atwood's machine in a famous example in classical mechanics. when its equation of motion is obtained by Lagrange method, it is assumed that the pulley if frictionless.
However, i didn't find any reference that calculates the dynamics of the machine with friction in the pulley.
Dear Abbas Ehsaniseresht,
If you to take friction into account, then you need to insert additional torque caused by friction force.
I b = M, where I - moment of inertia, b - angular acceleration and M - the torque caused by tension forces + friction force.
Also you use standard equations for the second Newton's law for the both bodies and the relating between angular acceleration and tangential acceleration (a = b R, R - radius of the pulley).
Thus, you recieve a system of 4 equations to solve.
Assuming the frictional force is related to just rotation without any sliding you can do what Dr. Abbas said. But if you have sliding friction, you should take this into consideration since it will be more dominant.
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