Working with elastic string

Two equal particles are connected by an elastic string which is just taut on a smooth table, the string being such that the weight of either particle would produce in it an extension $a$ . If one of the particle is projected with velocity $u$ directly away from the other, the distance each particle would have travelled when the string first return to is natural length can be represented as

$2\pi u \sqrt{\dfrac{a}{\alpha g}}$

Where g is acceleration due to gravity, find the value of $\alpha$ .