   Chapter 16, Problem 51AP Physics for Scientists and Enginee...

10th Edition
Raymond A. Serway + 1 other
ISBN: 9781337553278

Solutions

Chapter
Section Physics for Scientists and Enginee...

10th Edition
Raymond A. Serway + 1 other
ISBN: 9781337553278
Textbook Problem

A pulse traveling along a string of linear mass density μ is described by the wave function y = [ A 0 e − k x ] sin   ( k x − ω t ) where the factor in brackets is said to be the amplitude. (a) What is the power P(x) carried by this wave at a point x? (b) What is the power P(0) carried by this wave at the origin? (c) Compute the ratio P(x)/P(0).

(a)

To determine

The power P(x) carried by this wave at a point.

Explanation

The wave function for string is given as,

y(x,t)=(A0ebx)sin(kxωt)

Formula to calculate the velocity of wave for small segment of string is,

v=dydt

Here, dy is the distance travel by wave in small segment and dt is the small time interval.

Substitute (A0ebx)sin(kxωt) for y.

v=d(A0ebx)sin(kxωt)dtv=ω(A0ebx)cos(kxωt)

Assume the mass presents in the small segment dx of string is,

dm=μdx

Here, μ is the linear mass density and dx is the small length of the string.

Formula to calculate the kinetic energy for small segment is,

dK=12dmv2        (1)

Here, dK is the kinetic energy for small segment and dm is the mass of small segment of string.

Substitute μdx for m and ω(A0ebx)cos(kxωt) for v in equation (1).

dK=12μdx(ω(A0ebx)cos(kxωt))2=12μω2(A0ebx)2cos2(kxωt)dx        (2)

Integrate the equation (2) over all the string elements in the wavelength of the waves for total kinetic energy.

dK=0λ12μω2(A0ebx)2cos2(kxωt)dxK=14μω2(A0ebx)2λ

Formula to calculate the potential energy for string in small segment is,

dp=12dmωy2        (3)

Here, dp is the potential energy present in the small segment of string.

Substitute μdx for m and (A0ebx)sin(kxωt) for y in equation (3)

(b)

To determine

The power P(0) carried by this wave at the origin.

(c)

To determine

The ratio of P(x)P(0).

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