Merry-go-round sound question?

Question

Two loudspeakerse are mounted on a merry go round whose radius is 9.01m. When stationary the speakers both play a tone whose frequency is 100 Hz. They are situated at opposite ends of a diameter. The speed of sound is 343 m/s and the merry go round revolves once every 20s. What is the beat frequency that is detected by the listener when the speakers are both pointing at him from equal distances?

Thanks for any help you can offer

Answer 1

speeds of loudspeakers
v = r * w = r* 2pi/T = 9.01*2*3.14/20 = 2.829 m/s
this is the speed of sound source (vs) having real frequency (n = 100Hz)
----------------------------------
the observer (v0=0) is at rest. when he is at equal distance from 2 sources of sound, he will find one (ahead) sound source receding away from him (with vs) and other (following) source approching him with vs. This will create beats because apprent freqs will be heard by him from 2 sources
---------------------------------
apparent freq n' = n *[v-v0] / [v-vs]
in this formula, essentially ALL V (sound), V0 and Vs must move in same direction. if any one is not then resolve it along
----------------------------------------
ahead speaker:
n=100, v0=0, vs = 2.829 m/s
but sound is travelling backwards, so resolve along as
v = - 343 m/s
n'(a) = 100*[-343] / [-343 -2.829] = 99.182 Hz
----------------------------
behind speaker:
n=100, v0=0, vs = + 2.829 m/s v = + 343 m/s
here all are approaching forward (+ x direction)
n'(b) = 100*[343] / [343 -2.829] = 100.832 Hz

Beat freq = n(beat) = n'(b) - n'(a) = 100.832 - 99.182
= 1.65 Hz

Answer 2

Merry Go Round Sound

Answer 3

The fact that the problem states:

What is the beat frequency that is detected by the listener when the speakers are both pointing at him from equal distances?

Makes this problem cake, because at that instant, one speaker is heading directly away from the listener, whilst the other is heading directly toward.

The speed of the speakers is r(omega), where omega is the angular speed or (1rev)/20s = (2pi)/20s = .314159/s. Thus the speed of the speakers are 2.83m/s. One is travelling away, the other toward. Use the doppler shift formula, one with + v, and the other with -v, and find the frequencies of the two speakers in motion. Then take their difference and this is the beat frequency.