Question

A whistle of frequency 564 Hz moves in a circle of radius 71.2 cm at an angular speed of 17.1 rad/s. What are (a) the lowest and (b) the highest frequencies heard by a listener a long distance away, at rest with respect to the center of the circle? (Take the speed of sound in air to be 343 m/s.)

Answer 1

Answer:

a) $f'=544.66 \textup{Hz}$

b) $f'=584.75 \textup{Hz}$

Explanation:

Given:

Frequency of the whistle, f = 564 Hz

Radius of the circle, r = 71.2 cm = 0.712 m

Angular speed, ω = 17.1 rad/s

speed of source, $v_s$ = rω = 0.712 × 17.1 = 12.1752 m/s

speed of sound, v = 343 m/s

Now, applying the Doppler's effect formula, we have

$f'=f\frac{v\pm v_d}{v\pm v_s}$

where,

$v_d$ = relative speed of the detector with respect to medium = 0

a) for lowest frequency, we have the formula as:

$f'=f\frac{v}{v+v_s}$

on substituting the values, we get

$f'=564\times\frac{343}{343+12.1752}$

or

$f'=544.66 \textup{Hz}$

b) for maximum frequency, we have the formula as:

$f'=f\frac{v}{v-v_s}$

on substituting the values, we get

$f'=564\times\frac{343}{343-12.1752}$

or

$f'=584.75 \textup{Hz}$