A cylinder (mass M, outer radius R, moment of inertia I, see diagram below) is o

Question

A cylinder (mass M, outer radius R, moment of inertia I, see diagram below) is on an incline of angle q, attached to a string on an inner radius r. The coefficient of kinetic friction between the cylinder and the incline is mk.

The string is pulled in such a way that the center of mass of the cylinder remains stationary. Derive an expression for the angular acceleration a of the cylinder in terms of M, g, R, r, I, q and mk .

Bonus: show quantitatively, and explain qualitatively how the answer would change if the string is rolled through the top of the inner radius, rather than the bottom.

Explanation / Answer

Forces along the incline:

Friction F=mk Mg cos q

due to string= T

due to weight = Mg sin q

since Center of mass is nt moving

Mgsinq - T -  mk Mg cos q =0

T = Mgsinq - mk Mg cos q

Now writing torque equation,

Tr - FR = I alpha

alpha= (Tr - FR)/I

=(Mg r sinq - mk Mg r cos q - mk MgR cos q) /I

angular acceleration, alpha = (Mg r sinq - mk Mg(R+r) cos q) /I

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