Question

A turntable 1.5m in diameter raotates at 75 rpm. Two speakers, each giving off sound of wavelength 31.3c., are attached to the rim of the table at opposite ends of a diameter. A listener stands in front of the turntable. (a)What is the greatest beat frequency the listener will receive from thsis system? (b)Will the listener be able to distinguish individual beats?

Answer 1

To find the greatest beat frequency, use the Doppler effect on frequencies emitted from speakers attached to a rotating table. The angular velocity of the table is converted to a linear velocity, and used to find Doppler-shifted frequencies for a listener standing in front. Human perception of beats as individual is possible up to about 20 Hz.

Step-by-step explanation:

The question is asking about the phenomenon known as the beat frequency which arises when two sound waves of slightly different frequencies interfere with each other.The greatest beat frequency that a listener will receive from this system can be found by considering the Doppler effect.

As one speaker moves towards the listener and the other moves away, the frequencies of the sounds they emit will be increased and decreased respectively, due to the Doppler shift. For a turning table spinning at 75 rpm (revolutions per minute) with a diameter of 1.5 m, we first convert the rpm to radians per second:

angular velocity (ω) = 75 rev/min × 2π rad/rev × 1 min/60 s = 7.85 rad/s. The linear velocity (v) at the rim of the turntable is given by: v = ω × r, where r is the radius (0.75 m).

v = 7.85 rad/s × 0.75 m = 5.89 m/s

The listener hears frequencies f' which are Doppler shifted from the original frequency (f) according to: f' = f × (v_sound ± v_emitter) / v_sound. Since the speed of sound (v_sound) is not provided, we'll use a typical value for air at room temperature, which is approximately 343 m/s.

The highest frequency is when the speaker is moving towards the listener: f_high = f × (343 + 5.89) / 343. The lowest frequency is when the speaker is moving away from the listener: f_low = f × (343 - 5.89) / 343.

The greatest beat frequency is the difference between these: Beat frequency = f_high - f_low. If the emitted sound from the speakers has a wavelength of 31.3 cm, we need the frequency (f) which we find using: .f = v_sound / wavelength.

f = 343 m/s / 0.313 m = 1096 Hz. Now we can calculate f_high and f_low and then find the beat frequency.

Whether the listener will be able to distinguish individual beats depends on the resulting beat frequency calculated above. Typically, humans can perceive beat frequencies up to approximately 20 beats per second (20 Hz) as individual beats.

If the calculated beat frequency is above this threshold, the listener will perceive it as a rough sound rather than distinct beats.

Answer 2

The greatest beat frequency the listener will receive from this system is approximately 5.471 Hz. Whether the listener can distinguish individual beats depends on their ability to perceive and distinguish sounds.

Step-by-step explanation:

The beat frequency is the difference between the frequencies of the two speakers. In this case, the wavelength of the sound produced by each speaker is given as 31.3m, and the diameter of the turntable is 1.5m. The speed of sound is approximately 343m/s. The combined frequency of the two speakers can be calculated using the formula:

`f_{beat` = 343 / (2 × 31.3) = 5.471 Hz

So, the greatest beat frequency the listener will receive from this system is approximately 5.471 Hz.

As for whether the listener will be able to distinguish individual beats, it depends on the listener's ability to perceive and distinguish sounds. If the beat frequency is within the listener's range of perception, they will be able to hear and identify the individual beats.

However, if the beat frequency is higher than what the listener can perceive, the beats may blend together and not be individually distinguishable.