Essentially that is sort of number theory problem, i.e.
With certain condition between the spring constant and the initial speed, the ball will just cover the same line over and over again on each circle (when 2* \pi * L0 / v happends to be integer time of the period of spring)
However, with other condition, the ball might cover different trace at different circle, thus will cover part of the circle of radius L (could it possibly to be all? maybe not due to some residue theorem from number theory)
It sound like an interesting problem, I will see if I can come up with something we I got some spare time. Sorry, can not type Chinese in office