Question

A whistle emitting a sound of frequency $300\text{Hz}$ is tied to a string of $2\text{m}$ length and is rotated with an angular velocity of $15\text{rad/s}$ in the horizontal plane. Find the range of frequencies heard by the observer stationed at a large distance from the whistle.

[Given, velocity of sound in air $=330\text{m/s}$. Assume, medium is stationary and also that the observer and the source are on the same horizontal plane.]

• A. $75\text{Hz}-330\text{Hz}$
• B. $275\text{Hz}-430\text{Hz}$
• C. $225\text{Hz}-2800\text{Hz}$
• D. $275\text{Hz}-330\text{Hz}$

Answer 1

The correct option is D $275\text{Hz}-330\text{Hz}$

Given:
Radius of circle $r=2\text{m}$
Frequency of whistle $f_0=300\text{Hz}$
Angular velocity of rotation $\omega =15{\text{rad s}}^{-1}$

From this, speed of source $v_s=r\omega =30{\text{ms}}^{-1}$

Velocity of sound in air, $v=330\text{m/s}$

When the source is rotating:

The observer will receive maximum frequency when the source is moving towards the observer.

$f_{max}=\left(\frac{v}{v-v_s}\right)f_0$

The observer will receive minimum frequency when the source is moving away from the observer.

$f_{min}=\left(\frac{v}{v+v_s}\right)f_0$

From the data given in the question,

$f_{max}=\frac{330}{330-30}\times 300=330\text{Hz}$
$f_{min}=\frac{330}{330+30}\times 300=275\text{Hz}$

$\therefore$ Frequency range is given by $275\text{Hz}-330\text{Hz}$