Q: A 70 kg skier starts from rest and glides down a frictionless 50 m slope inclined at 10°. What is the skier's speed at the bottom of the slope? (Use $$g = 9.8\ \mathrm{m/s^2$}.)$$

$$v = \sqrt{2g\Delta s\sin\theta} = \sqrt{2 \times 9.8 \times 50 \times \sin 10^\circ} \approx 13.3\ \mathrm{m/s}$$

Q: A 10 kg box is pulled 5 m across a horizontal floor by a constant force of 40 N at an angle of 30° above the horizontal. The coefficient of kinetic friction is 0.2. What is the net work done on the box?

$$W_{\text{net}} = (F\cos\theta - f_k)\Delta s = (40\cos 30^\circ - 0.2 \times 10 \times 9.8) \times 5 \approx 110\ \mathrm{J}$$

Q: A 150,000 kg rocket is launched vertically with a thrust of $$4.0 \times 10^6$ N. Ignoring air resistance, what is the rocket's speed after rising 500 m? (Use g = 9.8$$\ \mathrm{$$m/s^2$$}$$.)

$$v = \sqrt{\frac{2(F_{\text{thrust}} - mg)\Delta y}{m}} = \sqrt{\frac{2(4.0 \times 10^6$ - 1.47 \times 10^6$) \times 500}{150,000}} \approx 115\ \mathrm{m/s}$$

Q: A force $$\vec{F} = (3\hat{i} + 4\hat{j}) N $$acts on a particle that moves from the origin to the point $$(2, 1) m. $$What is the work done by the force?

$$W = \vec{F} \cdot \Delta \vec{s} = 3 \times 2 + 4 \times 1 = 10\ \mathrm{J}$$

Q: A 2.0 kg block is attached to a spring with $$k = 50\ \mathrm{N/m} on a $$horizontal surface ($$\mu_s = 0.60). $$The spring is stretched slowly until the block just begins to slip. How far is the spring stretched at that instant? ($$g = 9.8\ \mathrm{m/s^2$})$$

$$\Delta x = \frac{\mu_s mg}{k} = \frac{0.60 \times 2.0 \times 9.8}{50} = 0.24\ \mathrm{m}$$

Q: A 10 kg crate is pulled 3.0 m across a floor by a 30 N force. The coefficient of kinetic friction is 0.20. What is the increase in thermal energy of the crate-floor system?

$$\Delta E_{\text{th}} = f_k \Delta s = \mu_k mg \Delta s = 0.20 \times 10 \times 9.8 \times 3.0 = 59\ \mathrm{J}$$

Q: A force of 10 N acts at an angle of 60° to the direction of displacement of 5 m. What is the work done by the force?

$$W = F\Delta s \cos\theta = 10 \times 5 \times \cos 60^\circ = 25\ \mathrm{J}$$

Q: A 1500 kg car is towed from rest for 200 m by a variable force such that the total work done is 500,000 J. What is the car's final speed?

$$v = \sqrt{\frac{2W}{m}} = \sqrt{\frac{2 \times 500,000}{1500}} \approx 26\ \mathrm{m/s}$$

Question 1

A 70 kg skier starts from rest and glides down a frictionless 50 m slope inclined at 10°. What is the skier's speed at the bottom of the slope? (Use $$g = 9.8\ \mathrm{m/s^2$}.)$$

Accepted Answer

$$v = \sqrt{2g\Delta s\sin	heta} = \sqrt{2 	imes 9.8 	imes 50 	imes \sin 10^\circ} \approx 13.3\ \mathrm{m/s}$$

Question 2

A 10 kg box is pulled 5 m across a horizontal floor by a constant force of 40 N at an angle of 30° above the horizontal. The coefficient of kinetic friction is 0.2. What is the net work done on the box?

Accepted Answer

$$W_{	ext{net}} = (F\cos	heta - f_k)\Delta s = (40\cos 30^\circ - 0.2 	imes 10 	imes 9.8) 	imes 5 \approx 110\ \mathrm{J}$$

Question 3

A 150,000 kg rocket is launched vertically with a thrust of $$4.0 	imes 10^6$ N. Ignoring air resistance, what is the rocket's speed after rising 500 m? (Use g = 9.8$$\ \mathrm{$$m/s^2$$}$$.)

