Question

A whistle of frequency 540 Hz is moving in a circle of radius 2 ft. at a constant angular speed of 15 $rad/s$. If the listener is at rest, standing far away from the centre, then which of the following is/are true?
(Consider velocity of sound in air as 1100 ft/s)

• A. Lowest frequency heard by the listener is 515 Hz
• B. Lowest frequency heard by the listener is 525 Hz
• C. Highest frequency heard by the listener is 545 Hz
• D. Highest frequency heard by the listener is 555 Hz

Answer 1

Answer: (B), (D)

Highest frequency heard by the listener is 555 Hz

The whistle is moving along a circular path with constant angular velocity $\omega =15\frac{rad}{s}$ . The linear velocity of the whistle is given by $v_s=\omega R$



where $R$ is radius of the circle.
At points $A$ and $B$ , the velocity of the whistle is parallel or anti-parllel to line OP; i.e., with respect to observer at $P$, the whistle has maximum velocity away from $P$ at point $A$, and minimum velocity towards $P$ at point $B$.

Since distance $OP$ is large compared to radius $R$ , the whistle may be assumed to be moving along the line $OP$ . The observer, therefore, listens maximum frequency when the source is at $B$ moving towards the observer:

$f_{max}=f\frac{v}{v-v_s}$
where $v$ is speed of sound in air.

Similarly, the observer listens minimum frequency when the source is at $A,$ moving away from the observer:

$f_{min}=f\frac{v}{v+v_s}$
where $v$ is speed of sound in air.
For $f=540Hz,v_s=2ft\times 15rad/s=30ft/s$ and $v=1100ft/s$ .
we get (approx.) $f_{max}=555Hz$ and $f_{\text{min}}=525Hz$