Chapter 16, Problem 51AP

### Physics for Scientists and Enginee...

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

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,

â€‚Â y(x,t)=(A0eâˆ’bx)sin(kxâˆ’Ï‰t)

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

â€‚Â v=dydt

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

Substitute (A0eâˆ’bx)sin(kxâˆ’Ï‰t) for y.

â€‚Â v=d(A0eâˆ’bx)sin(kxâˆ’Ï‰t)dtv=âˆ’Ï‰(A0eâˆ’bx)cos(kxâˆ’Ï‰t)

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

â€‚Â dm=Î¼dx

Here, Î¼ is the linear mass density and dx is the small length of the string.

Formula to calculate the kinetic energy for small segment is,

Â Â Â Â dK=12dmv2Â Â Â Â Â Â Â Â (1)

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

Substitute Î¼dx for m and âˆ’Ï‰(A0eâˆ’bx)cos(kxâˆ’Ï‰t) for v in equation (1).

Â Â Â Â dK=12Î¼dx(âˆ’Ï‰(A0eâˆ’bx)cos(kxâˆ’Ï‰t))2=12Î¼Ï‰2(A0eâˆ’bx)2cos2(kxâˆ’Ï‰t)dxÂ Â Â Â Â Â Â Â (2)

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

Â Â Â Â âˆ«dK=âˆ«0Î»12Î¼Ï‰2(A0eâˆ’bx)2cos2(kxâˆ’Ï‰t)dxK=14Î¼Ï‰2(A0eâˆ’bx)2Î»

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

Â Â Â Â dp=12dmÏ‰y2Â Â Â Â Â Â Â Â (3)

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

Substitute Î¼dx for m and (A0eâˆ’bx)sin(kxâˆ’Ï‰t) 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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