A microphone is attached to a spring that is suspended from the ceiling, as the drawing indicates. Directly below on the floor is a stationary 624-Hz source of sound. The microphone vibrates up and down in simple harmonic motion with a period of 2.31 s. The difference between the maximum and minimum sound frequencies detected by the microphone is 2.26 Hz. Ignoring any reflections of sound in the room and using 343 m/s for the speed of sound, determine the amplitude (in m) of the simple harmonic motion.
To determine the amplitude of the simple harmonic motion, we can start by analyzing the relationship between the period of vibration, the frequency of the source sound, and the difference in sound frequencies detected by the microphone.
1. The period (T) of simple harmonic motion is related to the frequency (f) by the equation:
T = 1/f
2. The difference in sound frequencies detected by the microphone is given as 2.26 Hz. This difference in frequency is equal to twice the rate at which the microphone moves up and down, since the microphone detects the sound when it is moving towards the source and away from the source.
Difference in frequency = 2 * (rate of up and down motion)
Difference in frequency = 2 * (1 / T)
3. Given that the source sound has a frequency of 624 Hz, the difference in frequency is equal to the difference between the maximum and minimum sound frequencies detected by the microphone. Since this difference is 2.26 Hz, we can write:
Maximum frequency - Minimum frequency = 2.26 Hz
(624 + Maximum frequency) - (624 - Minimum frequency) = 2.26 Hz
4. Rearranging the equation above, we can express the maximum and minimum frequencies in terms of the source frequency:
Maximum frequency = 624 + 2.26/2 Hz
Minimum frequency = 624 - 2.26/2 Hz
5. The speed of sound is given as 343 m/s. The speed of sound is related to the frequency of sound (f) and the wavelength (λ) by the equation:
Speed of sound (v) = frequency (f) * wavelength (λ)
6. The wavelength of sound can be determined using the maximum and minimum frequencies detected by the microphone. We can assume that the microphone is at a position where it detects the maximum frequency when moving towards the source and the minimum frequency when moving away from the source. This means that the microphone is at the nodes (points of zero displacement) of the standing wave produced by the source sound.
The distance between adjacent nodes is equal to half a wavelength (λ/2). Hence, we can write:
λ/2 = speed of sound / Maximum frequency
λ/2 = speed of sound / Minimum frequency
7. Multiply both sides of the equations above by 2 to obtain the wavelength (λ):
λ = (2 * speed of sound) / Maximum frequency
λ = (2 * speed of sound) / Minimum frequency
8. The amplitude (A) of the simple harmonic motion of the microphone can be related to the wavelength (λ) by the equation:
Amplitude (A) = λ/4
Now, we can substitute the known values into the equations above and calculate the amplitude (A):
Maximum frequency = 624 + 2.26/2 Hz
Minimum frequency = 624 - 2.26/2 Hz
Speed of sound (v) = 343 m/s
λ = (2 * 343) / Maximum frequency
λ = (2 * 343) / Minimum frequency
A = λ/4
Calculating the values:
Maximum frequency = 624 + 2.26/2 = 625.13 Hz
Minimum frequency = 624 - 2.26/2 = 622.87 Hz
λ = (2 * 343) / 625.13 = 1.097 m
λ = (2 * 343) / 622.87 = 1.099 m
A = 1.097 / 4 = 0.274 m
A = 1.099 / 4 = 0.275 m
Therefore, the amplitude of the simple harmonic motion of the microphone is approximately 0.274 m or 0.275 m.
To determine the amplitude of the simple harmonic motion, we need to make use of the equation for the Doppler effect in sound. The frequency observed by the microphone can be calculated using the equation:
f_observed = f_source * (v_sound + v_microphone) / (v_sound - v_source)
Where:
- f_observed is the frequency observed by the microphone,
- f_source is the frequency of the source (624 Hz in this case),
- v_sound is the speed of sound (343 m/s in this case), and
- v_microphone is the velocity of the microphone due to its simple harmonic motion.
The microphone oscillates up and down in simple harmonic motion, which means that its velocity oscillates sinusoidally. We know the period of the motion (2.31 s), and we can use that information to calculate the angular frequency (ω) as follows:
ω = 2π / T
Where:
- ω is the angular frequency, and
- T is the period (2.31 s).
Once we have the angular frequency, we can find the maximum velocity (v_max) of the microphone using the following formula:
v_max = ω * A
Where:
- A is the amplitude of the simple harmonic motion.
Now, we know that the difference between the maximum and minimum sound frequencies detected by the microphone is 2.26 Hz. This means that when the microphone is at its highest velocity, the frequency detected will be f_observed + 1.13 Hz, and when it is at its lowest velocity, the frequency detected will be f_observed - 1.13 Hz.
Now, we can set up two equations using the Doppler effect equation:
f_observed + 1.13 = f_source * (v_sound + v_max) / (v_sound - v_source) (equation 1)
f_observed - 1.13 = f_source * (v_sound - v_max) / (v_sound - v_source) (equation 2)
We can rearrange these equations to solve for v_max:
v_max = (v_sound * (f_observed - f_source)) / (2 * f_source ± 1.13 * (f_observed + f_source) / f_source)
Now, we can substitute the values given in the question:
v_max = (343 * (2.26 Hz - 624 Hz)) / (2 * 624 Hz ± 1.13 * (2.26 Hz + 624 Hz) / 624 Hz)
Simplifying this equation will give us the value for v_max, which is equal to the maximum velocity of the microphone. To find the amplitude, we can use the formula:
A = v_max / ω
Substituting the value of v_max and the calculated value of ω will give us the amplitude in meters.
the Doppler shift is half of the frequency difference
... higher freq going down, lower going up
the speed indicated by the Doppler shift is the velocity of the microphone at the midpoint of its travel
using the period and max velocity, find the amplitude