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Chapter 18: Problem 17

An observer measures an intensity of \(1.13 \mathrm{~W} / \mathrm{m}^{2}\) at an unknown distance from a source of spherical waves whose power output is also unknown. The observer walks \(5.30 \mathrm{~m}\) closer to the source and measures an intensity of \(2.41 \mathrm{~W} / \mathrm{m}^{2}\) at this new location. Calculate the power output of the source.

Short Answer

Expert verified
The power output of the source is approximately \(792 W\).

Step by step solution

01

Express the intensities in terms of distances and power output

The intensity \(I\) at a distance \(r\) from a source of power \(P\) is given by: \(I = \frac{P}{4πr^{2}}\). Thus:Intensities at the two locations can be represented as follows:\(I_{1} = \frac{P}{4πr_{1}^{2}}\)\(I_{2} = \frac{P}{4πr_{2}^{2}}\)
02

Solve the equation for the power output

Divide the first equation by the second equation. The power output \(P\) cancels out, leaving:\(\frac{I_{1}}{I_{2}} = \frac{r_{2}^{2}}{r_{1}^{2}}\)We can rearrange this equation to get:\(\frac{r_{2}^{2}}{r_{1}^{2}} = \frac{I_{1}}{I_{2}}\)Remembering that \(r_{2} = r_{1} - 5.30 m\), we get:\(\frac{(r_{1} - 5.30)^{2}}{r_{1}^{2}} = \frac{1.13}{2.41}\)After cross-multiplying and simplifying, we obtain a quadratic equation for \(r_{1}\). We only consider the positive root, since distance cannot be negative:\(r_{1} ≈ 13.6 m\)
03

Calculate the power output

Substitute the obtained value for \(r_{1}\) into the first equation:\(P = 4πr_{1}^{2}I_{1} ≈ 792 W\), which is the power output of the source.

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Most popular questions from this chapter

The equation of a particular transverse wave on a string is $$ y=(1.8 \mathrm{~mm}) \sin [(23.8 \mathrm{rad} / \mathrm{m}) x+(317 \mathrm{rad} / \mathrm{s}) t] $$ The string is under a tension of \(16.3 \mathrm{~N}\). Find the linear mass density of the string.

The equation of a transverse wave traveling along a string is given by $$ y=(2.30 \mathrm{~mm}) \sin [(1822 \mathrm{rad} / \mathrm{m}) x-(588 \mathrm{rad} / \mathrm{s}) t] $$ Find \((a)\) the amplitude, \((b)\) the frequency, \((c)\) the velocity, \((d)\) the wavelength of the wave, and ( \(e\) ) the maximum transverse speed of a particle in the string.

A string \(2.72 \mathrm{~m}\) long has a mass of \(263 \mathrm{~g}\). The tension in the string is \(36.1 \mathrm{~N}\). What must be the frequency of traveling waves of amplitude \(7.70 \mathrm{~mm}\) in order that the average transmitted power be \(85.5 \mathrm{~W}\) ?

Determine the amplitude of the resultant wave when two sinusoidal waves having the same frequency and traveling in the same direction are combined, if their amplitudes are \(3.20 \mathrm{~cm}\) and \(4.19 \mathrm{~cm}\) and they differ in phase by \(\pi / 2 \mathrm{rad}\).

The equation of a transverse wave traveling along a very long string is given by $$ y=(6.0 \mathrm{~cm}) \sin [(2.0 \pi \mathrm{rad} / \mathrm{m}) x+(4.0 \pi \mathrm{rad} / \mathrm{s}) t] $$ Calculate \((a)\) the amplitude, \((b)\) the wavelength, \((c)\) the frequency, \((d)\) the speed, \((e)\) the direction of propagation of the wave, and \((f)\) the maximum transverse speed of a particle in the string.
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