Qu1_solution - Physics 325 Quiz#1 Solutions Wednesday 27...

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Physics 325 Quiz #1 Solutions Wednesday 27 September 2006 1a) In one dimension, U ( x ) = - Z F ( x ) dx = - A k sin kx + C It is the negative of the sine function, with a first minimum at x = π/ 2 k . The value of the integration constant C does not change any result of the problem, and it can be set to any number. 1b) All stable equilibrium points are of the form x n = 2 nπ/k + π/ 2 k = (4 n + 1) π/ 2 k , where n is an integer,positive, negative, or zero. All unstable equilibrium points are of the form x j = 3 π/ 2 k + 2 jπ/k = (4 j + 3) π/ 2 k , where j is a positive, negative, or zero integer. 1c) The frequency of small oscillations about each of the stable equilibrium points is the same as any of the others because U ( x ) is periodic. Let’s pick the point at x 0 = π/ 2 k . Let the deviation at x 0 be δx , then we can do a Taylor expansion around the point x 0 . F ( x ) = A cos kx 0 - Ak sin( kx 0 ) δx + ... ≈ - Akδx By the equation for angular frequency in harmonic oscillation, we have
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Qu1_solution - Physics 325 Quiz#1 Solutions Wednesday 27...

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