Q: A 950 kg elevator is ascending with an upward acceleration of $$0.25 \, \mathrm{m/s^2$}. $$What is the tension in the elevator cable?

$$T = 950 \times (9.8 + 0.25) = 10,307.5 \, \mathrm{N}$$

Q: A gasoline engine has a thermal efficiency of $$27.0\%. If $$the engine absorbs $$Q_h = 2,000 \, \mathrm{J} of $$heat in each cycle, how much work is done by the engine in each cycle?

$$W = 0.27 \times 2,000 = 540 \, \mathrm{J}$$

Q: A 10 kg aluminum cylinder at $$80^\circ \mathrm{C} is to be $$cooled to $$30^\circ \mathrm{C}. $$Specific heat of aluminum is $$900 \, \mathrm{J/(kg \cdot ^\circ C)}. $$How much heat must be removed?

$$Q = mc\Delta T = 10 \times 900 \times (80 - 30) = 450,000 \, \mathrm{J}$$

Q: How much heat must be added to $$2.0 \, \mathrm{kg} of $$ice at $$0^\circ \mathrm{C} to $$completely melt it? (Latent heat of fusion for ice is $$334,000 \, \mathrm{J/kg})$$

$$Q = mL = 2.0 \times 334,000 = 668,000 \, \mathrm{J}$$

Q: A child and sled with a combined mass of $$50 \, \mathrm{kg}$$ slide down a $$10.0 \, \mathrm{m}$$ high snow-covered hill. If friction does $$2,000 \, \mathrm{J} of $$work, what is the speed at the bottom?

$$v = \sqrt{2(g h - W_f/m)} = \sqrt{2(9.8 \times 10 - 2,000/50)} = \sqrt{2(98 - 40)} = \sqrt{116} \approx 10.8 \, \mathrm{m/s}$$

Q: A $$1.5 \, \mathrm{kg}$$ block at the top of a $$20.0 \, \mathrm{m}$$ incline slides down. If $$100 \, \mathrm{J} is $$lost to friction, how much kinetic energy does the block have at the bottom?

$$KE = mgh - W_f = 1.5 \times 9.8 \times 20 - 100 = 294 - 100 = 194 \, \mathrm{J}$$

Q: A $$0.50 \, \mathrm{kg}$$ ball is dropped from a height of $$15 \, \mathrm{m}. $$How much work is done by friction if the ball reaches the ground with a speed of $$10 \, \mathrm{m/s}$$?

$$W_f = mgh - \frac{1}{2}mv^2$ = 0.5 \times 9.8 \times 15 - 0.5 \times 0.5 \times 10^2$ = 73.5 - 25 = 48.5 \, \mathrm{J}$$

Q: A $$0.40 \, \mathrm{kg}$$ ball is attached to a spring (spring constant $$k = 80 \, \mathrm{N/m}) $$and compressed $$0.15 \, \mathrm{m}. $$What is the speed of the ball when the spring returns to equilibrium?

$$v = \sqrt{\frac{kx^2$}{m}} = \sqrt{\frac{80 \times 0.15^2$}{0.40}} = \sqrt{4.5} \approx 2.12 \, \mathrm{m/s}$$

Q: A $$0.50 \, \mathrm{kg}$$ ball is thrown vertically upward with an initial speed of $$8.0 \, \mathrm{m/s}. $$What is the maximum height reached by the ball?

$$h = \frac{v^2$}{2g} = \frac{8.0^2$}{2 \times 9.8} = \frac{64}{19.6} \approx 3.27 \, \mathrm{m}$$

Q: A $$100 \, \mathrm{kg}$$ ball is dropped from a window $$25.0 \, \mathrm{m}$$ above the ground. If $$1,000 \, \mathrm{J} of $$energy is lost to air resistance, what is the speed of the ball just before it hits the ground?

$$v = \sqrt{2(g h - W_f/m)} = \sqrt{2(9.8 \times 25 - 1,000/100)} = \sqrt{2(245 - 10)} = \sqrt{470} \approx 21.7 \, \mathrm{m/s}$$

Question 1

A 950 kg elevator is ascending with an upward acceleration of $$0.25 \, \mathrm{m/s^2$}. $$What is the tension in the elevator cable?

Accepted Answer

$$T = 950 	imes (9.8 + 0.25) = 10,307.5 \, \mathrm{N}$$

Question 2

A gasoline engine has a thermal efficiency of $$27.0\%. If $$the engine absorbs $$Q_h = 2,000 \, \mathrm{J} of $$heat in each cycle, how much work is done by the engine in each cycle?

