Threshold of Pain. You are investigating the report of a UFO landing in an isolated portion of New Mexico, and you encounter a strange object that is radiating sound waves uniformly in all directions. Assume that the sound comes from a point source and that you can ignore reflections. You are slowly walking toward the source. When you are \({\bf{7}}.{\bf{5}}{\rm{ }}{\bf{m}}\)from it, you measure its intensity to be\(0.11 {W \mathord{\left/

{\vphantom {W {{m^2}}}} \right.

\kern-\nulldelimiterspace} {{m^2}}}\). An intensity of \(1.0 {W \mathord{\left/

{\vphantom {W {{m^2}}}} \right.

\kern-\nulldelimiterspace} {{m^2}}}\)is often used as the “threshold of pain.” How much closer to the source can you move before the sound intensity reaches this threshold?

Step by step solution

01

Given data

\(\begin{aligned}{l}{I_1} = 0.11\,{W \mathord{\left/

{\vphantom {W {{m^2}}}} \right.

\kern-\nulldelimiterspace} {{m^2}}}\\{r_1} = 7.5\,m\\{I_2} = 1.0\,{W \mathord{\left/

{\vphantom {W {{m^2}}}} \right.

\kern-\nulldelimiterspace} {{m^2}}}\end{aligned}\)

02

Concept/ Formula used 

For a Point source

\(I = \frac{P}{{4\pi {r^2}}}\)

\(\frac{{{I_1}}}{{{I_2}}} = \frac{{r_2^2}}{{r_1^2}}\)

03

Threshold distance  

\(\begin{aligned}{c}\frac{{{I_1}}}{{{I_2}}} = \frac{{r_2^2}}{{r_1^2}}\\{r_2} = {r_1}\sqrt {\frac{{{I_1}}}{{{I_2}}}} \\ = 7.5\sqrt {\frac{{0.11{W \mathord{\left/

{\vphantom {W {{m^2}}}} \right.

\kern-\nulldelimiterspace} {{m^2}}}}}{{1.0\,{W \mathord{\left/

{\vphantom {W {{m^2}}}} \right.

\kern-\nulldelimiterspace} {{m^2}}}}}} \\ = 2.5\,m\end{aligned}\)

So it is possible to move\({r_1} - {r_2} = 7.5\,m - 2.5 = 5.0\,m\)closer to source.

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