## Underdamped Oscillation

1. The problem statement, all variables and given/known data
Show that the local maximum or minimum for the displacement of an underdamped oscillation does not occur halfway between the times at which the mass passes its equilibrium point.

2. Relevant equations
$$x = e^{-\frac{ct}{2m}}(A cos(wt) + B sin(wt))$$
$$x = K e^{-\frac{ct}{2m}} sin(wt + P)$$

$$T = \frac{2\pi}{w}$$

w = angular frequency = $$\frac{\sqrt{4mk - c^2}}{2m}$$
Because the system is underdamped, 4mk > c^2

T = period

3. The attempt at a solution
I said $$sin(wt + P)$$ attains its maximum when $$wt + P = (4k + 1)\frac{\pi}{2}$$ and min when $$wt + P = (2k + 1)\frac{\pi}{2}$$
Likewise, it crosses the equilibrium when $$wt + P = k\pi$$.

Thus the period for a pass by equilibrium is simply $$\frac{pi}{w}$$

So I did $$k\pi + \frac{\pi}{2w} = (2k + \frac{1}{w})\frac{\pi}{2}$$
But this does cross a min if w = 1... where did I go wrong?

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 I also have a follow up question. Given a general equation, how do I estimate the parameters c, m, k
 Sorry for bumping this, but I'm actually not even sure if I understand the question fully. If anyone could at least tell me whether I'm correctly interpreting the question...