Elementary Qualifier Examination October 10, 2005 NAME CODE: [ ]

1. Elementary Qualifier Examination October 10, 2005 NAME CODE: [ ] • Instructions: • Do any ten (10) of the twelve (12) problems on the following pages. • Indicate on this page (below right) which 10 problems you wish to have graded. • If you need more space for any given problem, write on the back of that problem’s page. • Mark your name code on all pages. • Be sure to show your work and explain what you are doing. • A table of integrals is available from the proctor. Possibly useful information: Planck constant ħ = h/(2) = 6.583  10-16 eV·sec Stefan-Boltzmann constant  = 5.670  10-8 J·K-4m-2s-1 Speed of light, c = 3.00 10 8 m/sec Permeability, 0 = 410-7 Tm/A Gas constant R = 8.3144 J /(molK) ag = 9.8 m/sec2 at the earth’s surface me = 9.109  10-31 kg = 0.511 MeV/c2 m = 1.883  10-28 kg = 105.6 MeV/c2 m0 = 2.407  10-28 kg = 135.0 MeV/c2 mK0 = 8.872  10-28 kg = 497.7 MeV/c2 Atomic weight N = 14.00674 u 1u = 1.660 10-27 kg = 931.5 MeV/c2 Harmonic Oscillator Relativistic kinematics E = g moc2 E2 = p2c2+mo2c4 hc = 1240 eV·nm Check the boxes below for the 10 problems you want graded Problem Number Score 1 2 3 4 5 6 7 8 9 10 11 12 Total

2. Problem 1 A bucket half-filled with water (total mass 10 kg) is swung smoothly in a circular path of radius 0.8 meters at a uniform rate of 8 revolutions per 10 seconds. a. Calculate and compare the amount of force the person holds the bucket with just to keep it moving through the very top and bottom positions along this circular path. b. What minimum uniform rotational speed is necessary to keep the water from falling from the bucket? With what force must the person hold the bucket then at the top and bottom positions? a. top: bottom: = 4.02 m/sec = 10 kg (20.2 m/sec2 – 9.8 m/sec2) = 104.0 N = 10 kg (20.2 m/sec2 + 9.8 m/sec2) = 300.0 N b. top: bottom: = 2.8 m/sec = 0 2mg = 196 N

3. Problem4 • 38.6 grams of N2 (a real gas that behaves like an ideal • gas) is allowed to expand reversibly, and isothermally. • The temperature during expansion is 425 K and the • volume changes from 14 to 27 liters. • How much work is produced during the expansion? • How much heat is produced during the expansion? Name code

4. Binoculars and microscopes are frequently made with coated optics. A thin layer of transparent material is added to the lens surface as shown below right. Multiple choice: For the first two questions select the best answer from the options that follow. Problem 9 Name code 1 2 • The desired condition is for • a. constructive interference between light rays 1 and 2. • b. the coating be more transparent than the lens. • c. destructive interference between rays 3 and 4. • d. the speed of light in the coating to be less than in the lens. • e. destructive interference between rays 1 and 2. • A necessary condition is for • nair < n1 < n2. • nair < n2 < n1. • n1 < nair < n2. • nair < n1 = n2. • Show your work in calculating the following: • If  denotes the central value of the wavelengths of • incident light (in air), and the coating has a refractive • index of n1 and the lens n2, what is the thinnest • possible the coating should be? Air nair MgF2n1 Glass n2 3 4

5. Problem 10 A neutral kaon decays, at rest, to a pair of neutral pions: a. Find the momentum (in units of MeV/c) of each pion in the lab frame. b. Find the speed of each pion in the lab frame. c. The mean lifetime of a pion at rest is 8.410-17 seconds. What mean path length would the pions coming from this kaon decay be expected to travel in the lab frame? • Conservation of momentum: • Conservation of energy: • b. • c. = 2.53  108 m/sec or = 2.53  108 m/sec

6. Problem 11 Experimentally the equilibrium spacing between nuclei in the HClmolecule is found to be 1.27 Å (for the Cl35 isotope) and the force constant approximating its molecular bond is measured to be 513 N/m. a. What is its ground state vibrational energy (in eV) ? b. What is the separation of its vibrational energy levels ? c. Transitions between vibrational levels result in the emission of what frequency of light? Approximately where in the electromagnetic spectrum (mR IR visible UV) does this lie? Name code

7. Problem 12 • The figure below right illustrates Compton scattering of a photon off an electron. • Using the labeled quantities in the figure, write an expression • for the momentum of the scattered electron (in terms of the • photon momenta). • b. stating conservation of energy for this system of photon and electron. • Starting separately from a. and b. above, find two independent expressions for pe2 • simplifying so that all vector quantities are expressed only with their amplitudes. me  p1  p2 d. 0.30 MeV X-rays are Compton scattered in the backward direction. What is the kinetic energy of the recoil electrons? and

8. Problem 12 • The figure below right illustrates Compton scattering of a photon off an electron. • Using the labeled quantities in the figure, write an expression • for the momentum of the scattered electron (in terms of the • photon momenta). • b. stating conservation of energy for this system of photon and electron. • Starting from a. above, find an expressions for pe2 (simplifying so that vector • quantities are expressed only with their amplitudes, i.e. . me  p1  p2 d. 0.30 MeV X-rays are Compton scattered in the backward direction. What is the kinetic energy of the recoil electrons? and