Use sum of product of vectors T and V

A small ring of mass \(m\) can slide on a friction less rod. It is connected to a block of mass \(M\). The rod is at a distance \(d\) from the pulley. Initially the ring is at a distance \(d\) from point \(O\). Find the velocity of the block when the ring is at a distance \(d/2\) from point \(O\).

The answer is in form

\({ V }^{ 2 }=\frac { Mgd(a\sqrt { a } -\sqrt { b } ) }{ M+bm } \)

Find \(a+b\).

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