Question

A small source of sound vibrating at frequency 500 Hz is rotated in a circle of radius 100/π cm at a constant angular speed of 5.0 revolutions per second. A listener situation situates himself in the plane of the circle. Find the minimum and the maximum frequency of the sound observed. Speed of sound in air = 332 m s⁻¹.

Answer 1

Given:
Speed of sound in air v = 332 ms⁻¹
Radius of the circle r = $\frac{100}{\mathrm{\pi }}$ cm = $\frac{1}{\mathrm{\pi }}$ m
Frequency of sound of the source ^$f_0$ = 500 Hz
Angular speed $\omega$ = 5 rev/s
Linear speed of the source is given by:
$v=\omega r$
⇒ $v=5\times \frac{1}{\mathrm{\pi }}=\frac{5}{\mathrm{\pi }}=1.59\mathrm{m}/\mathrm{s}$

∴ velocity of source ^$v_s$ = 1.59 m/s

Let X be the position where the observer will listen at a maximum and Y be the position where he will listen at the minimum frequency.



Apparent frequency $\left(f_1\right)$ at X is given by:

$f_1=\left(\frac{v}{v-v_s}\right)f_0$

On substituting the values in the above equation, we get:
$f_1=\left(\frac{332}{332-1.59}\right)\times 500\approx 515\mathrm{Hz}$

Apparent frequency $\left(f_2\right)$ at Y is given by:

$f_2=\left(\frac{v}{v+v_s}\right)f_0$

On substituting the values in the above equation, we get:
$f_2=\left(\frac{332}{332+1.59}\right)\times 500\approx 485\mathrm{Hz}$