Non-ideal pulley

Question

In a non-ideal pulley (with friction at axle to be negligible) the tension on both side of rope is not equal because friction is present between rope and pulley, but when we write torque on the pulley then we only write:

$$ T_1R - T_2R= I\alpha $$

where $T_1$, $T_2$ are tension on either side of rope hanging on the pulley and $I$, $\alpha$ are moment of inertia of pulley and angular acceleration of pulley respectively. why we are not including torque due to fiction in the equation.

Answer 1

The friction force acting on the rope by the pulley is the force that is responsible for the "Tension Difference" and we can write $$\tau = \int_0^\pi R \cdot df = (T'-T)\cdot R = I\alpha$$

We are indeed including the torque due to friction, which is the only torque acting on the pulley.