Question

A whistle of frequency 540 Hz is moving in a circle of radius 2 ft at a constant angular speed of 15 rad/s. What are the lowest and highest frequencies heard by a listener standing at rest, a long distance away from the centre of the circle?

(velocity of sound in air is 1100 ft/sec.)

Answer 1

As the whistle is moving along a circular path let its angular velocity be $\omega$

Therefore Iinear velocity

$v_s=\omega R=2\times 15=30ft/s$

At points A and B, the velocity $v_s$ of whistle is parallel to line OP; i.e. with respect to observer at P, whistle has maximum velocity $v_s$ away from P at point A, and towards P at point B. (Since distance OP is large compared to radius R, whistle may be assumed to be moving along line OP)

Observer, therefore, listens maximum frequency when source is at B moving towards observer:

$f_{max}=\frac{v}{v-v_s}f=\frac{1100}{1100-30}\times 540=555Hz$

minimum frequency when source is at A moving away from the observer

$f_{min}=\frac{v}{v+v_s}f=\frac{1100}{1130}\times 540=525Hz$