Question

A whistle of frequency 540Hzmoves in a circle of radius 60.0cmat an angular speed of 15.0rad/s. What are the (a) lowest and (b) highest frequencies heard by a listener a long distance away, at rest with respect to the center of the circle?

Short answer

1. The lowest frequency heard by a listener a long distance away, at rest with respect to the center of the circle is, 526Hz.
2. The highest frequency heard by a listener a long distance away, at rest with respect to the center of the circle is, 555Hz.

Step-by-step solution

The given data

1. Frequency of a whistle is, f = 540Hz
2. Angular speed of the whistle is,$\omega =15.0rad/s$
3. Radius of circle is, R = 60.0cm or 0.60m.
4. The listener is at rest with respect to the center of the circle, i.e., v_D = 0

Understanding the concept of the Doppler Effect

We can find the velocity of the source from its angular speed and radius of the circle. By inserting all the values in the formulae for the lowest and highest frequency heard by the listener according to the Doppler Effect, we can find their values.

Formulae:

The transverse speed of a body in motion,

$\mathit{v}\mathbf{=}\mathit{R}\mathit{\omega }$ …(i)

The lowest frequency as per the Doppler Effect,

role="math" localid="1661745748258" $\mathit{f}\mathbf{\text{'}}\mathbf{=}\mathit{f}\frac{(v+v_D)}{(v+v_s)}$ …(ii)

The highest frequency as per the Doppler Effect,

$\mathit{f}\mathbf{\text{'}}\mathbf{=}\mathit{f}\frac{(v+v_D)}{(v-v_s)}$ …(iii)

a) Calculation of lowest frequency

Speed of the whistle using equation (i) can be given as:

$\begin{array}{rcl}{v}_s& =& \left(0.60m\right)\left(15rad/s\right)\\ & =& 9m/s\end{array}$

Speed of the sound is given as:

v = 343m/s

According to the Doppler Effect, the lowest frequency heard by the listener using equation (ii) is given as:

role="math" localid="1661746065085" width="232" height="91">f'=540Hz×343m/s+0343m/s+9m/s=526.19Hz□526Hz

Therefore, the lowest frequency heard by a listener is 526Hz.

b) Calculation of highest frequency

According to the Doppler Effect, the highest frequency heard by the listener using equation (iii) is given as:

$\begin{array}{rcl}f\text{'}& =& 540Hz\times \frac{\left(343m/s+0\right)}{\left(343m/s-9m/s\right)}\\ & =& 554.6Hz\\ & & \square 555Hz\end{array}$

Therefore, the highest frequency heard by the listener is, 555Hz.