Q: A space shuttle of mass $$2.7\ \mathrm{kg} is $$launched vertically from rest and reaches a speed of $$190\ \mathrm{km/h} in$$ $$8.5\ \mathrm{s}. $$What is the apparent weight of the shuttle during this phase? (Assume $$g = 9.8\ \mathrm{m/s^2$}.)$$

$$W = m(g + a) = 2.7 \times (9.8 + \frac{190/3.6}{8.5}) = 2.7 \times (9.8 + 6.22) = 43.6\ \mathrm{N}$$

Q: A flywheel on a stationary bike has a moment of inertia $$I = 0.5\ \mathrm{kg\cdot m^2$}$$ and is spinning at $$120\ \mathrm{rad/s}. If $$the brakes apply a constant torque of $$0.6\ \mathrm{N\cdot m}$$, how long will it take for the flywheel to come to a stop?

$$\Delta t = \frac{I \omega}{\tau} = \frac{0.5 \times 120}{0.6} = 100\ \mathrm{s}$$

Q: A skier starts from rest at the top of a frictionless hill of height $$h_1 = 45$$\ \mathrm{m}$$ and slows down as she climbs a second hill of height h_2$ = 5.0\ \mathrm{m}. $$What is the minimum initial speed required to just make it over the second hill?

$$v_0$$ = \sqrt{2g($$h_1$$ - $$h_2$$)} = \sqrt{2 \times 9.8 \times (45 - 5)} = \sqrt{784} = 28\ \mathrm{m/s}$$

Q: Your roommate leaves a $$120\ \mathrm{W}$$ fan running in your apartment for $$8$$ hours. How much thermal energy does the fan add to the air?

$$Q = P \times t = 120 \times 8 \times 3600 = 3,456,000\ \mathrm{J}$$

Q: A $$5.0\ \mathrm{m}-$$diameter garden pond holds $$1.5 \times 10^4$\ \mathrm{kg} of $$water. Solar energy is incident on the pond at an average rate of $$400\ \mathrm{W/m^2$}. If $$the water absorbs all the solar energy and does not exchange energy with its surroundings, how many hours will it take to warm the water from $$12\ ^\circ\mathrm{C} to$$ $$15\ ^\circ\mathrm{C}$$? (Specific heat of water $$c = 4186\ \mathrm{J/kg\cdot K})$$

$$Q = mc\Delta T = 1.5 \times 10^4$ \times 4186 \times 3 = 1.89 \times 10^8$\ \mathrm{J}$$; \newline P = 400 \times \pi $$(2.5)^2 = 7$$,854\ \mathrm{W}$$; \newline t = \frac{1.89 \times 10^8$}{7,854} = 24,070\ \mathrm{s} = 6.7\ \mathrm{h}$$

Q: A gas is compressed from $$850\ \mathrm{cm^3$} to$$ $$225\ \mathrm{cm^3$} at a $$constant pressure of $$480\ \mathrm{kPa}. At $$the same time, $$155\ \mathrm{J} of $$heat energy is transferred out of the gas. What is the change in thermal energy of the gas during this process?

$$W = P \Delta V = 480,000 \times (0.225 - 0.85) = -300\ \mathrm{J}$$; \newline \Delta E_{int} = Q - W = -155 - (-300) = 145\ \mathrm{J}$

Q: A $$100\ \mathrm{cm^3$}$$ block of aluminum ($$\rho = 2700\ \mathrm{kg/m^3$}) is $$suspended in ethyl alcohol. What is the tension in the string holding the block?

$$T = mg - \rho_{fluid} V g = 0.27 \times 9.8 - 789 \times 0.0001 \times 9.8 = 2.646 - 0.773 = 1.87\ \mathrm{N}$$

Question 1

A space shuttle of mass $$2.7\ \mathrm{kg} is $$launched vertically from rest and reaches a speed of $$190\ \mathrm{km/h} in$$ $$8.5\ \mathrm{s}. $$What is the apparent weight of the shuttle during this phase? (Assume $$g = 9.8\ \mathrm{m/s^2$}.)$$

Accepted Answer

$$W = m(g + a) = 2.7 	imes (9.8 + \frac{190/3.6}{8.5}) = 2.7 	imes (9.8 + 6.22) = 43.6\ \mathrm{N}$$

Question 2

A flywheel on a stationary bike has a moment of inertia $$I = 0.5\ \mathrm{kg\cdot m^2$}$$ and is spinning at $$120\ \mathrm{rad/s}. If $$the brakes apply a constant torque of $$0.6\ \mathrm{N\cdot m}$$, how long will it take for the flywheel to come to a stop?

