Question

A whistle of frequency 500 Hz tied to the end of a string of length 1.2 m revolves at 400 rev/min. A listener standing some distance away in the plane of rotation of whistle hears frequencies in the range (v=340 m/s)
A. 486 to 586 Hz
B. 436 to 586 Hz
C. 436 to 574 Hz
D. 445 to 570 Hz

Answer 1

In case of doppler effect if source is moving towards observer, observer feels more frequency and if source moves away from the observer, he feels less frequency and same if observer is moving towards observer, he feels more frequency and if he moves away from source, he feels less frequency.
Formula used:
\eqalign{ & f = {f_o}\left( {\dfrac{v}{{v - {v_s}}}} \right) \cr & f = {f_o}\left( {\dfrac{v}{{v + {v_s}}}} \right) \cr & {v_s} = l\omega \cr}

Complete answer:
When an object is in circular motion and it is rotating with the angular velocity of $\omega$and it has radius $l$ then its linear velocity will be ${v_s} = l\omega$.
In this case $\omega = 400rev/\min = \dfrac{{400 \times 2\pi }}{{60}}rad/\sec = 41.90rad/\sec$
And $l$=1.2m hence ${v_s} = l\omega = 41.9 \times 1.2 = 50.28m/s$
In case of doppler effect the maximum frequency one can observe is $f = {f_o}\left( {\dfrac{v}{{v - {v_s}}}} \right)$ and the minimum frequency one can observe is $f = {f_o}\left( {\dfrac{v}{{v + {v_s}}}} \right)$
In this case ${f_o}$=500Hz and $v$= 340 m/s and ${v_s}$=$50.28m/s$
Hence we have got all the values to get the minimum and the maximum frequency.
Maximum frequency observed will be $f = {f_o}\left( {\dfrac{v}{{v - {v_s}}}} \right) = 500\left( {\dfrac{{340}}{{340 - 50.28}}} \right) = 586$
Minimum observed frequency will be $f = {f_o}\left( {\dfrac{v}{{v + {v_s}}}} \right) = 500\left( {\dfrac{{340}}{{340 + 50.28}}} \right) = 436$

So, the correct answer is “Option B”.

Additional Information:
Velocity of sound will be different in different media. If the velocity of sound is higher in one medium and lesser in another medium then the first one is called rarer medium and the second one is called denser medium. If a source or observer is moving then due to doppler effect there will be change in observed frequency. If source and velocity are not moving then even if media change then frequency remains constant.

Note:
We can see this doppler effect in our everyday life. When an ambulance is approaching a traffic constable standing at a point then during approaching he feels the siren sound louder than when the ambulance is leaving him but actually the ambulance has constant frequency and its all due to doppler effect.