Elementary Qualifier Examination October 13, 2003 NAME CODE: [ ]

1. Elementary Qualifier Examination October 13, 2003 NAME CODE: [ ] • Instructions: • Do any ten (10) of the twelve (12) problems of 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: e = 1.60 10-19 C me = 511 keV/c2 1 atm = 1.013105 N/m2 g = 9.8 m/sec2 mn = 1.6710-27 kg 1u = 1.66 10-27 kg Compton wavelength, C = 2.43 10-12 m Coulomb’s constant, k=1/(4o)=9.0109N·m2/c2 Planck constant h = 6.62610-34 J·sec = 4.136 10-15 eV·sec Speed of light, c = 3.00 108 m/sec hc = 1240 eV·nm Permeability, 0 = 410-7 Tm/A Compton Scattering - = C(1-cos) Relativistic kinematics E = g moc2 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 Name code A neutron traveling with velocity v = 1107 m/sec collides head-on with a nucleus (radius, r  10-14 m). Assuming the neutron decelerates uniformly between the outer radius of the nucleus and its center where it comes to rest and is trapped: a. estimate the magnitude of the average net force stopping the neutron. b. find the time to complete this reaction.

3. Problem 2 Name code  L R m Consider a “conic pendulum” : a mass, m, suspended by a cord of length, L, at an angle  from the vertical as it traces a circular path of radius R. a. Find the acceleration due to gravity, g, in terms of L, , and the pendulum’s period, T. b. Given L = 1m,  = 45o, T = 1.69 sec, compute g.

4. Problem 3 Name code For each bounce: H vi ℓ vf ℓ vh vh • A ball bounces down stairs, striking the center of each step, and bouncing each • time to the same height H above the step. The stair height equals its depth, ℓ, • and the coefficient of restitution e = -vf/vi is assumed given. vfand viare the • vertical velocities just after and before a bounce (see diagram above). • Show that the vertical components of the ball’s velocity immediately before • and after each bounce can be written: • Find an expression for the horizontal velocity vh in terms of only g, ℓ, and e.

5. Problem 4 Name code Consider an ideal diatomic gas enclosed in an insulated chamber with a movable piston. The values of the initial state variables are P1 = 8 atm, V1 = 4 m3 and T1 = 400 K. The final value of the pressure after an adiabatic expansion is P2 = 1 atm. Find V2 , T2 , W (the work done by the gas in expanding) and U (the change in the gas’ internal energy).Recall that for an ideal diatomic gas . V2 = T2 = W = U =

6. Problem 5 Name code • After the circuit shown in the figure at right has reached • the steady state, switch S1 is opened and S2 closed. • Calculate: • the frequency of oscillation. • the energy in the circuit. • the maximum current. 5mF 4mH S2 R=2 S1 + -  = 80v

7. Problem 6 Name code Consider the four charges shown, at the corners of a square with side, a. Calculate the energy in eV necessary to remove one of the charges to infinity. a = 2.810-10 m -e +e 2 1 a 3 4 a -e +e

8. Problem 7 Name code An electric field of 1.5 kV/m and a magnetic field of 0.40T act on a moving electron to produce no net force. Calculate the minimum speed of the electron. Draw a diagram of the vectors

9. Problem 8 Name code • Consider the circuit shown in the figure. • a. If the current in the straight wire is i, find the • the magnetic flux through the rectangular loop. • If the current i decreases uniformly from 90A • to zero in 15 msec, calculate the magnitude and direction of the induced current in the loop. The resistance of the loop is 5m. i ℓ = 50cm 10 cm 5 cm

10. Problem 9 Name code • A particle has total energy 1.123 MeV and momentum 1.00 MeV/c. • What is the particle’s (rest) mass? • Find the total energy of this particle in a reference frame in which its momentum is 2 MeV/c. • Find the particle’s velocities in the first and second frames.

11. Problem 10 Name code • The 12C16O molecule absorbs infrared radiation of frequency 6.421013 Hz. • The atomic masses are MC = 12 u and MO = 16 u. Assuming that the system • is a harmonic oscillator, find: • the ground-state vibrational energy of the CO molecule in eV. • the molecular force constant. • the classical amplitude of the ground-state vibrations.

12. Problem 11 Name code An electron is described by the 1-dimensional wavefunction where C is a constant. (a) Find the value of C that normalizes . (b) For what value of x is the probability for finding the electron largest? (c) Calculate the expectation value of x for this electron and comment on any difference you find between it and the most likely position.

13. Problem 12 Name code • X-rays, produced in a cathode tube of voltage 62 kV, undergo • Compton scattering in the backward direction. • What are the wavelengths of the incident and scattered X-rays? • What is the momentum of the recoil electrons? • What is the kinetic energy of the recoil electrons?