A gong that makes damped oscillation

[kolua]

Homework Statement

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.

A. If the loudness drops to 85 % of its original value in 5.0 s , what is the time constant of the damped oscillation?
B. How long does it take for the sound to be 20 % as loud as it was at the start?
C. What fraction of the original loudness remains after 1.0 min?

Homework Equations

X_max(t)=Ae^-t/2τ
E(t)=E₀e^-t/τ

The Attempt at a Solution

0.85E₀=E₀e^-5/τ
τ=-5/ln0.85=-30.8s

t=τ⋅ln0.2=49.5s

E_1min/E₀=e^60/30.8=0.14

 

[Simon Bridge]

Please show your reasoning...

 

[kolua]

Please show your reasoning...

I am not sure if the formulas that I used here are correct. The data seem to can be plugged in straight away. or not?

 

[Simon Bridge]

This is where your reasoning comes in. If you were thinking: "I have a bunch of formulas, I'll just try them out and hope for the best." then this will only be correct by accident.
Being correct by accident will get you marks but it is much better to be correct on purpose.

You need to read about what the formulas mean and where they come from.
Better yet, use your understanding of the physics to derive your own: when you understand the physics, you don't need to memorise the formulas.

What does damped harmonic motion usually look like?
ie. What is the shape of the envelope of the oscillations?