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Unformatted text preview: 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 is fixed at both ends. Part 1 [1 point] : What is the angular frequency 1 of the first mode (n=1)? Using the now familiar expression n = ( n / L ) T / , one has that 1 = L T . 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 x L cos ( n t + n ) , where n = n L T . Part 3 [2 points] : Suppose that the is released at time t = 0 with the following displace- ment: y ( x , t = ) = 4 h L x x < L /4 2 h- 4 h L x L /4 x < L /2 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.) Lets calculate the coefficients of the Fourier series,...
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