Accepted Answer

$$v = \sqrt{\frac{2(F_{	ext{thrust}} - mg)\Delta y}{m}} = \sqrt{\frac{2(4.0 	imes 10^6$ - 1.47 	imes 10^6$) 	imes 500}{150,000}} \approx 115\ \mathrm{m/s}$$

Question 4

A force $$\vec{F} = (3\hat{i} + 4\hat{j}) N $$acts on a particle that moves from the origin to the point $$(2, 1) m. $$What is the work done by the force?

Accepted Answer

$$W = \vec{F} \cdot \Delta \vec{s} = 3 	imes 2 + 4 	imes 1 = 10\ \mathrm{J}$$

Question 5

A 2.0 kg block is attached to a spring with $$k = 50\ \mathrm{N/m} on a $$horizontal surface ($$\mu_s = 0.60). $$The spring is stretched slowly until the block just begins to slip. How far is the spring stretched at that instant? ($$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$\Delta x = \frac{\mu_s mg}{k} = \frac{0.60 	imes 2.0 	imes 9.8}{50} = 0.24\ \mathrm{m}$$

Question 6

A 10 kg crate is pulled 3.0 m across a floor by a 30 N force. The coefficient of kinetic friction is 0.20. What is the increase in thermal energy of the crate-floor system?

Accepted Answer

$$\Delta E_{	ext{th}} = f_k \Delta s = \mu_k mg \Delta s = 0.20 	imes 10 	imes 9.8 	imes 3.0 = 59\ \mathrm{J}$$

Question 7

A force of 10 N acts at an angle of 60° to the direction of displacement of 5 m. What is the work done by the force?

Accepted Answer

$$W = F\Delta s \cos	heta = 10 	imes 5 	imes \cos 60^\circ = 25\ \mathrm{J}$$

Question 8

A 1500 kg car is towed from rest for 200 m by a variable force such that the total work done is 500,000 J. What is the car's final speed?

Accepted Answer

$$v = \sqrt{\frac{2W}{m}} = \sqrt{\frac{2 	imes 500,000}{1500}} \approx 26\ \mathrm{m/s}$$

Question 9

A light bulb is rated at 60 W. What does this rating mean?

Accepted Answer

The bulb dissipates energy at a rate of 60 J/s.

Question 10

A force $$\vec{F}$$ acts perpendicular to the displacement $$\Delta \vec{s} of a $$particle. What is the work done by the force?

Accepted Answer

$$W = \vec{F} \cdot \Delta \vec{s} = F\Delta s \cos 90^\circ = 0$$

Question 11

A 35 kg motor is lifted 3.0 m onto a truck in 8.0 s at constant speed. What is the average power output required? ($$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$P = \frac{mg\Delta y}{\Delta t} = \frac{35 	imes 9.8 	imes 3.0}{8.0} = 129\ \mathrm{W}$$

Question 12

A spring with $$k = 100\ \mathrm{N/m} is $$compressed 0.20 m from equilibrium. What is the work done by the spring as it returns to equilibrium?

Accepted Answer

$$W = \frac{1}{2}k(\Delta s_i^2$ - \Delta s_f^2$) = \frac{1}{2} 	imes 100 	imes (0.20^2$ - 0^2$) = 2.0\ \mathrm{J}$$

Question 13

A skier is gliding down a slope at constant speed. What energy transformation is taking place?

Accepted Answer

$$U ightarrow E_{	ext{th}}$$

Question 14

A box is pulled at constant speed across a rough floor. What can you say about the work done by friction?

Accepted Answer

The work done by friction is always negative.

Question 15

A force $$\vec{F}$$ acts on a particle moving with velocity $$\vec{v}. $$What is the instantaneous power delivered to the particle?

Accepted Answer

$$P = \vec{F} \cdot \vec{v}$$

Question 16

A car turns a quarter circle on a level road. What is the work done by static friction during the turn?

Accepted Answer

Zero, because the force is perpendicular to the displacement at every instant.

Question 17

Ball A has half the mass and eight times the kinetic energy of ball B. What is the ratio of their speeds $$v_A/v_B$$?

Accepted Answer

$$v_A/v_B = 4$$

Question 18

Which of the following best describes the energy transfer when a tow rope pulls a skier up a slope at constant speed?

Accepted Answer

Work is transferred into gravitational potential energy: $$W \rightarrow U$$

Chapter 9: Work and Kinetic Energy – Physics with Calculus Study Notes

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