronaut moves away at 3 m/s, what is the velocity of the satellite?

Accepted Answer

$$v_{sat} = -\frac{m_{astr}}{m_{sat}} v_{astr} = -\frac{60}{180} 	imes 3 = -1 \ \mathrm{m/s}$$

Question 10

A 0.5 kg ball moving at 4 m/s collides elastically with a stationary 0.5 kg ball. What are their velocities after the collision?

Accepted Answer

The first ball stops, and the second ball moves at 4 m/s.

Question 11

A 2 kg object is rotating at 10 rad/s with a radius of 0.5 m. What is its angular momentum?

Accepted Answer

$$L = mvr = m(\omega r)r = m\omega r^2$ = 2 	imes 10 	imes 0.5^2$ = 5 \ \mathrm{kg \cdot m^2$/s}$$

Question 12

A disk of mass 4 kg and radius 0.3 m rotates at 20 rad/s. What is its rotational kinetic energy?

Accepted Answer

$$KE_{rot} = \frac{1}{2} I \$$omega^2$$ = \frac{1}{2} 	imes \frac{1}{2} m r^2$ 	imes \$$omega^2$$ = \frac{1}{4} 	imes 4 	imes 0.09 	imes 400 = 36 \ \mathrm{J}$$

Question 13

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

Accepted Answer

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

Question 14

A solid sphere of mass 2 kg and radius 0.1 m rolls without slipping down a 2 m high incline. What is its speed at the bottom?

Accepted Answer

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

Question 15

A 1.5 kg mass is attached to a spring with a force constant $$k = 200 \ \mathrm{N/m}. $$What is the period of oscillation?

Accepted Answer

$$T = 2\pi \sqrt{\frac{m}{k}} = 2\pi \sqrt{\frac{1.5}{200}} \approx 0.545 \ \mathrm{s}$$

Question 16

A pendulum of length 0.8 m swings with small amplitude. What is its period?

Accepted Answer

$$T = 2\pi \sqrt{\frac{L}{g}} = 2\pi \sqrt{\frac{0.8}{9.8}} \approx 1.79 \ \mathrm{s}$$

Question 17

A 0.3 kg mass oscillates on a spring with amplitude 0.2 m and period 0.5 s. What is the maximum speed of the mass?

Accepted Answer

$$v_{max} = \omega A = \frac{2\pi}{T} 	imes A = \frac{2\pi}{0.5} 	imes 0.2 \approx 2.51 \ \mathrm{m/s}$$

Physics with Calculus: Force, Motion, and Dynamics (Chapters 5-8) – Study Guide

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