If the loudness drops to 90 % of its original value in 5.0 s , what is the time constant of the damped oscillation

This question is incomplete, the complete question is;

A gong makes a loud noise when struck. The noise gradually gets less and less loud until it fades below the sensitivity of the human ear. The simplest model of how the gong produces the sound we hear treats the gong as a damped harmonic oscillator. The tone we hear is related to the frequency f of the oscillation, and its loudness is proportional to the energy of the oscillation.

If the loudness drops to 90 % of its original value in 5.0 s , what is the time constant of the damped oscillation

the time constant of the damped oscillation is 47.44s

Explanation:

Given that;

t = 5.0s

Lets say Ao is the amplitude of initial loudness and later A(t) = 0.9 Ao

EXPRESSION for amplitude is  A(t) = Ao e^-t / T

t is time while T is time constant

so

0.9Ao = Ao e^-t / T  

0.9 = e^ -t/T

So we take the natural log of both the sides

ln (0.9) = -t/T

-0.1054 =  -t/T

0.1054 =  t/T

WE now substitute our value of t

0.1054 =  t/T

0.1054T =  5.0

T = 5 / 0.1054

T = 47.44s

therefore the time constant of the damped oscillation is 47.44s

as we know our atmosphere or air have different levels. every level has other gasses, and every gas has another permeability. when the light comes from lower permeability to higher permeability, it changes its direction. so that means that when the light comes from the moon to the earth, it changes it's direction many more times. it basically means that the shape of the light part of the moon is changed on the earth.

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