Q27E
Question
Energy Output. By measurement you determine that sound waves are spreading out equally in all directions from a point source and that the intensity is \(0.026 {W \mathord{\left/
{\vphantom {W {{m^2}}}} \right.
\kern-\nulldelimiterspace} {{m^2}}}\) at a distance of \(4.3 m\) from the source. (a) What is the intensity at a distance of \(3.1 m\) from the source? (b) How much sound energy does the source emit in one hour if its power output remains constant?
Step-by-Step Solution
Verified(a) \(0.050\,{W \mathord{\left/
{\vphantom {W {{m^2}}}} \right.{{m^2}}}\)
\(\begin{aligned}{l}{I_1} = 0.0261\,{W \mathord{\left/
{\vphantom {W {{m^2}}}} \right.
{{m^2}}}\\{r_1} = 4.3\,m\\{r_2} = 3.1\,m\end{aligned}\)
For point source,
\(I = \frac{P}{{4\pi {r^2}}}\)
\(\frac{{{I_1}}}{{{I_2}}} = \frac{{r_2^2}}{{r_1^2}}\)
(a)
\(\begin{aligned}{c}\frac{{{I_1}}}{{{I_2}}} = \frac{{r_2^2}}{{r_1^2}}\\{I_2} = {I_1}{\left( {\frac{{{r_1}}}{{{r_2}}}} \right)^2}\\ = \left( {0.0261\,{W \mathord{\left/
{\vphantom {W {{m^2}}}} \right.
{{m^2}}}} \right){\left( {\frac{{4.3\,m}}{{3.1\,m}}} \right)^2}\\ = 0.050\,{W \mathord{\left/
{\vphantom {W {{m^2}}}} \right.
{{m^2}}}\end{aligned}\)