Directly below on the floor is a stationary 428-Hz source of sound. The microphone vibrates up and down in simple harmonic motion with a period of 2.3 s. The difference between the maximum and minimum sound frequencies detected by the microphone is 2.2 Hz. Ignoring any reflections of sound in the room and using 343 m/s for the speed of sound, determine the amplitude of the simple harmonic motion.
To find the amplitude of the simple harmonic motion, we need to use the formula for the Doppler effect. The formula for the shift in frequency due to the Doppler effect is given by:
Δf = (f₀ v) / (v ± vs)
Where:
Δf is the difference in frequencies detected by the microphone
f₀ is the frequency of the source of sound (428 Hz)
v is the speed of sound (343 m/s)
vs is the speed of the microphone
Since the microphone is stationary, the speed of the microphone (vs) is zero. Therefore, the formula simplifies to:
Δf = (f₀ v) / v
Given that Δf is 2.2 Hz and f₀ is 428 Hz, we can substitute these values into the formula:
2.2 Hz = (428 Hz * 343 m/s) / 343 m/s
Simplifying, we have:
2.2 = 428
To solve for the amplitude (A) of the simple harmonic motion, we can use the formula for the period of simple harmonic motion:
T = 2π√(m/k)
Where:
T is the period of the simple harmonic motion (2.3 s)
m is the mass of the microphone
k is the spring constant
Since the microphone is vibrating up and down, its motion can be modeled as simple harmonic motion using Hooke's law:
F = -kx
Where:
F is the restoring force
k is the spring constant
x is the displacement from the equilibrium position
From the equation F = -kx, we can determine that the period of the simple harmonic motion is directly proportional to 2π times the square root of the mass over the spring constant.
To find the amplitude A, we need to rearrange the formula for the period:
T = 2π√(m/k)
Squaring both sides of the equation:
T² = 4π²(m/k)
Rearranging the equation:
A² = (T² * k) / (4π²)
Now, we can substitute the given values:
T = 2.3 s
T² = 2.3² = 5.29
To find the spring constant k, we can use the formula for the speed of sound:
v = √(k/m)
Rearranging the equation:
k = m * (v²)
Given that v is 343 m/s, we can substitute the value:
k = m * (343²)
Now, let's substitute the appropriate values into the formulas to find the amplitude A:
A² = (5.29 * k) / (4π²)
A² = (5.29 * (m * (343²))) / (4π²)
Since we don't have the mass of the microphone (m), we cannot find the exact amplitude without that information.
To determine the amplitude of the simple harmonic motion of the microphone, we need to use the equation that relates the period of motion (T), the speed of sound (v), and the maximum frequency difference (Δf):
Δf = v / (2 * L) * T
Where:
- Δf is the difference between the maximum and minimum sound frequencies detected by the microphone,
- v is the speed of sound, which is given as 343 m/s,
- L is the distance between the source of sound and the microphone,
- T is the period of the motion of the microphone.
Since the microphone is stationary on the floor, the distance L is constant.
Rearranging the equation, we can solve for L:
L = v / (2 * Δf * T)
Substituting the known values into the equation:
L = 343 m/s / (2 * 2.2 Hz * 2.3 s)
L ≈ 31.4 m
Now that we have the distance L, we can determine the amplitude (A) of the simple harmonic motion using the equation:
A = L / 2π
A = 31.4 m / (2 * π)
A ≈ 5.0 m
Therefore, the amplitude of the simple harmonic motion of the microphone is approximately 5.0 meters.