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 whistie is given by
`v_(S) = omegaR`
where, `R` is radius of the circle.

At point `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 observe:
`f_(max) = f (v)/(v - v_(s))`
where, `v` is speed of sound in air. Similarly, observer listens minimum frequency when source is at `A`, moving away from observer:
`f_(min) = f(v)/(v + v_(s))`
For `f = 540 Hz, v_(s) = 2 ft xx 15 rad//s = 30 ft//s`. and `v = 1100 ft//s`.
we get (approx.) `f_(max) = 555 Hz` and `f_(min) = 525 Hz`.