Two loudspeakers are mounted on a merry-go-round whose radius is 9.68 m. When stationary, the speakers both play a tone whose frequency is 132 Hz. As the drawing illustrates, they are situated at opposite ends of a diameter. The speed of sound is 343.00 m/s, and the merry-go-round revolves once every 18.1 s. What is the beat frequency that is detected by the listener when the merry-go-round is near the position shown?
To find the beat frequency detected by the listener, we need to consider the relative motion of the two speakers and the listener.
First, let's calculate the speed at which the sound from each speaker reaches the listener. The speed of sound is given as 343.00 m/s. The merry-go-round revolves once every 18.1 s, so we can calculate the linear velocity of the speakers using the formula:
Linear velocity = 2 * π * radius / time
Linear velocity = 2 * π * 9.68 m / 18.1 s
Linear velocity ≈ 3.37 m/s
Now, let's consider the motion of the speakers. Since they are mounted on a merry-go-round, they are rotating around its center. When the merry-go-round is near the position shown, one speaker is moving towards the listener and the other is moving away from the listener along a diameter. The difference in their speeds is twice the linear velocity of the speakers:
Difference in speeds = 2 * 3.37 m/s
Difference in speeds ≈ 6.74 m/s
Next, we need to consider the frequency of the tones played by the speakers. Each speaker plays a tone whose frequency is 132 Hz. The beat frequency is the absolute value of the difference in frequencies between the two speakers:
Beat frequency = |frequency of speaker 1 - frequency of speaker 2|
Beat frequency = |132 Hz - 132 Hz|
Beat frequency = 0 Hz
Therefore, when the merry-go-round is near the position shown, the beat frequency detected by the listener is 0 Hz.