Dr. Gundersen Phy 206 Test 2 March 6, 2013

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1 Signature: Idnumber: Name: You must do all four questions. There are a total of 100 points. Each problem is worth 25 points and you have to do ALL problems. A formula sheet is provided on the LAST page which you can tear off. TO GET PARTIAL CREDIT, SHOW YOUR WORK ON PROBLEMS CHECK DISCUSSION SECTION ATTENDED: [ ] Dr. Barnes 1P, 11:00-11:50 a.m. [ ] Dr. Barnes 1Q, 12:30-1:20 p.m. [ ] Dr. Zuo 1R, 2:00-2:50 p.m. [ ] Dr. Ghandour 1S, 3:30-4:20 p.m. [ ] Dr. Ghandour 1T, 5:00-5:50 p.m. [ ] Dr. Zuo 2P, 11:00-11:50 a.m. [ ] Dr. Van Vliet 2R, 2:00-2:50 p.m

2 [1.] This problem has five multiple choice questions. Circle the best answers. [1A.] An ideal gas may expand from an initial pressure p i and volume V i to a final volume V f isothermally, adiabatically, or isobarically. For which type of process is the heat that is added to the gas the largest? (Assume that p i, V i, and V f are the same for each process.) [A] isothermal [B] adiabatic [C] isobaric [E] not enough information [1B.] A refrigerator operates by [D] all the processes have the same heat flow [A] doing work to move heat from a low-temperature thermal reservoir to a high-temperature thermal reservoir. [B] doing work to move heat from a high-temperature thermal reservoir to a low-temperature thermal reservoir. [C] using thermal energy to produce useful work. [D] moving heat from a low-temperature thermal reservoir to a high-temperature thermal reservoir without doing work. [E] moving heat from a high-temperature thermal reservoir to a low-temperature thermal reservoir without doing work. [1C.] Which of the following statements about the Carnot cycle is incorrect? [A] The maximum efficiency of a Carnot engine is 100% since the Carnot cycle is an ideal process. [B] The Carnot cycle consists of two isothermal processes and two adiabatic processes. [C] The Carnot cycle consists of two isothermal processes and two constant entropy (isentropic) processes. [D] The efficiency of the Carnot cycle depends solely on the temperature of the two thermal reservoirs. [E] The Carnot cycle is reversible. [1D.] Which of the following processes always results in an increase in the internal energy of the system? [A] The system loses heat and does work on the surroundings. [B] The system gains heat and does work on the surroundings. [C] The system loses heat and has work done on it by the surroundings. [D] The system gains heat and has work done on it by the surroundings. [E] None of the above will always increase the internal energy of the system. [1E.] Which statement is not true about a string instrument? [A] Longer strings give rise to lower resonant frequencies. [B] The resonant frequencies are proportional to the square root of the string s tension. [C] Strings with higher linear mass densities yield lower resonant frequencies. [D] The resonant frequency is proportional to the wave speed. [E] The resonant frequency is proportional to the amplitude of the string s motion.

3 [2.] A three step heat engine cycle consists of an adiabatic expansion (process AB), an isobaric compression (process BC) and an isochoric process (process CA). The working substance consists of n moles of an ideal gas with a molar heat capacity at constant volume of C V and a molar heat capacity at constant pressure C p. The cycle operates between pressures (p 1,p 2 ) and volumes (V 1,V 2 ) as shown in the figure below. [a] Label the points on graph with A, B, or C, connect the points with properly shaped lines representing each process, and indicate arrows on the lines showing the direction of the cycle. (7 points) [b] Calculate the heat Q, the work W, and the change in internal energy U for each process in the cycle and enter your results in the table below. Express your answers in terms of p 1, p 2, V 1, V 2, C p, C V and the gas constant R. (18 points)

5 [3.] This problem concerns the cycle described in problem 2. [a] Compute the efficiency of the cycle described in problem 2. Express your answer in terms of the compression ratio r = V 2 /V 1 and γ = C p /C V. (10 points) [b] Compute the change in entropy for each of the processes shown in problem 2, i.e. S AB, S BC, and S CA. Express your answer in terms of n, C p, C V, r, and γ.(9 points) [c] Show that the change in entropy for the complete cycle is 0. (6 points)

7 [4.] A string is attached to a wall at x = 0 and x = L. The string is oscillating in its fourth harmonic and its shape is described by the standing wave y(x,t) = 3sin(5x)sin(7t). The numbers in the wave function are in terms of standard MKS units. Give units for your answers in parts b) and c). [a] In the diagram above, draw the standing wave pattern at times t = π/14 seconds and t = 3π/14 seconds and indicate the positions of the nodes and antinodes. (9 points) [b] As described in class, standing waves consist of the sum of two counter-propagating traveling waves. What is the amplitude, wavelength, frequency, period and speed of the traveling waves that make up the standing wave? (10 points) [c] What is the length (L) of the string, what is its transverse velocity, v y (x,t), and what is its transverse acceleration, a y (x,t)? (6 points)

9 Fluid Mechanics: ρ = m V, p = df da, p 2 p 1 = ρg(y 2 y 1 ), p = p 0 + ρgh, B = m disp g A 1 v 1 = A 2 v 2, dv dt = Av, p 1 + ρgy ρv2 1 = p 2 + ρgy ρv2 2 Temperature and Heat: T C = 5 9 (T F 32 ), T K = T C , L = αl 0 T, V = βv 0 T, Q = mc T, Q = nc T, Q = ±ml, H = dq dt = kat H T C L Thermal Properties of Matter: F A = Y α T = ka dt dx, H net = Aeσ(T 4 T 4 s ) m tot = nm, pv = nrt, M = N A m, K tr = 3 2 nrt, 1 2 m(v2 ) av = 3 2 kt, C V = #DOF R 2 v rms = 3kT 3RT (v 2 ) av = m = M, f(v) = 4π( m 2πkT )3/2 v 2 e mv 2 /2kT W = V2 First Law of Thermodynamics: V 1 pdv, U = Q W, U = nc V T, C p = C V + R, γ = C p C V T 1 V γ 1 1 = T 2 V γ 1 2, p 1 V γ 1 = p 2V γ 2, W = C V R (p 1V 1 p 2 V 2 ) Second Law of Thermodynamics: e = W = 1 + Q C = 1 Q C Q H Q H Q H, e Carnot = 1 T C, K = Q C T H W K Carnot = T 2 C dq, S rev = T H T C 1 T, S = k lnw Mechanical Waves: v = λf = ω/k, f = ω/2π = 1/T, λ = 2π/k, y(x,t) = Acos(kx ωt), v = = Q C Q H Q C 2 y(x,t) x 2 = 1 v 2 2 y(x,t) t 2 F µ, P(x,t) = µfω 2 A 2 sin 2 (kx ωt), P av = 1 µfω 2 A 2, v y (x,t) = y(x,t) 2 t a y (x,t) = v y(x,t), y(x,t) = y 1 (x,t)+y 2 (x,t), y(x,t) = A SW sinkxsin ωt, λ n = 2L t n, f n = nv 2L

A) 120 degrees B) 90 degrees C) 60 degrees D) 45 degrees E) 30 degrees

A) 120 degrees B) 90 degrees C) 60 degrees D) 45 degrees E) 30 degrees Phys10 - First Major 071 Zero Version Q1. Two identical sinusoidal traveling waves are sent along the same string in the same direction. What should be the phase difference between the two waves so that