Elementary Qualifier Examination October 13, 2003 NAME CODE: [ ] Instructions: (a)Do any ten (10) of the twelve (12) problems of the following pages. (b)Indicate on this page (below right) which 10 problems you wish to have graded. (c)If you need more space for any given problem, write on the back of that problem’s page. (d)Mark your name code on all pages. (e)Be sure to show your work and explain what you are doing. (f)A table of integrals is available from the proctor. Possibly useful information: e = 1.60  C m e = 511 keV/c 2 1 atm =  10 5 N/m 2 g = 9.8 m/sec 2 m n = 1.67  kg 1u = 1.66  kg Compton wavelength, C = 2.43  m Coulomb’s constant, k=1/(4  o )=9.0  10 9 N·m 2 /c 2 Planck constant h =  J·sec =  eV·sec Speed of light, c = 3.00  10 8 m/sec hc = 1240 eV·nm Permeability,  0 = 4  10  7 T  m/A Compton Scattering  = C (1-cos  ) Relativistic kinematics E =  m o c 2 Check the boxes below for the 10 problems you want graded Problem Number Score Total

Name code Problem 1 A neutron traveling with velocity v = 1  10 7 m/sec collides head-on with a nucleus (radius, r  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.

Name code Problem 2 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,  = 45 o, T = 1.69 sec, compute g. R L  m

Name code Problem 3 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 =  v f /v i is assumed given. v f and v i are the vertical velocities just after and before a bounce (see diagram above). A.Show that the vertical components of the ball’s velocity immediately before and after each bounce can be written: B.Find an expression for the horizontal velocity v h in terms of only g, ℓ, and e. ℓ ℓ H For each bounce: vivi vfvf vhvh vhvh

Name code Problem 4 Consider an ideal diatomic gas enclosed in an insulated chamber with a movable piston. The values of the initial state variables are P 1 = 8 atm, V 1 = 4 m 3 and T 1 = 400 K. The final value of the pressure after an adiabatic expansion is P 2 = 1 atm. Find V 2, T 2, 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. V 2 = T 2 = W =  U =

Name code Problem 5 After the circuit shown in the figure at right has reached the steady state, switch S 1 is opened and S 2 closed. Calculate: the frequency of oscillation. the energy in the circuit. the maximum current F5F  = 80v 4mH R=2  S2S2 S1S1

Name code Problem 6 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  m e+e +e+e ee ee a a

Name code Problem 7 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

Name code Problem 8 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. b.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 . 5 cm 10 cm ℓ = 50cm i

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

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

Name code Problem 11 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.

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