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FINAL EXAM, PHYSICS 5306, Fall, 2005Dr. Charles W. Myles

Take Home Final Exam: Distributed, Tuesday, December 7

Due, in my office or mailbox, 5pm, Thurs., Dec. 15.NO EXCEPTIONS!ExamRULE: You are allowed to use almostany resources (library books, etc.) to solve these

problems.EXCEPTION: YouMAY NOT COLLABORATE WITH ANY OTHERPERSONin solving them! If you have questions or difficulties with these problems, consultwith me, notwith fellow students (whether or not they are in this class!), with post-docs, or withother 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 thesemay lower your grade!!

1. PLEASE write on one side of the paper only!! This may waste paper, but it makes mygrading 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. Organizedwork, 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 inthis exam!

5. PLEASE clearly mark your final answers and write neatly. If I cannot read or find youranswer, 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 gradingthan 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 andworth 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_______________________________

Signature

1. Work Problem 29 of Chapter 5 in the book by Goldstein. Go as far as you can towardsobtaining a solution to the equations of motion.

2.

a. Work Problem 18 of Chapter 7 in the book by Goldstein.b. Work Problem 26 of Chapter 7 in the book by Goldstein.

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

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3. Note: The following problems are independent of one another! Parts a & b are aboutrelativity. Part c is about rigid body rotation!a. The 3 kilometer long Stanford Linear Accelerator accelerates electrons (rest energy mec2 = 0.5

MeV) to energy 50 GeV(50 109 eV). Use relativistic kinematics to calculate (PUT INNUMBERS!) the speed of the electrons at the end.

b. A free neutron (rest energy 939.6 MeV) is unstable & decays into a proton (rest energy mpc2 =938.3 MeV) & an electron (rest energy mec2 = 0.5 MeV). Use relativistic kinematics to calculate(PUT IN NUMBERS!) the energy released in this process.

c. See Figure. A pendulum consists of a uniform rod of length& mass m. It is suspended in a vertical plane by one end.At the other end, a uniform disk of mass M is attached sothat it can rotate freely in its own plane, which is the verticalplane. The rod & the disk each have negligible thickness. Setup the equations of motion for this system in the Lagrangianformalism.

4. Parts a-c concern a particle of mass m moving in one dimension under the influence of a time& position dependent force given by F(x,t) = (kx-2)e-t/, where k& are positive constants.

a. Set up the Lagrangian & derive Lagranges Equation of motion.b. Set up the Hamiltonian & derive Hamiltons Equations of motion. Show that these are

equivalent to the equation of motion obtained in part a.c. Compare the Hamiltonian & the total mechanical energy. Are these the same? Is energy

conserved for this system?

See Figure. Parts d-fconcern a plane pendulum of mass m attached to amassless, rigid rod. After the pendulum is set into motion, the length isshortened at a constant rate . That is, (d/dt) = - . At time t = 0, the rodlength is 0.The suspension point remains fixed. (Hint: The condition on (d/dt) isa time-dependent constraint which fixes the time dependence of).

d. How many degrees of freedom are there for this system? Set up theLagrangian & derive Lagranges Equation of motion.

e. Set up the Hamiltonian & derive Hamiltons Equations of motion. Show that these areequivalent to the equation of motion obtained in part d.

f. Compare the Hamiltonian & the total mechanical energy. Are these the same? Is energyconserved for this system?

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

5. See Figure. A massless ideal spring of relaxed length b & springconstant connects two particles of masses m1 & m2. The systemrests on a frictionless table & may oscillate (is its time

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dependent length while it is oscillating)AND rotate in the plane of the table ( is the angle ofrotation in the plane).The figure looks down on the plane of the table.

a. How many degrees of freedom are there for this system? Set up the Lagrangian for thissystem & derive the Lagrange Equations of motion.

b. Set up the Hamiltonian for this system & derive Hamiltons Equations of motion.c. Compare the Hamiltonian & the total mechanical energy. Are these the same?d. Is energy conserved for this system? Are there any other constants of the motion (physical

quantities which are conserved)?

6. A plane triatomic molecule consists of equal masses m at the vertices of a triangle. If themasses were at rest, the masses would be on an equilateral triangle of side b. The molecule isheld together by forces that are harmonic for small oscillations & the force constants are eachequal to k. Consider motion in the plane of the molecule only.a. Write expressions for the kinetic energy, the potential energy, & the Lagrangian of the

system.

b. Derive the equations of motion using Lagranges equations.c. Assume small oscillations. Set up & solve the solve the secular equation for the normal

mode eigenfrequencies & eigenvectors. How many normal modes are there? Do any ofthem correspond to = 0?

Answer the following question using mostly WORDS, in complete, grammatically correctEnglish sentences! You may also make a sketch if that helps your discussion.

d. Qualitatively discuss the motion of the 3 masses in each of the normal modes.