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FINAL EXAM, PHYSICS 5306, Fall, 2002Dr. Charles W. Myles

Take Home Final Exam: Distributed, Wednesday, December 4DUE, IN MY OFFICE OR MAILBOX, 5PM, TUES., DEC. 10 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!) or with other faculty. You are bound by the TTU Codeof Student Conduct not to violate this rule! Anyone caught violating this rule will, at aminimum , receive an F on this exam!

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

5. PLEASE clearly mark your final answers and write neatly. If I cannot read or find your answer, youcan'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 thedetailed 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 andworth 20 points for a total of 100 points on this exam. Note! The problems from Goldstein arefrom the 3rd Edition ! If you have the 2 nd Edition, the problem numbering MIGHT be different !

Please sign this statement and turn it in with your exam:I have neither given nor received help on this exam

_______________________________ Signature

1. Work Problem 20 of Chapter 5 in the book by Goldstein ( 3 rd Edition !). Go as far as you cantowards obtaining a solution to the equations of motion.

2. Work Problem 13 of Chapter 6 in the book by Goldstein ( 3 rd Edition !). Go as far as you can

towards obtaining the eigenfrequencies and the normal mode eigenvalues for smalloscillations.

3. a. Work Problem 7 of Chapter 7 in the book by Goldstein ( 3 rd Edition !). b. Work Problem 8 of Chapter 7 in the book by Goldstein ( 3 rd Edition !).

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NOTE! WORK ANY 5 (FIVE) OUT OF THE 6 PROBLEMS!4. See the figures. A homogeneous cube, each edge of which has a length , is initially in a

position of unstable equilibrium with one edge in contact with a horizontal plane. The cube isthen given a small displacement and allowed to fall. Find the angular velocity of the cube atthe instant it strikes the plane if:

a.The edge cannot slide on the plane. See figure

b. Sliding without friction can occur on the plane. See figure

Hint: in both cases will only depend on g, and geometric factors. The best way to solve this isto use conservation of energy. You will need to know (or to calculate) the moment of inertia of acube about an axis perpendicular to one face and passing through that face center.

5. See the figure. The large mass, M , is constrained to move on a smooth,horizontal frictionless track. A small mass, m , is connected to M by amassless, inextensible string of length .

a. Set up the Lagrangian and derive Lagranges Equations of motion.b. Under the assumption of small oscillations, set up and solve the secular

equation for the normal mode eigenfrequencies and eigenvalues.

6. A particle of mass M and 4-momentum P decays into two particles of masses

m 1 and m 2.a. Use conservation of 4-momentum in the form P = p1 + p2 (where p1 and p2 are the 4

momenta of m 1 and m 2, respectively) to show that the total energy of m 1 in the restframe of the decaying particle is given by: (E 1/c2) = [M 2 + (m 1)2 (m 2)2]/(2M)and that E 2 is obtained from this expression by interchanging m 1 and m 2.

b. Show that the kinetic energy of the ith (i=1,2 ) particle in the same frame isT i = M[1 (m i/M) - ( M/M)], where M = M - m 1 m 2 is the mass excess or Q value of the process.

c. The charged pi-meson ( Mc 2 = 139.6 MeV ) decays into a mu-meson ( m 1c2 = 105.7MeV ) and a neutrino ( m 2 0). Calculate ( PUT IN NUMBERS!! ) the kinetic energiesof the mu-meson and the neutrino in the pi-mesons rest frame.