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Course: PHYS 521, Fall 2008
School: UVA
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Word Count: 549

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521 Phys Final Exam 10 December 2008 This is a closed book, closed notes exam, to be taken in a single three-hour period. The problems should be worked on separate pages and attached to this sheet when completed. There are six problems, which will be weighted equally. For full credit, be sure to show and explain all your work. Name: Signature: Some possibly useful formulas: cos( + ) = cos cos - sin sin...

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521 Phys Final Exam 10 December 2008 This is a closed book, closed notes exam, to be taken in a single three-hour period. The problems should be worked on separate pages and attached to this sheet when completed. There are six problems, which will be weighted equally. For full credit, be sure to show and explain all your work. Name: Signature: Some possibly useful formulas: cos( + ) = cos cos - sin sin sin( + ) = sin cos + cos sin 1 0 1 0 dx = 1.311 1 - x4 1 - x4 dx = 0.874 1 1 1 m x2 + y 2 + z 2 = m 2 + 2 2 + z 2 = m r2 + r2 2 + r2 sin2 2 2 2 2 T = 1 1. A bead of mass m slides without friction on a wire in the shape of a parabola, described by y = -a + x2 . 2a In addition, a spring with spring constant k stretches from the mass to a fixed point at the origin x = y = 0. For simplicity, take the unstretched length of the spring to be zero. (a) Choose a suitable generalized coordinate and calculate the Lagrangian. (b) Determine the equation of motion. (c) Find all equilibrium points and calculate the frequency for small oscillations about each. 2. For the system of problem 1, (a) Use conservation of energy to obtain an integral form for the solution of the motion. (b) For values of the energy slightly above the minimum possible value, use the integral from (a) to determine the period of the motion explicitly, as a function of the energy, m, k, and a. Check that your answer is consistent with what you obtained for problem 1(c). 3. The normal modes of a stretched string satisfy the Sturm-Liouville equation - d dx d dx + v = 2 for functions (x), v(x), and (x). Here x ranges a from to b. For boundary conditions d/dx|a = (a) and d/dx|b = (b), prove that two modes n and m are orthogonal according to b n (x)m (x)(x)dx = 0 a 2 2 as long as n = m . 2 4. A pendulum is constructed from a massless string of length a attached to a sphere of mass m and radius a, as shown. Calculate the two frequencies exhibited for small displacements from equilibirium. Recall that the moment of inertia for a sphere about its center is (2/5)ma2 . a a 5. Parabolic coordiates (, , ) are defined in terms of cylindrical coordinates (, z, ) by z= 1 ( - ) 2 and = . The angular coordinate is the same in both systems. (a) Calculate the Lagrangian using parabolic coordinates, for a particle of mass m moving in the central potential V = /r = / 2 + z 2 . (b) Calculate the Hamiltonian H(p, q), expressed in terms of the canonical momenta (p , p , p ). (c) Show that the Hamilton-Jacobi equation H S q , {q } , t + S =0 t is separable in this case, and write out the ...

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