# 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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