# Friction in pulleys

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 ?

• Your equation is dimensionally incorrect. – Farcher May 11 at 15:35

It is the friction between the pulley and rope due to which the factor $$T_1R-T_2R$$ appears in the equation. Due to friction,the tensions in the ropes on either ends cannot be same[ The relation is actually $$T_1= T_2 e^{\mu \theta }; \mu \rightarrow$$ coefficient of friction, $$\theta \rightarrow$$ angle of wrap of rope around pulley] The friction prevents slipping between rope and pulley and actually causes pulley to turn with the rope. This implies these tensions can't be same. In absence of this friction, the pulley wouldn't rotate. The rope would just slip over the pulley and the tension is same throughout the rope. The friction at the axle, on the other hand, opposes the rotation of pulley - a completely opposite effect as compared to that by the friction between rope and pulley.