Auniform solid cylinder of radius r roll's over a horizontal planepassing into an inclined plane forming an @ with the horizontal.what is the maximum velocity v which still permit the cylinder to roll onto the inclined plane section without a jump.the sliding isassumed to be absent

Dear Mayank,

 have drawn the situation explained the question as follows -

To paraphrase it again, cylinder is rolling from position 1 to position 2 as explained in the diagram. Take some time to understand position 2.

cylinder looses some potential energy because of drop in height. the resulting heigh h will be rsin(90-@) or rcos@. (1)

second thing to understand is that there will be rotational motion around point P as shown in the figure. That is, the centre of the mass of the cylinder is rotating around P. (2)

the third thing to understand is that the total energy will be conserved while going from position 1 to position 2. (3)

And lastly, for cylinder not to jump the normal force of the cylinder (lets say it N), has to be greater than zero. (4)

Use condition 3 to get one equation [potential energy + kinetic energy at position 1 is equal to potential energy + kinetic energy at position 2]

And use condition 2 to get the equation at point P [the rotational motion, normal force and the potential energy]

And finally condition 4, i.e. N >0. For limit condition, put N = 0. You will get the final answer

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