Q: A submarine is submerged at a depth where the water pressure is $$3.5 \times 10^6$ Pa. If the atmospheric pressure at the surface is 1.0$$ \times $$10^5 Pa $$and the density of seawater is $$1025\ \mathrm{kg/m^3$}$$, what is the depth of the submarine? (Assume $$g = 9.8\ \mathrm{m/s^2$})$$

$$h = \dfrac{P - P_0$}{\rho g} = \dfrac{3.5 \times 10^6$ - 1.0 \times 10^5$}{1025 \times 9.8} \approx 335\ \mathrm{m}$$

Q: A gas is compressed isothermally from a volume of $$4.0\ \mathrm{L} to$$ $$2.0\ \mathrm{L} at a $$temperature of $$300\ \mathrm{K}. If $$the initial pressure is $$1.0\ \mathrm{atm}$$, what is the final pressure?

$$P_2$$ = $$P_1$$ \dfrac{$$V_1$$}{$$V_2$$} = 1.0 \times \dfrac{4.0}{2.0} = 2.0\ \mathrm{atm}$$

Q: A steel bridge is $$500\ \mathrm{m}$$ long at $$20^\circ\mathrm{C}. If $$the coefficient of linear expansion for steel is $$1.2 \times 10$$^{-5}$$\ \mathrm{K}^{-1}$$, how much longer is the bridge on a hot day at $$40^\circ\mathrm{C}$$?

$$\Delta \ell = \alpha \$$ell_0$$ \Delta T = 1.2 \times 10$$^{-5}$$ \times 500 \times 20 = 0.12\ \mathrm{m}$$

Q: A piston contains $$0.5\ \mathrm{mol} of an $$ideal gas at $$300\ \mathrm{K}. $$What is the average kinetic energy per molecule?

$$K = \dfrac{3}{2}kT = \dfrac{3}{2} \times 1.38 \times 10$$^{-23}$$ \times 300 \approx 6.21 \times 10$$^{-21}$$\ \mathrm{J}$$

Q: A standing wave on a string fixed at both ends has a length $$\ell = 1.2\ \mathrm{m}$$ and a wave speed $$v = 60\ \mathrm{m/s}. $$What is the frequency of the third harmonic?

$$f_3$$ = \dfrac{3v}{2\ell} = \dfrac{3 \times 60}{2 \times 1.2} = 75\ \mathrm{Hz}$$

Q: A $$0.1\ \mathrm{m^3$}$$ tank contains $$2.0\ \mathrm{mol} of an $$ideal gas at $$400\ \mathrm{K}. $$What is the pressure inside the tank?

$$P = \dfrac{nRT}{V} = \dfrac{2.0 \times 8.31 \times 400}{0.1} = 66,480\ \mathrm{Pa}$$

Question 1

A submarine is submerged at a depth where the water pressure is $$3.5 	imes 10^6$ Pa. If the atmospheric pressure at the surface is 1.0$$ 	imes $$10^5 Pa $$and the density of seawater is $$1025\ \mathrm{kg/m^3$}$$, what is the depth of the submarine? (Assume $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$h = \dfrac{P - P_0$}{ho g} = \dfrac{3.5 	imes 10^6$ - 1.0 	imes 10^5$}{1025 	imes 9.8} \approx 335\ \mathrm{m}$$

Question 2

A gas is compressed isothermally from a volume of $$4.0\ \mathrm{L} to$$ $$2.0\ \mathrm{L} at a $$temperature of $$300\ \mathrm{K}. If $$the initial pressure is $$1.0\ \mathrm{atm}$$, what is the final pressure?

Accepted Answer

$$P_2$$ = $$P_1$$ \dfrac{$$V_1$$}{$$V_2$$} = 1.0 	imes \dfrac{4.0}{2.0} = 2.0\ \mathrm{atm}$$

Question 3

A steel bridge is $$500\ \mathrm{m}$$ long at $$20^\circ\mathrm{C}. If $$the coefficient of linear expansion for steel is $$1.2 	imes 10$$^{-5}$$\ \mathrm{K}^{-1}$$, how much longer is the bridge on a hot day at $$40^\circ\mathrm{C}$$?

Accepted Answer

$$\Delta \ell = \alpha \$$ell_0$$ \Delta T = 1.2 	imes 10$$^{-5}$$ 	imes 500 	imes 20 = 0.12\ \mathrm{m}$$

Question 4

A piston contains $$0.5\ \mathrm{mol} of an $$ideal gas at $$300\ \mathrm{K}. $$What is the average kinetic energy per molecule?

Accepted Answer

$$K = \dfrac{3}{2}kT = \dfrac{3}{2} 	imes 1.38 	imes 10$$^{-23}$$ 	imes 300 \approx 6.21 	imes 10$$^{-21}$$\ \mathrm{J}$$

Question 5

A $$2.0\ \mathrm{kg}$$ block of ice at $$0^\circ\mathrm{C} is $$placed in contact with a $$5.0\ \mathrm{kg}$$ block of copper at $$100^\circ\mathrm{C}. $$The specific heat of copper is $$386\ \mathrm{J/(kg\cdot K)}$$ and the latent heat of fusion for ice is $$3.34 	imes 10^5$\ \mathrm{J/kg}. $$What is the final state of the system?

Accepted Answer

All the ice melts, and the final temperature is below $$100^\circ\mathrm{C}.$$

Question 6

A standing wave on a string fixed at both ends has a length $$\ell = 1.2\ \mathrm{m}$$ and a wave speed $$v = 60\ \mathrm{m/s}. $$What is the frequency of the third harmonic?

