Question

A whistle of frequency $540Hz$ is moving in a circle of radius $2$ feet at a constant angular speed of $15$$rad/\sec $. What are the lowest and highest frequencies heard by a listener standing at rest, a long distance away from the center of the circle? (Velocity of sound in air is $11000ft/\sec $)

Answer 1

Hint: Here we are using the formula of Doppler Effect and substitute the given values in that formula and to find the highest frequency and lowest frequency heard by the observer.
Doppler Effect: The apparent difference between the frequency at which sound or light waves leave a source and that at which they reach an observer, caused by the relative motion of the observer and the wave source.

Formula Used:
We will be using the formula of Doppler Effect. It is given by,
$f = \dfrac{{v \pm {v_D}}}{{v \pm {v_s}}} \times f$
Where, $v$- Speed of sound in air
${v_D}$-Speed of the detector relative to the medium $({v_D} = 0)$
${v_s}$- Speed of the source
For lowest frequency ${f_{\min }} = \dfrac{v}{{v + r\omega }} \times f$
For highest frequency ${f_{\max }} = \dfrac{v}{{v - r\omega }} \times f$

Complete step by step solution:
Let the angular velocity be$\omega $. As the whistle is moving along the circular path. Hence linear velocity of the whistle
$v = \omega \times r$
$ = $$2 \times 15 = 30ft/s$
The maximum frequency is heard when the source approaches the observer. The highest frequency heard by the observer is
$\Rightarrow$ ${f_{\max }} = \dfrac{v}{{v - {v_s}}} \times f = \dfrac{{1100}}{{1100 - 30}} \times 540$ $ = 555Hz$
The minimum frequency is heard when the source moves away from the observer. The lowest frequency heard by the listener is
$\Rightarrow$ ${f_{\min }} = \dfrac{v}{{v + {v_s}}} \times f = \dfrac{{1100}}{{1100 + 30}} \times 540$ $ = 525Hz$

The lowest frequency heard by the observer is $525Hz$ and The highest frequency heard by the observer is $555Hz$.

Note: The Doppler Effect is observed whenever the source of waves is moving with respect to an observer. When a sound object moves towards you, the frequency of the sound waves increases. Otherwise, if it moves away from you, the frequency of the sound waves decreases. It is mostly used by astronomers who use the information about the shift in frequency of electromagnetic waves.