PHY 06 SPRING 006 Prof. Massimiliano Galeazzi Midterm # March 8, 006 NAME: Problem # SIGNAURE: UM ID: Problem # Problem # otal Some useful relations: st lat of thermodynamic: U Q - W Heat in an isobaric process: Q n C P Heat in an isochoric process: Q n C Adiabatic processes: γ p const., const. Work: dw p d Ideal gas equation of state: p n R Engine efficiency: e W Entropy: Wave equation: Speed of propagation of a wave on a string: Q H dq ds y( v µ y( v t
PHY 06 Spring 006 Midterm # March 8, 006 Problem # A quantity of n moles of a diatomic ideal gas is initially enclosed in a volume o and its pressure is p o. he gas expands adiabatically to three times its initial volume, it is then compressed isothermally to the original volume, and finally it is brought back to its initial state at constant volume. All processes are reversible. [NOE: Write the results only in terms of p o, o, and n a) What is the initial temperature of the gas? b) Draw the processes on the p- diagram to the left. c) What is the temperature of the gas during the isothermal compression? d) Calculate the heat transfer for each of the three processes. e) What is the efficiency of an engine operating through this cycle? f) What is the total change in entropy of the system for one cycle? p a) p oo nro, o o o nr γ c) Process ab is adiabatic, therefore const. γ ( ) p oo γ o o, C nr γ nr, with γ P C d) Q ab 0 (adiabatic) c poo poo Q bc Wbc nr ln nr ln ln b nr p oo poo Q ca nc ( a c ) n R γ p oo nr nr poo ln Q e) C Qbc ln e + + Q γ H Qca p oo f) he cycle is fully reversible, therefore S 0 p o o 7 γ ( ) ( ) o R R 7 γ ( )
PHY 06 Spring 006 Midterm # March 8, 006 Problem # One mole of an ideal gas at temperature o fills one third of a cylinder. he gas is kept in there through a thin partition. he rest of the cylinder is empty. he cylinder is also provided with a piston and is in thermal contact with a heat reservoir much bigger than the cylinder (see figure). a) he thin partition is broken and the gas expands freely occupying the whole volume. What are the work W, the heat transfer Q, and the change in internal energy U of the gas? b) Calculate the entropy change S in the gas. c) he gas is then compressed using the piston to its initial volume. he temperature is maintained constant through the contact with the heat reservoir. he reservoir is big enough that its temperature does not change. Calculate the change in entropy of the gas for this process. d) he processed described in a) and c) bring the gas from an initial state back to the same state, therefore representing a cyclical process. What is the change in entropy of the Universe for the full cycle? e) Is your answer to question d) in agreement with the second law of thermodynamic? Explain in no more than one sentence. f) Since the process is cyclical, you could restore the thin partition and start a new cycle again. Could you use this cycle either as thermal engine or a refrigerator? Qualitatively explain what happens to work and heat transfer in the cycle. a) Isothermal (there is the heat reservoir) U 0 ; Free expansion W0; (first law of D) Q0. b) Process is irreversible, use reversible isothermal dq Q W nr ln i S R ln w N f w Also, i S kb ln kb ln NkB ln nr ln R ln wi wi f nr ln dq Q W c) As in the first method in b): i S R ln R ln d) Suniverse S gas + Sreservoir S + S + Sreservoir Sreservoir, f
PHY 06 Spring 006 Midterm # March 8, 006 f nr ln dq Q Qgas W reservoir gas i Sreservoir R ln R ln e) YES, the entropy of the universe is bigger than zero and it is always supposed to be bigger or equal to zero. f) NO, first of all, there is only heat reservoir and we have seen that engines and refrigerators need. Second, the net result of the process is to convert work into heat, therefore this is not a refrigerator, nor an engine. 4
PHY 06 Spring 006 Midterm # March 8, 006 Problem # A very long string is fixed at one end on a wall and on the other end is wrapped around a wheel of negligible mass and then attached to a mass M to provide the tension in the string (see figure). he string has mass m and length L. Assume that the string is long enough to neglect reflections at the wall. A transverse wave is generated on the string. If the origin if the x-axis is placed at the wheel, the wave can be described by the function: ( Asin [ B( Cx y for x>0 and t>0, With A, B, and C constant. [Write your results in terms of M, m, L, A, B, and x o. You may need the trigonometric expression for a generic angle θ: sin θ sinθ cosθ. a) Show that the wave function satisfies the wave equation. b) What is the value of the constant C? c) What is the maximum displacement of the particles in the string? d) What is the maximum transverse speed v y of the particles in the string? e) If you measure the speed and the acceleration of the particles at a point with coordinate x o, at what time t does the string reach the maximum displacement at the point? f) At what time t does it reach the maximum transverse speed at that point of coordinate x o? a) y( y( v t y ABC sin[ B( Cx cos B( Cx ABC sin B Cx t y AB C cos[ B( Cx y ABsin[ B( Cx cos[ B( Cx ABsin B( Cx t y AB cos[ B( Cx t AB C cos[ B( Cx AB cos[ B( Cx v Satisfied for C C v v [ [ ( ) [
PHY 06 Spring 006 Midterm # March 8, 006 b) See a), or remember than wave equation is satisfied for y y( x ± v and considering that y( Asin [ B( Cx Asin [ BC( x C v C c) y is max for sin [ ( Cx ± B y max A y v y t v y is max for sin [ B ( Cx ± v max AB d) ABsin[ B( Cx cos[ B( Cx ABsin[ B( Cx t Cxo B B Cxo t,, t Cxo 4B e) y is max for sin [ B ( Cx ± ( Cx,,,... f) y taking the first value: ( Cx B o B o v is max for sin [ B ( Cx ± ( ),... B o taking the first value: ( Cx 6