A small source of sound S of frequency 500 Hz is attached to the end of a light string and is whirled in a vertical circle of radius 1.6 in. The string just remains tight when the source is at the highest point. (a) An observer is located in the same vertical plane at a large distance and at the same height as the centre of the circle. The speed of sound in air `= 330 m s^-1 and `g = 10ms^-2`. Find the maximum frequency heard by the observer. (b) An observer is situated at a large distance vertically above the centre of the circle. Find the frequencies heard by the observer corresponding to the sound emitted by the source When it is at the same height as the centre. ltbr.

Explanation

The string remains just tight at highest point i.e., speed of source at lowest point is `sqrt(5 g R)`, where `R` is radius of circle.
`v_(s) = sqrt(5 g R) = sqrt(5 xx 10 xx 1.6) = 8.9 m//s`
(a) Maximum frequency will be heard when source is in position as shown

`f_(max) = f((v)/(v - v_(s))) = 500 ((330)/(330 - 8.9)) = 514 Hz`
(b) When string becomes horizontal, speed of source
`v'_(s) = v_(s) ^(2) - 2 g R = 5 g R - 2 g R = 3 g R`
`v'_(s) = sqrt(3 g R) = sqrt(3 xx 10 xx 1.6) = 6.9 m//s`

Maximum frequency will be heard when source is in position (1)
`f_(max) = f((v)/(v - v'_(s))) = 500((330)/(330 - 6.9)) = 511 Hz`
Minimum frequency will be heard when source is in position (2)
`f_(min) = f((v)/(v + v'_(s))) = 500((330)/(330 - 6.9)) = 490 Hz`