Question

A source of sound (velocity of sound = 330 ms^-1) is moving along a circular orbit, of radius 3m, 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 m. The frequency, of oscillations, of the detector, is 5/π per second.

The source emits sound waves of frequency 340 Hz. The maximum and minimum values, of the source frequency, as recorded by the detector, would equal (nearly)

(1) 453 Hz and 284 Hz

(2) 453 Hz and 255 Hz

(3) 442 Hz and 255 Hz

(4) 442 Hz and 284 Hz

Answer 1

(3) 442 Hz and 255 Hz

The linear velocity of the source, in its circular path is

v = rω = 3 x 10 m / s 30 m/s

Its time period, of rotation is

The detector (frequency being (5/π)rad /s), has a time period of (π/5) s i.e, the same as that of the source.

The detector (executing a SHM), has a continuously varying velocity. However, its velocity amplitude is

The source, and the detector, are having the same time period. We can, therefore, have situations where the source and detector are both 

(i) approaching each other 

(ii) moving away from each other.

In these two cases, (when the detector has its maximum velocity), the detector would record the maximum and the minimum (apparent) frequencies of the source. We have