Whether or not the particle performs complete oscillations?

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If you have a particle that is attached to two elastic strings moving with SHM. How can I determine whether or not the particle performs complete oscillations?

For example I have a particle that's at a point equidistant from A and B (which are the ends of the two strings), and say that the the point where the particle is at equilibrium is 2m from point A (and 3m from point B). If the particle is given a speed V when its at the midpoint of A and B towards the point of equilibrium, say it performs SHM with acceleration a=-(g/k)x. How can I determine using the information I have what I need for the particle to perform complete oscillations? Like, for example, if u^2<3gk then the particle will perform complete oscillations. (The values I used are examples and aren't correct.)
 
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What do you mean by "complete" oscillations?
 
Nevermind, I got it. :smile:
 

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