logarithmic decrement

Jamin2112

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

Basically, I have LQ''(t) + RQ'(t) + (1/C)Q(t)=0, and I'm supposed to

"Show that the ration of the charge Q between two successive maxima is given by exp(RT_d/2L), where T_d is the time between two successive maxima. The natural logarithm of this ration is called the logarithmic decrement.


2. Relevant equations

Dunno

3. The attempt at a solution

So I got a solution Q(t)=e^(-Rt)/(2L) [ C1cos( (√(R²-4L/C) )/(2L)t) + C2sin( (√(R²-4L/C) )/(2L)t).

But I can't figure out how to find T_d. I mean, I could always find t when dQ/dt=0; but then I'd have to plug two values of t back into Q(t) and find the difference, and ............ So what's the right way to do this?

lanedance

First notice the sin & cos terms have the same argument & the choice of c1 & c2 will just choose an overall phase. So for this argument set c2 = 0.

Then the maxima will just be where cos is maximum and successive maxima will occur where the argument of cos has changed by 2pi

LCKurtz

Not exactly. The max's don't agree with the max's of the cosine, but the right idea. To the OP, just look at e^-btcos(at+c).

lanedance

good pickup thanks - They will be pretty close when the natural frequency is much larger that the decay constant, but you do need to take the exponential into account