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 \(\alpha\).

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