Cylinders don't jump and slip!

An uniform solid cylinder of mass \(M\) and radius \(R\) rolls without slipping on horizontal passing into an inclined plane which makes an angle \(\theta \) with the vertical as shown in figure.

Find the maximum value of the velocity \({v}_{o}\) which still permits the cylinder to roll onto the inclined plane section without a jump.

If it's maximum value can be expressed as

\[{v}_{o_\text{max}} = \sqrt {\cfrac {gR}{a} (b\sin \theta - c)}\text{,} \]

then find the value of \(a+b+c\).

Details and Assumptions

\(\bullet \) There is sufficient friction on the entire surface (with coefficient of friction \(\mu\)).

\(\bullet \) Here \(a, b, c\) are positive integers such that \(\text{gcd}(a, b, c)=1\).

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