ASSIGNMENT 2 Physics 218 Due Sep 10 Fall 2004 1. Consider the piano string of Prob. 2 in Assignment 1. That is, consider a steel piano wire D = 1 mm in diameter, ` =0 . 604 m long, with ρ = 8 g/cm 3 , and stretched with tension τ0 = 628 N, so it is tuned to middle C (261.6 Hz). If the string is oscillating in only its fundamental mode with A = 2 mm, what is the total energy of vibration? 2. Derive Eqs. (1.8.28), the orthogonality relations for sin( ω n t ) and cos( ω n t ). Recall that T 1 is the period of the fundamental, i.e. , T 1 =1 /f 1 =2 π/ω 1 . 3. (a) Calculate ∂η ( x, t ) /∂t for the Fourier series in Eq. (1.8.24). (b) Use the ∂η ( x, t ) /∂t that you calculated in Part (a) to derive Eq. (1.8.30) for the average kinetic energy of a vibrating string ±xed at x = 0 and x = ` : h K i = `λ0 8 ∞ X n =1 ω 2 n ( a 2 n + b 2 n ) (c) Combine Eqs. (1.8.29) and (1.8.30) to obtain the average total energy h E i in Eq. (1.8.31)
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