Accepted Answer

$$f_3$$ = \dfrac{3v}{2\ell} = \dfrac{3 	imes 60}{2 	imes 1.2} = 75\ \mathrm{Hz}$$

Question 7

A sound wave in air has an intensity of $$1.0 	imes 10$$^{-6}$$\ \mathrm{W/m^2$}. $$What is the sound level in decibels (dB)? (Reference intensity $$I_0 = 1.0$$ 	imes $$10^{-12}$$\ \mathrm{$$W/m^2$$}$$)

Accepted Answer

$$\beta = 10 \log_{10} \left( \dfrac{I}{I_0$} \right ) = 10 \log_{10} (10^6$) = 60\ \mathrm{dB}$$

Question 8

A refrigerator removes $$500\ \mathrm{J} of $$heat from its interior and expels $$700\ \mathrm{J} to $$the room. What is the coefficient of performance (COP) of the refrigerator?

Accepted Answer

$$COP = \dfrac{|Q_L|}{|Q_H| - |Q_L|} = \dfrac{500}{700 - 500} = 2.5$$

Question 9

A thin soap film ($$n = 1.33) is $$illuminated with white light at normal incidence. What is the minimum nonzero thickness for constructive interference in the reflected light (air-film-air)?

Accepted Answer

$$t = \dfrac{\lambda}{4n}$$

Question 10

A $$0.1\ \mathrm{m^3$}$$ tank contains $$2.0\ \mathrm{mol} of an $$ideal gas at $$400\ \mathrm{K}. $$What is the pressure inside the tank?

Accepted Answer

$$P = \dfrac{nRT}{V} = \dfrac{2.0 	imes 8.31 	imes 400}{0.1} = 66,480\ \mathrm{Pa}$$

Question 11

A Carnot engine operates between $$T_H = 500\ \mathrm{K}$$ and $$T_L = 300\ \mathrm{K}. $$What is its maximum possible efficiency?

Accepted Answer

$$e_{ideal} = 1 - \dfrac{T_L}{T_H} = 1 - \dfrac{300}{500} = 0.4$$

Question 12

A pipe open at both ends has a length of $$0.85\ \mathrm{m}. $$What is the wavelength of the second harmonic?

Accepted Answer

$$\$$lambda_2$$ = \dfrac{2\ell}{2} = \ell = 0.85\ \mathrm{m}$$

Question 13

A beam of light passes from air ($$n_1 = 1.00$) into glass (n_2$ = 1.50) at an $$angle of $$30^\circ to $$the normal. What is the angle of refraction in the glass?

Accepted Answer

$$n_1$$ \sin \$$theta_1$$ = $$n_2$$ \sin \$$theta_2$$ \implies \sin \$$theta_2$$ = \dfrac{1.00 	imes \sin 30^\circ}{1.50} = \dfrac{0.5}{1.5} = 0.333 \implies \$$theta_2$$ \approx 19.5^\circ$$

Question 14

A $$0.5\ \mathrm{m}$$ long copper rod ($$\alpha = 1.7 	imes 10$$^{-5}$$\ \mathrm{K}^{-1}) is $$heated from $$20^\circ\mathrm{C} to$$ $$120^\circ\mathrm{C}. $$What is the change in length?

Accepted Answer

$$\Delta \ell = \alpha \$$ell_0$$ \Delta T = 1.7 	imes 10$$^{-5}$$ 	imes 0.5 	imes 100 = 0.00085\ \mathrm{m}$$

Question 15

A $$0.2\ \mathrm{kg}$$ aluminum plate ($$c = 900\ \mathrm{J/(kg\cdot K)}) at$$ $$100^\circ\mathrm{C} is $$dropped into $$0.5\ \mathrm{kg} of $$water at $$20^\circ\mathrm{C}. $$What is the final equilibrium temperature? (Assume no heat loss to the environment.)

Accepted Answer

$$m_{Al}c$$_{Al}(T_f - 100) + $$m_{H_2O}c$$_{H_2O}(T_f - 20) = 0 \implies T_f \approx 25.2^\circ\mathrm{C}$$

Question 16

A $$1.0\ \mathrm{m}$$ long string has a mass of $$0.01\ \mathrm{kg}$$ and is under a tension of $$100\ \mathrm{N}. $$What is the speed of a wave on the string?

Accepted Answer

$$v = \sqrt{\dfrac{F_T}{\mu}} = \sqrt{\dfrac{100}{0.01/1.0}} = \sqrt{10000} = 100\ \mathrm{m/s}$$

Question 17

A double-slit experiment uses slits separated by $$0.5\ \mathrm{mm}$$ and a screen $$2.0\ \mathrm{m}$$ away. If the first-order bright fringe is observed $$2.4\ \mathrm{mm}$$ from the central maximum, what is the wavelength of the light?

Accepted Answer

$$\lambda = \dfrac{y_1$ d}{\ell} = \dfrac{2.4 	imes 10$$^{-3}$$ 	imes 0.5 	imes 10$$^{-3}$$}{2.0} = 6.0 	imes 10$$^{-7}$$\ \mathrm{m}$$

Question 18

A $$0.5\ \mathrm{kg}$$ block slides at $$2.0\ \mathrm{m/s} on a $$frictionless surface and compresses a spring. If all kinetic energy is converted to spring potential energy, what is the maximum compression if $$k = 200\ \mathrm{N/m}$$?

Accepted Answer

$$\dfrac{1}{2}mv^2$ = \dfrac{1}{2}kx^2$ \implies x = \sqrt{\dfrac{mv^2$}{k}} = \sqrt{\dfrac{0.5 	imes 4}{200}} = 0.1\ \mathrm{m}$$

PHY222 Midterm 2: Fluids, Thermodynamics, Waves, and Light – Equation Sheet Study Guide

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