Qualifying Examination

Mechanics and Statistical Mechanics

October 1, 2012

Instructions: Work three of the four mechanics problems (1-4), and threeof the four statistical mechanics problems (5-8). Please mark clearly whichproblems shall be graded.

1. A thin rod of uniform mass density and weight W rests upon two points at eachend. One of these points is suddenly removed. What is the force on the other pointimmediately after this removal when the angular displacement, θ, is very small?

2. Three particles, two of mass m and one of mass M , are confined to move on a hori-zontal circle of constant radius r and are connected by identical springs lying on thecircle, each of spring constant k and unstretched length a. The springs are of equallength b when the system is at rest. Consider in the following motions that stretchthe springs only by a small amount from b. Assume perfect springs and neglect fric-tion.

(a) Write down the equations of motion.(b) Find the normal coordinates and normal frequencies of the system for the caseM = m.(c) For the phase space and Hamiltonian associated with this problem, consider ageneric dynamic variable of interest, f(. . . ), which is not explicitly time dependent.Discuss how you would obtain the time evolution of f(. . . ) in terms of Poisson Brack-ets.

1

3. Two particles of equal mass, m, are constrained to move without friction on the sur-face of a sphere. The radius of the sphere is a known function of time, i.e., R(t) isgiven. Between the two particles the Newtonian force of gravity is active.

(a) Describe the configuration manifold, Q. Explain the nature of the constraints.(b) Derive the Lagrangian, L, and Hamiltionian, H. Derive the canonical equationsof motion. (Hint: Treat the potential energy as a generic function of appropriatevariables, but don’t express it in its full glory.)(c) Discuss any applicable conservation laws.(d) State the dimensionality of the associated phase space. (e) Give an explicit ex-pression for the symplectic matrix of this system.

4. Show that the production of a positron-electron pair by a photon γ → e+ + e

−

requires the presence of a nucleus of mass M and compute the minimum energy ofthe photon as a function of the involved masses. Positron and electron masses m areequal.

5. (a) A few minutes after the Big Bang, the universe consisted of trace amounts ofnucleons (protons and neutrons) coupled to a reservoir of photons, neutrinos, andelectron-positron pairs. At temperatures above 1010 K, the neutron-to-proton ratioof this primordial soup is held in equilibrium via bi-directional charged current weakinteractions interconverting protons (rest mass 938.6 MeV) and neutrons (rest mass939.6 MeV); thus, the nucleons can be thought of as a 2-state system. As the Uni-verse cooled to 1010 K (kT ∼1 MeV), these reactions fizzled out and the equilibriumn-to-p ratio at this temperature was frozen/locked in and is essentially todays cosmicvalue. From statistical mechanical principles, estimate this n-to-p ratio.

(b) Consider a 6-spin paramagnet system where an individual spin up leads to acontribution to total system energy of −µB and an individual spin down leads to acontribution to a total system energy of +µB. The system begins in a state withno applied B field. A magnetic field is applied and the system ends in a state withEtot = −4µB. What is the change in (Gibbs) entropy?The system begins in an initialstate with no applied B field.”

6. Calculate the entropy of mixing for a system of two monatomic ideal gases, A and B,whose relative proportion, x, is arbitrary. Let N be the total number of moleculesand let x be the fraction of these that are of species B. Assume that the species,which have the same mass m and thermal energy U , are initially segregated in volumeV . Then the partition is removed and the particles mix. Use the entropy for an ideal

2

gas (Sackur-Tetrode equation)

S = Nk

�ln

�V

N

�4πmU

3Nh2

�3/2�

+52

�

7. Consider a cyclic heat engine (Rankine), where water at low pressure (1) is pumpedto high pressure (2), then flows into a boiler, where heat is added at constant pres-sure and the water turns to steam (3). The steam hits a turbine, where it expandsadiabatically, cools, and ends up at the original pressure (4). Finally the partiallycondensed fluid (water + steam) is cooled further in a condenser.

(a) Draw a PV diagram indicating the cycle including points (1) through (4), andthe approximate phase boundaries for water and steam.(b) Define an efficiency e and explain how to calculate it with appropriate approxi-mations (Hint: Enthalpy.)

8. Consider a molecule that is best described by a rigid, massless, ideal spring (Hookeslaw, constant k) that connects two identical point particles (mass m). The moleculevibrates with very small amplitude, and rotates freely in space. Consider more ofsuch molecules, confined in a box of Volume, V , in thermal equilibrium with a heatbath at temperature, T .

Calculate the ”specific heat”, Cp, as a function of temperature, for regimes in whichthe molecules(a) move freely, but rotational-vibrational states are not excited, and(b) when vibrations and rotations are successively accessible.

3