Problem 4

 
4) A whistle of frequency f₀= 512 Hz moves in a circle of radius 1 m. The frequency of revolution (for the circular motion) is 3.0 rev/s. What are (a) the lowest (12.5 pts) and (b) the highest frequencies (12.5 pts) heard by a listener a long distance away at rest with respect to the center of the circle? Take the speed of sound to be v = 330 m/s.  

Answer:
The first step is to find the velocity that the shistle has as it goes around the circle. To do this we say
$v = r \omega$
$v = r (2 \pi f)$
$v = (1 m) (2) \pi (3 rev/sec)$
$v = 6\pi m/s$
$v \simeq 18.9 m/s$
Now, the whistle will be moving this velocity away from the observer when the whistle is at a right angle to the observer from the center of rotation. Similarly, when the whistle is on the other side of the circle, it will have its maximum velocity away from the observer. To find out the frequency heard by the observer, we say that
$f = f' (\frac{v}{v \pm v_{s}})$
For the maximum frequency heard, when the whistle is moving with a velocity right toward the observer, we have
$f = f' (\frac{v}{v - 18.9})$
$f = (512)(\frac{330}{311.1})$
f = 543 Hz
For the minimum frequency heard, we want the case when the whistle is moving directly away from the observer.
$f = f' (\frac{v}{v + 18.9})$
$f = (512)(\frac{330}{348.9})$
f = 484 Hz