Q26E
Question
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
Verified\(5.0\,m\) closer to source.
\(\begin{aligned}{l}{I_1} = 0.11\,{W \mathord{\left/
{\vphantom {W {{m^2}}}} \right.
\kern-\nulldelimiterspace} {{m^2}}}\\{r_1} = 7.5\,m\\{I_2} = 1.0\,{W \mathord{\left/
{\vphantom {W {{m^2}}}} \right.
\kern-\nulldelimiterspace} {{m^2}}}\end{aligned}\)
For a Point source
\(I = \frac{P}{{4\pi {r^2}}}\)
\(\frac{{{I_1}}}{{{I_2}}} = \frac{{r_2^2}}{{r_1^2}}\)
\(\begin{aligned}{c}\frac{{{I_1}}}{{{I_2}}} = \frac{{r_2^2}}{{r_1^2}}\\{r_2} = {r_1}\sqrt {\frac{{{I_1}}}{{{I_2}}}} \\ = 7.5\sqrt {\frac{{0.11{W \mathord{\left/
{\vphantom {W {{m^2}}}} \right.
\kern-\nulldelimiterspace} {{m^2}}}}}{{1.0\,{W \mathord{\left/
{\vphantom {W {{m^2}}}} \right.
\kern-\nulldelimiterspace} {{m^2}}}}}} \\ = 2.5\,m\end{aligned}\)
So it is possible to move \({r_1} - {r_2} = 7.5\,m - 2.5 = 5.0\,m\) closer to source.