A source of sound is moving along a circular orbit of radius `3 meter` with an angular velocity of `10 rad//s`. A sound detector located far away from the source is executing linear simple harmonic motion along the line `BD` with an amplitude `BC = CD = 6 meters`. The frequency of oscillation of the detector is `(5)/(pi)` per second. The source is at the point `A` when the detector is at the point `B`. If the source emits a continuous sound wave of frequency `340 Hz`, Find the maximum and the minimum frequencies recorded by the detector.

Explanation

Speed of source: `v_(s) = R omega = 3 xx 10 = 30 m//s`
Maximum speed of detector will be when it is in mean position i.e. at `C`
Maximum speed of detector
`v' = a omega, = 6 xx 2 pi xx (pi)/(5) = 60 m//s`
Speed of sound `v = 330 m//s`
Time period of circular motion `T = (2 pi)/(omega) = (pi)/(5) s`
Time period of SHM `T' = (2 pi)/(omega') = (2 pi)/(2 pi xx (5)/(pi)) = (pi)/(5) s`
The frequency received by detector is minimum when source and detector recede each other (i.e., source at `E` and detector at `C` after time `T//4`, respectively)
`f'_(min) = f((v - v')/(v + v_(s))) = (340) ((330 + 60)/(330 - 30))`
`= 255 Hz`
(b) The frequency received by detector is maximum when source and detector approach each other (i.e., source is at `F` and detector at `C` after time `3T//4` and `3T'//4`, respectively)
`f'_(max) = f((v + v')/(v - v_(s))) = (340) ((330 + 60)/(330 - 30))`
`= 442 Hz`