Question

A whistle of frequency $540\text{Hz}$ rotates in a circle of radius $2\text{m}$ at an angular speed of $15\text{rad/sec}$. What will be the frequency heard by a listener a long distance away at rest with respect to the centre of the circle when the source is at $\text{B}$. (Speed of sound $=330\text{m/s}$)

• A. $495\text{Hz}$
• B. $594\text{Hz}$
• C. $540\text{Hz}$
• D. $600\text{Hz}$

Answer 1

The correct option is A $495\text{Hz}$

Given,
whistle frequency, $f_s=540\text{Hz}$,
radius of circular path, $r=2\text{m}$
angular speed $\omega =15\text{rad/sec}$
speed of sound, $C=330\text{m/s}$

In case of circular motion $V=r\omega$.
So here velocity of source,

$V_{\text{source}}=2\times 15=30\text{m/s}$

As the detector is at rest so its velocity, $V_0=0$.

The frequency $f$ heard by the listener, where source is moving away can be calculated by using,

$f=f_s\left(\frac{C-V_0}{C+V_s}\right)$

So, when source is at $B$ and moving away from the listener,
$f=f_s\left[\frac{C}{C+V_s}\right]$

Substituting the values in the above rquation we get,
$f=540\left[\frac{330}{330+30}\right]$

$\therefore f=495\text{Hz}$

Hence, option (a) is correct answer.

$\begin{array}{c}\text{Caution:}\\ \text{Sign convention:}\\ \text{Direction from the source to the listener is taken positive}\end{array}$