An overdamped oscillator with natural frequency ω and damping coefficient γ starts out at position x0 > 0. What is the maximum initial speed (directed toward the origin) it can have and not cross the origin?

I do not really know where to start with this question. What does the question mean by the speed not crossing the origin?

I just don't know how to begin to solve this in general.


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When you have an overdamped oscillator and release the mass from rest with some deflection, it will return to zero without ever crossing (no oscillation). For a lightly damped oscillator, you would see oscillations (zero crossings) whose amplitude becomes smaller with time.

Now if your mass is launched towards zero (the equilibrium) with sufficient speed, it will cross zero (once) before the damping slows it down. The question asks about the maximum speed that just doesn't cause a zero crossing (and which therefore "throws" the mass to the equilibrium position in the shortest time, most likely).

A diagram might help to explain:

enter image description here

  • $\begingroup$ In the case of a critically damped oscillator would would the desired solution look similar? And by throwing the mass to equilibrium without passing equilibrium be quite slow? (if the goal is not to pass equilibrium) $\endgroup$ – user96828 Oct 28 '15 at 3:03
  • $\begingroup$ The graphs look similar but the equations are different. Do you know the general solution for overdamped system? Here is a link. I do not understand your second sentence (And by... be quite slow?). $\endgroup$ – Floris Oct 28 '15 at 3:09

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