The red carpets unrolls, show time

A \(\color \red{red~carpet}\) of mass \(M\) made of inextensible material is rolled along its length in the form of a \(cylinder\) of Radius \(R\) and is kept on a rough floor. The carpet starts unrolling without sliding on the floor when a negligibly small push is given to it.The \(horizontal~velocity\) of the axis of the cylindrical part of the carpet when its radius reduces to \(\dfrac{R}{2}\) is

\( \large{\sqrt{\dfrac{a\times gR}{b}}}\)

where \(a,b\) are \(co~ prime~integers\) and \(g\) is acceleration due to gravity

Find \(\large{a+b}\)

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