The moment equation of a pulley with a rope applying tension on both sides is as follows:
$$I\alpha = f + T_1R - T_2R$$
( $I$ - moment of inertia; $\alpha$ - angular acceleration; $f$ - friction between axle and the axle holder; $T$ - Tension; $R$ - perpendicular distance from centre of axle to tension)
The L.H.S and the first term of R.H.S equates to zero as the pulley is 'massless and frictionless'.
But there is no mention about friction between rope and pulley. Why is that ?
If we consider a pulley to have friction between itself and the ropes, how would its moment equation be ?