HW2_solution473

HW2_solution473 - ECE 473/TAM 413 Homework Assignment #2...

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Homework Assignment #2 Solutions - 1 ECE 473/TAM 413 Homework Assignment #2 Solutions 1. The tension in a string is provided by hanging a 3-kg mass at one end; the other end is rigidly fixed. The length of the string is 2.5 m and its mass is 50 g. Determine the phase speed of waves on the string. Tension in the string is T = F = mg = (3 kg)(9.81 m/s 2 ) = 29.4 N Mass per unit length is ! L = m L = 0.05 kg 2.5 m = 0.02 kg /m Phase speed is c = T ! L = 29.4 N 0.02 kg / m = 38.3m /s 2. Problem 2.3.1 in Kinsler et al. (show all work). Put your final answers in the form of a wave equation. (a) Assume the linear density varies with position, that is, ! L = ! L x ( ) . The tension force on each element is given by: df y = ! ! x T ! y ! x " # $ % & dx = T ! 2 y ! x 2 dx (see Eq. 2.3.4) Using Newton’s 2nd Law: f y = ma = m ! 2 y ! t 2 , but the mass for an element is: dm = ! L x ( ) dx for some element at x, giving df y = dm ( ) a = ! L x ( ) " 2 y " t 2 dx . Equating forces T ! 2 y ! x 2 dx = " L x ( ) ! 2 y ! t 2 dx and rearranging yields: ! 2 y ! t 2 = T " L x ( ) ! 2 y ! x 2 . (b) Assume the string hangs vertically supported only at the upper end. Therefore, the tension is given by the force of gravity and the mass of the string hanging below the position x for a dx element at x. Thus, the tension at x is given by: T = m x ( ) g = ! L xg. From (2.3.4), df y = ! ! x T ! y ! x " # $ % & dx = ! ! x ( L xg ! y ! x " # $ % & dx = ( L g ! ! x x ! y ! x " # $ % & dx Using Newton’s 2nd Law: df y = dm ( ) a = ! L " 2 y " t 2 dx , where dm = ! L dx . Equating forces ! L g " " x x " y " x # $ % & ( dx = ! L " 2 y " t 2 dx and rearranging yields: ! 2 y ! t 2 = g ! ! x x ! y ! x " # $ % &
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Homework Assignment #2 Solutions - 2 3. Problem 2.5.1 in Kinsler et al. y x,t ( ) moves 5 cm per second, that is, c = 5 cm/s 4. Given a finite-length string driven at x = 0 under steady-state conditions by Fe j ! t and supported rigidly at the other end at x = L, determine (a) the instantaneous input power !
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HW2_solution473 - ECE 473/TAM 413 Homework Assignment #2...

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