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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. If the listener is at rest, standing far away from the centre, then which of the following is/are true?
(Consider velocity of sound in air as 1100 ft/s)

A
Lowest frequency heard by the listener is 515 Hz
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B
Lowest frequency heard by the listener is 525 Hz
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C
Highest frequency heard by the listener is 545 Hz
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D
Highest frequency heard by the listener is 555 Hz
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Solution

The correct options are
B Lowest frequency heard by the listener is 525 Hz
D Highest frequency heard by the listener is 555 Hz
The whistle is moving along a circular path with constant angular velocity ω=15 rads . The linear velocity of the whistle is given by vs=ωR



where R is radius of the circle.
At points A and B , the velocity of the whistle is parallel or anti-parllel to line OP; i.e., with respect to observer at P, the whistle has maximum velocity away from P at point A, and minimum velocity towards P at point B.

Since distance OP is large compared to radius R , the whistle may be assumed to be moving along the line OP . The observer, therefore, listens maximum frequency when the source is at B moving towards the observer:

fmax=fvvvs
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:

fmin=fvv+vs
where v is speed of sound in air.
For f=540 Hz,vs=2 ft×15 rad/s=30 ft/s and v=1100 ft/s .
we get (approx.) fmax=555 Hz and fmin =525 Hz

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