FINAL EXAM, PHYSICS 5306, Fall, 2003

Dr. Charles W. Myles

Take Home Final Exam: Distributed, Monday, December 8

DUE, IN MY OFFICE OR MAILBOX, 5PM, MON., DEC. 15. NO EXCEPTIONS!

TAKE HOME EXAM RULE: You are allowed to use almost any resources (books from the

library, etc.) to solve these problems. THE EXCEPTION is that you MAY NOT COLLABORATE WITH ANY OTHER PERSON in solving them! If you have questions or difficulties with these problems, you may consult with me, but not with fellow students (whether or not they are in this class!), with post-docs, or with other faculty. You are bound by the TTU Code of Student Conduct not to violate this rule! Anyone caught violating this rule will, at a minimum, receive an “F” on this exam!

INSTRUCTIONS: Please read all of these before doing anything else!!! Failure to follow these

may lower your grade!!

  1. PLEASE write on one side of the paper only!! This may waste paper, but it makes my grading easier!
  2. PLEASE do not write on the exam sheets, there will not be room! Use other paper!!
  3. PLEASE show all of your work, writing down at least the essential steps in the solution of a problem. Partial credit will be liberal, provided that the essential work is shown. Organized work, in a logical, easy to follow order will receive more credit than disorganized work.
  4. PLEASE put the problems in order and the pages in order within a problem before turning in this exam!
  5. PLEASE clearly mark your final answers and write neatly. If I cannot read or find your answer, you can't expect me to give it the credit it deserves and you are apt to lose credit.
  6. NOTE!!! the setup (THE PHYSICS) of a problem will count more heavily in the grading than the detailed mathematics of working it out.

PLEASE FOLLOW THESE SIMPLE DIRECTIONS!!!! THANK YOU!!!

NOTE! WORK ANY 5 OF THE 6 PROBLEMS! Each problem is equally weighted and worth 20 points for a total of 100 points on this exam.

Please sign this statement and turn it in with your exam:

I have neither given nor received help on this exam

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1.Consider the symmetric top, discussed in detail in Sect. 5.7 of Goldstein, in the special case in which the axis of rotation is vertical (i.e., the z-axis is vertical in Fig. 5.7, so that =0.) Show that the motion is either stable or unstable, depending on whether the quantity [4I1Mg]/[(I3)2(3)2] is less than or greater than unity. Hint: To do this, you have to consider the behavior of the top for values of  0. That is you need to find conditions under which =0 will be a minimum or a maximum of the effective potential V´() of Eq. 5.60. If the motion is stable & a small perturbation is applied, the top will perform small oscillations about =0. If it is unstable & a small perturbation is applied, the top will begin nutational motion. A sketch of V´() near =0 under various conditions might help. ALSO answer the following in complete, grammatically correct English sentences (NOT WIITH MATH!): If the top is set vertically spinning in the stable =0 configuration, what happens as, after some time, friction slowly reduces the value of 3? This is called the “sleeping” top.

NOTE!!!! WORK ANY 5 OUT OF THE 6 PROBLEMS!

2.See figure. A particle of mass m is attached to a rigid support by a spring of spring constant . At equilibrium, the spring hangs vertically downward. To this mass-spring combination is attached a 2ndidentical mass-spring oscillator. The spring of the 2nd oscillator is connected to the mass of the 1st oscillator. Consider vertical motion only.

a.Set up the Lagrangian and the Hamiltonian for the system. (Hint:These oscillators are obviously coupled, so the potential must have a coupling term in it!)

b.Compute the eigenfrequencies, the eigenvectors, and the normal modes of small oscillation the system.

c.How (if at all) does the gravitational force affect the eigenfrequencies?

3.Note: Parts a and b are obviously independent of each other!

  1. See figure. A neutron (mnc2 = 939.6 MeV) at rest, decays into a proton (mpc2 = 938.3 MeV), an electron (mec2 = 0.5 MeV), and an antineutrino (m 0). The three particles emerge at symmetrical angles in a plane, 120º apart. Using relativistic kinematics, find (put in numbers!) the (3-vector) momentum and the kinetic energy of each particle. I want NUMBERS (with proper units!) here, not just symbols!
  2. A particle of mass m, charge q, and kinetic energy T is moving at a relativistic velocityv (near c) perpendicular to a magnetic field B. The particle’s moves path is circular with radius r. Derive an expression for r in terms of m, q, T, and B.

4.Two masses, m1and m2are attached to the ends of a massless spring of spring constant k and zero relaxed length. The system oscillates AND ROTATES freely in a horizontal plane so that the gravitational potential energy does not enter the problem.

  1. How many degrees of freedom are there? Is there a constraint? If so, write the equation of constraint.
  2. Set up the Lagrangian for the system.Using Lagrange’s Equations, derive the equations of motion for the system.
  3. Set up the Hamiltonian for the system. Using Hamilton’s Equations, derive the equations of motion for the system. Show that these are equivalent to the equations of motion obtained in part b.
  4. Make the small oscillation approximation both for the displacements of the masses due to the spring and for their angular displacements as the system rotates. In this approximation find the eigenfrequencies, eigenvectors, and normal modes.

NOTE!!!! WORK ANY 5 OUT OF THE 6 PROBLEMS!

5.A particle of mass m is moving at a relativistic velocity v with respect to an specific inertial frame. The motion is confined to the x-axis. The particle is subject to a conservative force described by the potential energy V(x) = b(x3)/3, where b is a positive constant.

  1. Use the relativistic Lagrangian for a single particle, discussed in Ch. 7 of Goldstein, along with Lagrange’s Equation, to derive the equation of motion for the particle.
  2. Use the relativistic Hamiltonian for a single particle, discussed in Ch. 8 of Goldstein, along with Hamilton’s Equations, to derive equations of motion for the particle.
  3. Show that the equations of motion found in part b are equivalent to the equation of motion found in part a.
  4. Start with the equation of motion found in part a, and assume that 2 = (v2/c2) is small enough that only lowest order terms in 2need to be kept in this equation. Assuming initial conditions that x = 0 and v = v0 at time t = 0, go as far as you can in trying to solve this lowest order equation for v(t) and x(t). You might need to leave the solution in terms of messy integrals that need to be done numerically.

6.Part a is obviously independent of parts b, c and d!

a.Consider a force-free and torque-free symmetric body (principal moments of inertia I1 = I2 I3). Let x3 be the symmetry axis of the body and x3´ be the space axis in the vertical direction. Consider the case where the angular momentum vector L is parallel to x3´ and the angular velocity vector makes an angle  with the x3axis. Let both L and be initially in the x2 -x3 plane. Compute the angular velocity of precession (d/dt, where  is the usual Euler angle) as a function of I1, I3, , and .

See figure. A mass m is suspended by a massless string of length  from a massless support S and oscillates through an angle  in a vertical plane containing the x-axis. The support S slides without friction back and forth along the x-axis according to the equation x = A cos(t) where A is a constant.

b.Set up the Lagrangian and derive Lagrange’s Equation of motion. Don’t try to solve it!

c.Set up the Hamiltonian and derive Hamilton’s Equations of motion. Don’t try to solve them!

d.Answer the following in complete, grammatically correct English sentences. Is the Hamiltonian equal to the total energy? Why or why not? Is the Hamiltonian conserved? Why or why not? Is the total energy conserved? Why or why not?