Accepted Answer

$$\Delta t = \frac{I \omega}{	au} = \frac{0.5 	imes 120}{0.6} = 100\ \mathrm{s}$$

Question 3

A spring with an unstretched length of $$11\ \mathrm{cm}$$ exerts a restoring force of $$F = 2.7\ \mathrm{N}$$ when stretched to $$15\ \mathrm{cm}. $$What is the spring constant $$k$$?

Accepted Answer

$$k = \frac{F}{\Delta x} = \frac{2.7}{0.04} = 67.5\ \mathrm{N/m}$$

Question 4

Two objects collide and bounce apart. Object 1 has an initial momentum $$\vec{p}_1 = (3, 2)\ \mathrm{kg\cdot m/s}$$ and object 2 has $$\vec{p}_2 = (-2, 1)\ \mathrm{kg\cdot m/s}. $$After the collision, object 2 has a final momentum $$\vec{p}_2' = (1, 4)\ \mathrm{kg\cdot m/s}. $$What is the final momentum of object 1?

Accepted Answer

Question 5

A skier starts from rest at the top of a frictionless hill of height $$h_1 = 45$$\ \mathrm{m}$$ and slows down as she climbs a second hill of height h_2$ = 5.0\ \mathrm{m}. $$What is the minimum initial speed required to just make it over the second hill?

Accepted Answer

$$v_0$$ = \sqrt{2g($$h_1$$ - $$h_2$$)} = \sqrt{2 	imes 9.8 	imes (45 - 5)} = \sqrt{784} = 28\ \mathrm{m/s}$$

Question 6

Your roommate leaves a $$120\ \mathrm{W}$$ fan running in your apartment for $$8$$ hours. How much thermal energy does the fan add to the air?

Accepted Answer

$$Q = P 	imes t = 120 	imes 8 	imes 3600 = 3,456,000\ \mathrm{J}$$

Question 7

An engine operates at maximum theoretical efficiency with a cold reservoir temperature of $$T_c = 15\ ^\circ\mathrm{C}$$ and is $$35\%$$ efficient. By how much should the temperature of the hot reservoir be increased to raise the efficiency to $$75\%$$?

Accepted Answer

$$\eta = 1 - \frac{T_c}{T_h} \implies T_h = \frac{T_c}{1 - \eta}$$; 
ewline T_{h,1} = \frac{288}{1 - 0.35} = 443\ \mathrm{K}$$, 
ewline T_{h,2} = \frac{288}{1 - 0.75} = 1152\ \mathrm{K}$$, 
ewline \Delta T = 1152 - 443 = 709\ \mathrm{K}$

Question 8

A $$5.0\ \mathrm{m}-$$diameter garden pond holds $$1.5 	imes 10^4$\ \mathrm{kg} of $$water. Solar energy is incident on the pond at an average rate of $$400\ \mathrm{W/m^2$}. If $$the water absorbs all the solar energy and does not exchange energy with its surroundings, how many hours will it take to warm the water from $$12\ ^\circ\mathrm{C} to$$ $$15\ ^\circ\mathrm{C}$$? (Specific heat of water $$c = 4186\ \mathrm{J/kg\cdot K})$$

Accepted Answer

$$Q = mc\Delta T = 1.5 	imes 10^4$ 	imes 4186 	imes 3 = 1.89 	imes 10^8$\ \mathrm{J}$$; 
ewline P = 400 	imes \pi $$(2.5)^2 = 7$$,854\ \mathrm{W}$$; 
ewline t = \frac{1.89 	imes 10^8$}{7,854} = 24,070\ \mathrm{s} = 6.7\ \mathrm{h}$$

Question 9

A gas is compressed from $$850\ \mathrm{cm^3$} to$$ $$225\ \mathrm{cm^3$} at a $$constant pressure of $$480\ \mathrm{kPa}. At $$the same time, $$155\ \mathrm{J} of $$heat energy is transferred out of the gas. What is the change in thermal energy of the gas during this process?

