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ph2a_quiz2_soln

# ph2a_quiz2_soln - Solution by T.A David Nichols Physics 2a...

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Solution by T.A. David Nichols Physics 2a Quiz #2 Solution October 29, 2008 Physics 2a Quiz #2 Solution A string with length L , tension T , and mass per unit length μ is fixed at both ends. Part 1 [1 point] : What is the angular frequency ω 1 of the first mode (n=1)? Using the now familiar expression ω n = ( n π / L ) T / μ , one has that ω 1 = π L T μ . Part 2 [1 point] : Write down the most general Fourier series for the transverse displace- ment y ( x , t ) of the string. Because both ends are fixed, the most general Fourier series is y ( x , t ) = n = 1 A n sin n π x L cos ( ω n t + δ n ) , where ω n = n π L T μ . Part 3 [2 points] : Suppose that the is released at time t = 0 with the following displace- ment: y ( x , t = 0 ) = 4 h L x 0 x < L /4 2 h - 4 h L x L /4 x < L /2 0 L /2 x L Write down an expression for the first four coefficients A n (for n = 1, 2, 3, 4) in the series expansion for y ( x , t ) . (Assume the string is released from rest.) Let’s calculate the coefficients of the Fourier series, A n , for a general integer n and eval- uate them for n = 1, 2, 3, 4 at the end (this ultimately will require less work). Since the

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