Friction between two faces of rotating cylinders

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Homework Statement


A rotating flywheel with radius r is spinning with initial angular velocity 'w' and has moment of inertia 'Ic' whose axis of rotation is in the vertical direction. Another cylinder is placed on top of it with no initial angular velocity, radius r, and moment of inertia 5*Ic. The two cylinders' faces have coefficient of friction 'mu'. Eventually, the cylinders both rotate without slip as one.


Homework Equations


I'm trying to determine the equation for the energy lost due to friction. I have a handle on the rest of the equation (conservation of momentum, energy balance), but I'm having trouble adapting the typical linear equation for friction energy (U_f = mu*Int(F_normal*dx)) to a rotational setup. Would the dx in the previous equation become r*dr*d(theta), in which theta is the amount of rotation traversed before no slip?

Clarification: the curved surfaces are not in contact with each other, but the top faces such that the axes of rotation are not just parallel but coincident.

The Attempt at a Solution

 
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