Simple problem: logarithmic decrement

joriarty

1. The problem statement, all variables and given/known data

Note this is exam revision rather than actual course work worth marks, so there is no need to be deliberately vague :)

The question comes in two parts, regarding a lightly damped harmonic oscillator with frequency 10 kHz and an amplitude that decays by 25% over 300 oscillations. First I am asked to calculate the logarithmic decrement, and then to make an expression that allows the amplitude to be calculated as a function of time elapsed.

2. Relevant equations

δ=(1/N)ln(A₀/A_N)

3. The attempt at a solution

The log decrement is 9.59x10^-4. Easy. For the second part, simply rearranging the log decrement formula gives A_N=A₀e^-Nδ. Knowing that N = 10,000*t, I get A(t)=A₀e^-9.59t.

What I do not understand is why my course notes give A(t)=e^-9.59t. Why is this answer not multiplied by A₀? Mathematically and physically, this does not make sense to me - the amplitude as a function of time definitely does depend on the initial amplitude! Am I right in thinking that is a mistake?

Thanks!

diazona

Yes, you're right. If nothing else, the units don't match, that tells you the formula in your notes can't be correct.

joriarty

Ah good, thanks for confirming that!