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
To determine the amplitude of the motion of the air molecules, we can use the equation for sound intensity:
I = ρvωA^2
Where:
I is the sound intensity (given as 1.2x10^-12 J/m^2s)
ρ is the density of air (given as 1.28 kg/m^3)
v is the speed of sound in air (given as 343 m/s)
ω is the angular frequency (2π times the frequency)
A is the amplitude of the motion of the air molecules (what we need to find)
We need to first calculate the angular frequency:
ω = 2πf
Where f is the frequency of the air molecules (given as 1000 Hz).
ω = 2π * 1000 = 2000π
Now, let's substitute all the known values back into the equation for sound intensity:
1.2x10^-12 = (1.28)(343)(2000π)(A^2)
Simplifying further:
1.2x10^-12 = 868.32πA^2
To solve for A, we can rearrange the equation:
A^2 = (1.2x10^-12) / (868.32π)
A^2 ≈ (1.2x10^-12) / (2726.54)
Taking the square root of both sides to find A:
A ≈ √(1.2x10^-12 / 2726.54)
Using a calculator:
A ≈ √(4.3993x10^-16)
A ≈ 6.6289x10^-9
Therefore, the amplitude of the motion of the air molecules is approximately 6.6289x10^-9 meters.