A whistle of frequency 540 Hz rotates in a circle of radius 2 m at a linear speed of 30 ms -1. What are the…
A whistle of frequency 540 Hz rotates in a circle of radius 2 m at a linear speed of 30 ms -1. What are the lowest and highest frequencies heard by an observer a long distance away at rest with respect to the centre of circle. Take speed of sound in air as 330 ms -1. Can the apparent frequency be ever equal to actual frequency?
Solution
Sol. Apparent frequency will be minimum when the source is at N and is moving away from the observer
$f_{\min}=\left(\frac{v}{v+v_s}\right)f=\left(\frac{330}{330+30}\right)(540)=495\ \text{Hz}$
Frequency will be maximum when source is at L and is approaching the observer.
$f_{\max}=\left(\frac{v}{v-v_s}\right)f=\left(\frac{330}{330-30}\right)(540)=594\ \text{Hz}$
Further, when source is at M and K, angle between velocity of source and line joining source and observer is 90° or $v_s\cos\theta=v_s\cos90^\circ=0$. So, there will be no change in the apparent frequency.
Answer: $495\ \text{Hz}$, $594\ \text{Hz}$