Accepted Answer

$$W = P \Delta V = 480,000 	imes (0.225 - 0.85) = -300\ \mathrm{J}$$; 
ewline \Delta E_{int} = Q - W = -155 - (-300) = 145\ \mathrm{J}$

Question 10

A $$100\ \mathrm{cm^3$}$$ block of aluminum ($$\rho = 2700\ \mathrm{kg/m^3$}) is $$suspended in ethyl alcohol. What is the tension in the string holding the block?

Accepted Answer

$$T = mg - ho_{fluid} V g = 0.27 	imes 9.8 - 789 	imes 0.0001 	imes 9.8 = 2.646 - 0.773 = 1.87\ \mathrm{N}$$

Question 11

A $$2.5\ \mathrm{cm}-$$diameter water line splits into two $$1.0\ \mathrm{cm}-$$diameter pipes. At point A, the water speed is $$2.5\ \mathrm{m/s}$$ and the gauge pressure is $$8.5\ \mathrm{kPa}. $$What is the gauge pressure at point B if the pipes are at the same elevation?

Accepted Answer

Using Bernoulli's equation and continuity: $$P_A + \frac{1}{2}ho v_A^2$ = P_B + \frac{1}{2}ho v_B^2$; A_A v_A = 2A_B v_B$; v_B$$ = \frac{A_A}{2A_B} v_A = \frac{$$(2.5/2)^2$$}{2 	imes $$(1/2)^2$$} 	imes 2.5 = 7.8\ \mathrm{m/s}$$; P_B = 8.5 + 0.5$$ 	imes 1000 	imes $$(2.5^2$$ - $$7.8^2)/1000 = 8.5 - 28.1 = -19.6$$\ \mathrm{kPa}$$

Question 12

A heavy steel ball is hung from a cord to make a pendulum. The ball is pulled to the side so that the cord makes a $$40^\circ$$ angle with the vertical and released. What time is required to reach the bottom of the swing? (Assume $$L = 2.0\ \mathrm{m}.)$$

Accepted Answer

$$t = \sqrt{\frac{L}{g}} \$$int_0$$^{\$$theta_0$$} \frac{d	heta}{\sqrt{\cos	heta - \cos\$$theta_0$$}}$$; 
ewline For small angles, $$t \approx \frac{T}{4} = \frac{\pi}{2} \sqrt{\frac{L}{g}} = 2.24\ \mathrm{s}$$

Question 13

A block with mass $$0.30\ \mathrm{kg} is $$attached to a horizontal spring. The block is pulled back from equilibrium until the spring exerts a force of $$1.5\ \mathrm{N}$$, then released. If the block oscillates with a frequency of $$1.5\ \mathrm{Hz}$$, how far was the block pulled back before being released?

Accepted Answer

$$F = kx$$, $$k = (2\pi f)^2$ m = (2\pi 	imes 1.5)^2$ 	imes 0.3 = 26.7\ \mathrm{N/m}$$; 
ewline $$x = \frac{F}{k} = \frac{1.5}{26.7} = 0.056\ \mathrm{m}$$

Question 14

A rock band playing an outdoor concert produces sound at $$127\ \mathrm{dB} at$$ $$10.0\ \mathrm{m}$$ from their single working loudspeaker. What is the sound intensity level $$30.0\ \mathrm{m}$$ from the speaker?

Accepted Answer

$$L_2$$ = $$L_1 + 10$$ \log_{10} \left(\frac{$$I_2$$}{$$I_1$$}\right) = 127 + 10 \log_{10} \left(\frac{$$1/30^2$$}{$$1/10^2$$}\right) = 127 + 10 \log_{10} (1/9) = 127 - 9.54 = 117.5\ \mathrm{dB}$$

Question 15

You are holding one end of an elastic cord that is fastened to a wall $$2.0\ \mathrm{m}$$ away. You begin shaking the end of the cord at $$2.2\ \mathrm{Hz}$$, creating a continuous sinusoidal wave of wavelength $$1.8\ \mathrm{m}. $$How much time will pass until a standing wave fills the entire length of the string?

Accepted Answer

$$v = f\lambda = 2.2 	imes 1.8 = 3.96\ \mathrm{m/s}$$; 
ewline t = \frac{2.0}{3.96} = 0.51\ \mathrm{s}$

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