410aex2soln - Final Exam Phys410 Fall 2011 Prof. Ted...

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Unformatted text preview: Final Exam Phys410 Fall 2011 Prof. Ted Jacobson www.physics.umd.edu/grt/taj/410a/ jacobson@umd.edu Point values are the numbers in brackets. Total points = 50 . Write clearly and explain your work to be eligible for partial credit. 1. Consider two blocks sliding on a frictionless surface in one dimension, one block with mass m and the other with mass 2 m . The blocks are connected to each other by a spring with spring constant k . The block on the left (right) is connected to a wall to its left (right) with a spring of the same spring constant k . Find the normal mode frequencies for this system. [10] Solution : Use the deviations x 1 , 2 from the equilibrium positions of the masses as the two generalized coordinates. Then L = 1 2 m x 2 1 + 1 2 2 m x 2 2- 1 2 k ( x 2 1 + ( x 2- x 1 ) 2 + x 2 2 ). m x 1 =- k [ x 1- ( x 2- x 1 )] =- k (2 x 1- x 2 ) and 2 m x 2 =- k [( x 2- x 1 ) + x 2 ] =- k (2 x 2- x 1 ). M x =- K x , with M = m 1 0 0 2 and K = k 2- 1- 1 2 . For a normal mode x =- 2 x , so ( K- 2 M ) x = 0, which has a solution for x if and only if det( K- 2 M ) = 0. In units with m = k = 1 we have det( K- 2 M ) = (2- 2 )(2- 2 2 )- 1 = 2 4- 6 2 + 3, whose roots are 2 = (3 3) / 2, hence = q (3 3) / 2 p k/m . 2. Suppose a positron with energy E annihilates with an electron at rest, producing two photons. (Electrons and positrons both have rest mass m e .) (a) Show that the two photons cannot both travel in the same direction. [2] (b) Assuming one of the photons travels in the same direction as the original positron, and the other photon travels in the opposite direction, determine the energies of the two photons. [7] (c) If E m e find the leading order approximations to the two photon energies. (Make sure your result makes sense!) [1] Solution : (a) The total incoming 4-momentum is the sum of two timelike 4-momenta, hence...
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This note was uploaded on 12/29/2011 for the course PHYSICS 410 taught by Professor Jacobson during the Fall '11 term at Maryland.

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410aex2soln - Final Exam Phys410 Fall 2011 Prof. Ted...

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