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 thrown vertically upward with an initial speed of 10 m/s. What is the maximum height it reaches? (Ignore air resistance, $$g = 9.8 \ \mathrm{m/s^2$})$$

$$h = \frac{v^2$}{2g} = \frac{10^2$}{2 \times 9.8} = 5.1 \ \mathrm{m}$$

Q: A force of 10 N is applied to a box for 4 seconds, causing it to accelerate from rest. What is the impulse delivered to the box?

$$J = F \Delta t = 10 \times 4 = 40 \ \mathrm{N \cdot s}$$

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?

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

Q: A solid disk (mass 2 kg, radius 0.5 m) rotates about its center at 4 rad/s. What is its rotational kinetic energy?

$$KE_{rot} = \frac{1}{2} I \$$omega^2$$ = \frac{1}{2} \times \frac{1}{2}mr^2$ \times 4^2$ = \frac{1}{2} \times 0.25 \times 16 = 2 \ \mathrm{J}$$

Q: A uniform beam of length 6 m and mass 30 kg is supported at its center. A 20 kg mass is hung 2 m from one end. What is the force exerted by the support?

$$F_{support} = (30 + 20)g = 50 \times 9.8 = 490 \ \mathrm{N}$$

Q: A satellite orbits Earth at a height where the acceleration due to gravity is $$6.0 \ \mathrm{m/s^2$}. If $$Earth's radius is $$6.4 \times 10^6$ \ \mathrm{m}$$, what is the satellite's orbital radius?

$$r = R_E \sqrt{\frac{g_E}{g}} = 6.4 \times 10^6$ \sqrt{\frac{9.8}{6.0}} = 8.1 \times 10^6$ \ \mathrm{m}$$

Q: A 5 kg object is moving in a circle of radius 2 m at a constant speed of 6 m/s. What is the net force acting on the object?

$$F = m \frac{v^2$}{r} = 5 \times \frac{36}{2} = 90 \ \mathrm{N}$$

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

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 3

A 0.5 kg ball is thrown vertically upward with an initial speed of 10 m/s. What is the maximum height it reaches? (Ignore air resistance, $$g = 9.8 \ \mathrm{m/s^2$})$$

Accepted Answer

$$h = \frac{v^2$}{2g} = \frac{10^2$}{2 	imes 9.8} = 5.1 \ \mathrm{m}$$

Question 4

A force of 10 N is applied to a box for 4 seconds, causing it to accelerate from rest. What is the impulse delivered to the box?

Accepted Answer

$$J = F \Delta t = 10 	imes 4 = 40 \ \mathrm{N \cdot s}$$

Question 5

A 3 kg object moving at 4 m/s collides elastically with a stationary 2 kg object. What is the velocity of the 3 kg object after the collision?

Accepted Answer

$$v_1$$' = \frac{$$m_1$$ - $$m_2$$}{$$m_1$$ + $$m_2$$} $$v_1$$ = \frac{3 - 2}{3 + 2} 	imes 4 = 0.8 \ \mathrm{m/s}$$

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

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

Question 7

A solid disk (mass 2 kg, radius 0.5 m) rotates about its center at 4 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}mr^2$ 	imes 4^2$ = \frac{1}{2} 	imes 0.25 	imes 16 = 2 \ \mathrm{J}$$

Question 8

A uniform beam of length 6 m and mass 30 kg is supported at its center. A 20 kg mass is hung 2 m from one end. What is the force exerted by the support?

Accepted Answer

$$F_{support} = (30 + 20)g = 50 	imes 9.8 = 490 \ \mathrm{N}$$

Question 9

A satellite orbits Earth at a height where the acceleration due to gravity is $$6.0 \ \mathrm{m/s^2$}. If $$Earth's radius is $$6.4 	imes 10^6$ \ \mathrm{m}$$, what is the satellite's orbital radius?

Accepted Answer

$$r = R_E \sqrt{\frac{g_E}{g}} = 6.4 	imes 10^6$ \sqrt{\frac{9.8}{6.0}} = 8.1 	imes 10^6$ \ \mathrm{m}$$

Question 10

A 5 kg object is moving in a circle of radius 2 m at a constant speed of 6 m/s. What is the net force acting on the object?

Accepted Answer

$$F = m \frac{v^2$}{r} = 5 	imes \frac{36}{2} = 90 \ \mathrm{N}$$

Question 11

A 0.1 kg bullet is fired into a 2 kg wooden block at rest on a frictionless surface. The bullet embeds in the block, and together they move at 3 m/s. What was the initial speed of the bullet?

Accepted Answer

$$v_{bullet} = \frac{(m_{block} + m_{bullet})v_{final}}{m_{bullet}} = \frac{(2 + 0.1) 	imes 3}{0.1} = 63 \ \mathrm{m/s}$$

Question 12

A 10 kg child sits on a seesaw 2 m from the pivot. Where should a 20 kg adult sit to balance the seesaw?

Accepted Answer

$$m_1$$ $$d_1$$ = $$m_2$$ $$d_2$$ \Rightarrow 10 	imes 2 = 20 	imes $$d_2$$ \Rightarrow $$d_2 = 1$$ \ \mathrm{m}$$

Question 13

A 0.5 kg mass is attached to a spring with a force constant $$k = 200 \ \mathrm{N/m}. $$What is the period of oscillation?

Accepted Answer

$$T = 2\pi \sqrt{\frac{m}{k}} = 2\pi \sqrt{\frac{0.5}{200}} = 0.314 \ \mathrm{s}$$

Question 14

A 1.5 kg object is moving at 8 m/s. What is its momentum?

Accepted Answer

$$p = mv = 1.5 	imes 8 = 12 \ \mathrm{kg \cdot m/s}$$

Question 15

A 0.3 kg ball is dropped from a height of 2 m and rebounds to a height of 1.5 m. What is the coefficient of restitution?

Accepted Answer

$$e = \sqrt{\frac{h_{rebound}}{h_{drop}}} = \sqrt{\frac{1.5}{2}} = 0.866$$

Question 16

A 4 kg object is rotating at 2 rad/s about an axis 0.5 m from its center. What is its angular momentum?

Accepted Answer

$$L = I\omega = (mr^2$)\omega = 4 	imes 0.5^2$ 	imes 2 = 2 \ \mathrm{kg \cdot m^2$/s}$$

Question 17

A 2 kg mass is attached to a string and swung in a vertical circle of radius 1 m. What is the minimum speed at the top of the circle to keep the string taut?

Accepted Answer

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

Force and Motion, Dynamics, and Motion in a Plane: Study Notes for Chapters 5-8

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