Determine the net torque of a frictionless mass pulley system with a pulley of a given radius, R, a given pulley mass, m1, and mass suspended from the system, m2. The net torque is equal to the angular acceleration of the mass suspended from the pulley, multiplied by the rotational inertia of the pulley.
Net Torque = Angular Acceleration * Inertia of Pulley
Angular Acceleration = (acceleration of mass, m2) / (radius of pulley)
Inertia of Pulley = (1/2 mass of pulley) * (radius of pulley)^2
Determine the applied, or observed, torque of the same system with friction. The calculation will be the exact same as above, however, the observed acceleration of the mass will be less due to the friction now added to the pulley.
Applied Torque = Angular Acceleration (with friction) * Inertia of Pulley
Find the frictional torque by subtracting the applied torque from the net torque.
Net Torque = Applied Torque + Frictional Torque
Frictional Torque = Net Torque - Applied Torque