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

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

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

Step-by-step 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(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(v)/(v+v_s)`

on substituting the values, we get

`f'=564*(343)/(343+12.1752)`

or

`f'=544.66 \textup{Hz}`

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

`f'=f(v)/(v-v_s)`

on substituting the values, we get

`f'=564*(343)/(343-12.1752)`

or

`f'=584.75 \textup{Hz}`