A small source of sound vibrating at a frequency $500 \mathrm{~Hz}$ is rotated along a circle of radius…
A small source of sound vibrating at a frequency $500 \mathrm{~Hz}$ is rotated along a circle of radius $\frac{100}{\pi} \mathrm{cm}$ at a constant angular speed of 5 revolutions per second. The minimum and maximum frequency of the sound observed by a listener situation in the plane of the circle is (speed of sound is $332 \mathrm{~ms}^{-1}$ )
338.5 Hz, 612.5 Hz
485.4 Hz, 535.6 Hz
435.3 Hz, 565.6 Hz
485.4 Hz, 515.5 Hz
Solution
The linear velocity of whistle, $v_s=r \omega$
$
\begin{aligned}
v_s & =\frac{100}{\pi} \mathrm{cm} \times 5 \mathrm{rev} / \mathrm{s} \\
& =\frac{1}{\pi} \mathrm{m} \times 2 \pi \times 5 \mathrm{rev} / \mathrm{s}=10 \mathrm{~m} / \mathrm{s}
\end{aligned}
$
When whistle approaches the listener, then the frequency of sound heard will be maximum and minimum.
$
\begin{aligned}
& n_{\max }=n\left(\frac{v}{v-v_s}\right)=500 \times \frac{332}{322}=515.5 \mathrm{~Hz} \\
& n_{\min }=n\left(\frac{v}{v+v_s}\right)=500 \times \frac{332}{342}=485.4 \mathrm{~Hz} .
\end{aligned}
$