Question

A whistle emitting a loud sound of frequency $540$Hz is whirled in a horizontal circle of radius $2$m and at a constant angular speed of $15rad/s$. The speed of sound is $330m/s$. The ratio of the highest to the lowest frequency heard by a listener standing at rest at a large distance from the center of the circle is$?$

• A. $1.0$
• B. $1.1$
• C. $1.2$
• D. $1.4$

Answer 1

The correct option is C $1.2$

Original frequency of the source $f_o=540$ Hz

Velocity of sound in air $v_{sound}=330m/s$

Angular velocity of the source $w=15rad/s$

Radius of circular orbit $r=2$m

Thus linear velocity of the source $v_{source}=rw$

$\therefore$ $v_{source}=2\times 15=30$ $m/s$

Using Doppler effect when a source moves towards the stationary observer.

Highest frequency heard by the observer $f_1=f_o[\frac{v_{sound}}{v_{sound}-v_{source}}]$

$\therefore$ $f_1=540[\frac{330}{330-30}]=540[\frac{330}{300}]$

Using Doppler effect when a source moves away from the stationary observer.

Lowest frequency heard by the observer $f_2=f_o[\frac{v_{sound}}{v_{sound}+v_{source}}]$

$\therefore$ $f_2=540[\frac{330}{330+30}]=540[\frac{330}{360}]$

Thus ratio of frequencies $\frac{f_1}{f_2}=\frac{(540\times 330)/300}{(549\times 330)/360}$

$⟹$ $\frac{f_1}{f_2}=\frac{360}{300}=1.2$