Accepted Answer

$$W = 0.27 	imes 2,000 = 540 \, \mathrm{J}$$

Question 3

A 10 kg aluminum cylinder at $$80^\circ \mathrm{C} is to be $$cooled to $$30^\circ \mathrm{C}. $$Specific heat of aluminum is $$900 \, \mathrm{J/(kg \cdot ^\circ C)}. $$How much heat must be removed?

Accepted Answer

$$Q = mc\Delta T = 10 	imes 900 	imes (80 - 30) = 450,000 \, \mathrm{J}$$

Question 4

How much heat must be added to $$2.0 \, \mathrm{kg} of $$ice at $$0^\circ \mathrm{C} to $$completely melt it? (Latent heat of fusion for ice is $$334,000 \, \mathrm{J/kg})$$

Accepted Answer

$$Q = mL = 2.0 	imes 334,000 = 668,000 \, \mathrm{J}$$

Question 5

A child and sled with a combined mass of $$50 \, \mathrm{kg}$$ slide down a $$10.0 \, \mathrm{m}$$ high snow-covered hill. If friction does $$2,000 \, \mathrm{J} of $$work, what is the speed at the bottom?

Accepted Answer

$$v = \sqrt{2(g h - W_f/m)} = \sqrt{2(9.8 	imes 10 - 2,000/50)} = \sqrt{2(98 - 40)} = \sqrt{116} \approx 10.8 \, \mathrm{m/s}$$

Question 6

A $$1.5 \, \mathrm{kg}$$ block at the top of a $$20.0 \, \mathrm{m}$$ incline slides down. If $$100 \, \mathrm{J} is $$lost to friction, how much kinetic energy does the block have at the bottom?

Accepted Answer

$$KE = mgh - W_f = 1.5 	imes 9.8 	imes 20 - 100 = 294 - 100 = 194 \, \mathrm{J}$$

Question 7

A $$0.50 \, \mathrm{kg}$$ ball is dropped from a height of $$15 \, \mathrm{m}. $$How much work is done by friction if the ball reaches the ground with a speed of $$10 \, \mathrm{m/s}$$?

Accepted Answer

$$W_f = mgh - \frac{1}{2}mv^2$ = 0.5 	imes 9.8 	imes 15 - 0.5 	imes 0.5 	imes 10^2$ = 73.5 - 25 = 48.5 \, \mathrm{J}$$

Question 8

A $$0.40 \, \mathrm{kg}$$ ball is attached to a spring (spring constant $$k = 80 \, \mathrm{N/m}) $$and compressed $$0.15 \, \mathrm{m}. $$What is the speed of the ball when the spring returns to equilibrium?

Accepted Answer

$$v = \sqrt{\frac{kx^2$}{m}} = \sqrt{\frac{80 	imes 0.15^2$}{0.40}} = \sqrt{4.5} \approx 2.12 \, \mathrm{m/s}$$

Question 9

A $$0.50 \, \mathrm{kg}$$ ball is thrown vertically upward with an initial speed of $$8.0 \, \mathrm{m/s}. $$What is the maximum height reached by the ball?

Accepted Answer

$$h = \frac{v^2$}{2g} = \frac{8.0^2$}{2 	imes 9.8} = \frac{64}{19.6} \approx 3.27 \, \mathrm{m}$$

Question 10

A $$100 \, \mathrm{kg}$$ ball is dropped from a window $$25.0 \, \mathrm{m}$$ above the ground. If $$1,000 \, \mathrm{J} of $$energy is lost to air resistance, what is the speed of the ball just before it hits the ground?

Accepted Answer

$$v = \sqrt{2(g h - W_f/m)} = \sqrt{2(9.8 	imes 25 - 1,000/100)} = \sqrt{2(245 - 10)} = \sqrt{470} \approx 21.7 \, \mathrm{m/s}$$

Question 11

A $$0.50 \, \mathrm{kg}$$ ball is thrown downward from a $$25.0 \, \mathrm{m}$$ window with an initial speed of $$2.0 \, \mathrm{m/s}. $$What is the speed just before it hits the ground? (Ignore air resistance.)

Accepted Answer

$$v = \sqrt{v_0$^2 + 2gh} = \sqrt{2.0^2$ + 2 	imes 9.8 	imes 25} = \sqrt{4 + 490} = \sqrt{494} \approx 22.2 \, \mathrm{m/s}$$

Question 12

A $$0.50 \, \mathrm{kg}$$ ball is thrown upward from a $$25.0 \, \mathrm{m}$$ window with an initial speed of $$5.0 \, \mathrm{m/s}. $$What is the speed just before it hits the ground? (Ignore air resistance.)

Accepted Answer

$$v = \sqrt{v_0$^2 + 2gh} = \sqrt{5.0^2$ + 2 	imes 9.8 	imes 25} = \sqrt{25 + 490} = \sqrt{515} \approx 22.7 \, \mathrm{m/s}$$

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