The sound intensity of 1.2x10-12J/m2s is the threshold of hearing for humans. What is the amplitude of the motion of the air molecules if the speed of sound in the air is 343m/s and the density of air is 1.28kg/m^3? Take the frequency of the air molecules as 1000 Hz
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To find the amplitude of the motion of the air molecules, we can use the formula for sound intensity:
I = 1/2 ρ v w^2 A^2
where:
I is the sound intensity,
ρ is the density of the medium (in this case, air),
v is the speed of sound in the medium (air),
w is the angular frequency of the sound wave, and
A is the amplitude of the motion of the air molecules.
Given values:
I = 1.2 × 10^(-12) J/m^2s
ρ = 1.28 kg/m^3
v = 343 m/s
f = 1000 Hz
First, we need to convert the frequency (f) to angular frequency (w) using the formula:
w = 2πf
w = 2π × 1000 = 2000π rad/s
Now, rearrange the formula for sound intensity to solve for A:
A^2 = (2I) / (ρ v w^2)
A^2 = (2 × 1.2 × 10^(-12)) / (1.28 × 343 × (2000π)^2)
Calculating this expression gives:
A^2 ≈ 2.906 × 10^(-21)
Taking the square root of both sides to isolate A, we get:
A ≈ √(2.906 × 10^(-21))
A ≈ 5.39 × 10^(-11) m
Therefore, the amplitude of the motion of the air molecules is approximately 5.39 × 10^(-11) meters.