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 1100ft/s)

Explanation

The whistle is moving along a circular path with constant angular velocity `omega`. The linear velocity of the whistle is given by
where R is radius of the circle.
At points A and B, the velocity `v_S` of whistle is parallel to line OP, i.e., with respect to observer at P, the 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, the whistle may be assumed to be moving along line OP.) The observer, therefore, listens maximum frequency when the source is at B moving towards the observer:
`f_(max)=f(v)/(v-v_S)`
Where v is speed of sound in air. Similarly, the observer listens minimum frequency when the source is at A, moving away from the observer:
`f_(min)=(fomega)/(v+v_S)`
for `f=540Hz`,`v_S=2ftxx15(rad)/(s)=30(ft)/(s)` and `v=1100(ft)/(s)`, we get (approx) `f_(max)=555Hz` and `f_(min)=525Hz`.