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Phys 2214 hw4 Solutions

Phys 2214 hw4 Solutions - Phys 2214 Homework#4 Solutions 1...

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Phys 2214 Homework #4 Solutions September 30, 2011 1. For a long, thin rod, the compressions created by a longitudinal wave traveling along the rod will change the length to a greater extent than the diameter. So, we can replace the equation for the bulk modulus, p = - B V/V ), by the equation for Young’s modulus, F/A = Y x/x ). s(x+Δx,t) x+Δx x s(x,t) (a) In general, the change in length of the segment is Δ L = [ x + Δ x + s ( x + Δ x, t )] - [ x + s ( x, t )] - Δ x = s ( x + Δ x, t ) + x + Δ x - x - Δ x - s ( x, t ) = s ( x + Δ x, t ) - s ( x, t ) . 1
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(b) The linear stress-strain relation is F A = Y Δ L L = Y s ( x, t ) - s ( x + Δ x, t ) Δ x , where the fact that Δ L is negative has been used and the original length is Δ x . (c) The force is given by F ( x, t ) = lim Δ x 0 Y A s ( x, t ) - s ( x + Δ x, t ) Δ x = - Y A ∂s ∂x . (d) The net force acting on the segment is F ( x, t ) - F ( x + Δ x, t ). Then Newton’s 2 nd law gives - Y A " ∂s ( x, t ) ∂x - ∂s ( x + Δ x, t ) ∂x # = ρ 0 A Δ x 2 s ∂t 2 . Dividing both sides by Δ x and taking the limit as Δ x 0 gives lim Δ x 0 h ∂s ( x,t ) ∂x - ∂s ( x x,t ) ∂x i Δ x = - 2 s ∂x 2 . We then get the wave equation 2 s ∂x 2 = ρ 0 Y 2 s ∂t 2 , which, upon comparison with the general wave equation, gives v = q Y/ρ 0 . 2. The fundamental frequency is given as f 1 = 150 Hz .
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