# Final Exam (Exam Iii) Physics 4304

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FINAL EXAM (EXAM III) PHYSICS 4304

MAY 6, 2004

Dr. Charles W. Myles

1. PLEASE write on one side of the paper only!! It wastes paper, but it makes my grading easier!
2. PLEASE don’t write on the exam sheets, there won’t be room! If you don’t have paper, I’ll give you some.
3. PLEASE show ALL work, writing down at least the essential steps in the problem solution. Partial credit will be liberal, provided that essential work is shown. Organized work, in a logical, easy to follow order will receive more credit than disorganized work.
4. The problem setup (PHYSICS) will count more heavily than the detailed mathematics of working it out.
5. PLEASE write neatly. Before handing in your solutions, PLEASE:a) number the pages and put them in numerical order, b) put the problem solutions in numerical order, and c) clearly mark your final answers. If I can’t read or find your answer, you can't expect me to give it the credit it deserves.

THANK YOU!!

NOTE!!!! Work any three (3) of the four problems. Each problem is equally weighted and worth 33 points. Everyone gets a 1 point “bonus” for a total of 100 points on this exam!

If you have read these instructions, please sign the line below, and I will add ten (10) points to your final exam grade to partially compensate for extra hours spent on evening exams and also for the fact that this class was held at the uncivilized hour of 8am!! To get this credit, remember to turn this page in with your exam solutions! Note that, those who have not signed this line will clearly not have read the instructions and they will not receive this extra 10 points! Have a good summer!

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NOTE!!!! Work any three (3) of the four problems!!

1. See figure. A mass m is attached to one end of an ideal, massless spring of spring constant k and relaxed length . The other end of the spring is attached to the ceiling, so that the system hangs vertically. The spring is simultaneously pulled to the side, as shown, and stretched past its relaxed length . The system is then released. The resulting motion is thus a superposition of plane pendulum and simple harmonic oscillator motions. Suggestion: Work this problem in plane polar coordinates!
1. Write expressions for the kinetic energy, the potential energy, and the Lagrangian of

the system. How many degrees of freedom are there for this system? (6 points)

1. Derive the equations of motion using Lagrange's equations. (7 points)
2. Derive expressions for the generalized momenta of the system. Write an expression for the Hamiltonian for the system. (7 points)
3. Derive the equations of motion using Hamilton's equations. Show that this is equivalent to the results obtained in part b. (7 points)
4. What are the constants of the motion? (3 points)
5. Under what conditions will this system reduce to a plane pendulum? Under what conditions will it reduce to a simple harmonic oscillator? (3 points)
1. A particle of mass m is subject to an attractive, time-dependent Central Force given by

F(r,t) = -[kr2]e-αt, where k and α are constants. Suggestion: Work this problem in plane polar coordinates!

1. Write expressions for the kinetic energy, the potential energy, and the Lagrangian ofthe system. How many degrees of freedom are there for this system? (6 points)
2. Derive the equations of motion using Lagrange's equations. (7 points)
3. Derive expressions for the generalized momenta of the system. Write an expression for the Hamiltonian for the system. (7 points)
4. Derive the equations of motion using Hamilton's equations. Show that this is equivalent to the results obtained in part b. (7 points)
5. Are there constants of the motion? If so, what are they? (3 points)
6. Is the Hamiltonian equal to the total energy? Is total energy conserved? (3 points)

NOTE!!!! Work any three (3) of the four problems!!

1. Note: Each of the following problems deal with a particle of mass μ moving in a central force field. Parts a, b, and c are independent ofparts d and e!

For parts a, b, and c the particle moves under the action of an attractive central force, which has a potential energy of the form (k is a positive constant): U(r) = -[k/r3].

1. For fixed angular momentum , write the effective potential energy V(r). Make a sketch of V(r) and qualitatively DISCUSS the radial motion for various particle energies E. Include cases of both E > 0 and E < 0 in this discussion! (7 points)
2. For a particular angular momentum 0, the particle is observed to be in a circular orbit of radius r0. (Recall that a circular orbit means that the total mechanical energy E0is equal to the minimum value of the effective potential V(r)). Calculate the angular momentum 0 for this circular orbit. (7 points)
3. For the circular orbit of part b, calculate the total mechanical energy E0, the particle speed v0 and the period T0. (7 points)

NOTE:The answers to parts b, and c should depend on (at most) k, μ, and r0.

For parts d, and e, the particle orbit is given by the equation r(θ) = A(θ)2, where A is a positive constant.

1. Calculate the force function F(r) and the corresponding potential energy function U(r). (HINT: The easiest way to solve this is to use the differential equation for the orbit, rather than the integral form.) (6 points)
2. Calculate r(t) and θ(t) for the particle. In addition to the time t, these will depend on μ, A, and . Don’t forget the integration constants! (6 points)
1. This problem requires that you know some details about elliptic orbits. In parts a –f, I want NUMBERS with proper UNITS!! Planet X has mass m = 1.5 1025 kg. It is in an elliptic orbit about the sun. The orbit eccentricity is ε = 0.25. The semi-major axis of the orbit is given as a = 7.0 1011 m. (All distances are measured from the center of the sun.) The sun’s mass is M = 2.0 1030 kg. The gravitational constant is G = 6.67 10-11 N m2/kg2.

Calculate:

1. The maximum and minimum distances of the planet from the sun (aphelion and perihelion), rmaxand rmin, and the semi-minor axis, b, of the orbit. (6 points)
2. The total angular momentum, , of the planet. (6 points)
3. The total mechanical energy, E, of the planet. (6 points)
4. The period of the orbit. (6 points)
5. The speed, vmaxof the planet when it is at perihelion (rmin). Its speed, vminwhen it is at aphelion (rmax). (6 points)
6. The speed, v of the planet when it is a distance r = 6.2 1011 m from the sun. (6 points)
7. Write the equation (with numerical factors explicitly evaluated) which describes the orbit r(θ). (3 points)