Cylinders don't jump and slip!

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

Find the maximum value of the velocity vo{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

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

then find the value of a+b+ca+b+c.


Details and Assumptions

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

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


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