How does frictional force of the pulley affect the acceleration of the system?

A larger frictional force would decrease the of the system.

We can relate frictional force and acceleration by the formula: F(net)=ma

Where, F(net) = Force##-##frictional force m = mass a = acceleration

Assuming we used a force of 8N, on an object with a mass of 2kg in two different pulleys with varying (say 1N and 3N).

For the first pulley with a frictional force of 1N, by substituting the numbers into the formula, F(net)=ma 8N##-##1N = 2kg x acceleration we will get an acceleration of ##3.5ms^-2##.

However, for the other pulley with a greater frictional force (2N), by using the same formula, 8N##-##3N = 2kg x acceleration we will get an acceleration of ##2.5ms^-2##

Therefore, it can be seen that a greater frictional force will reduce the acceleration in a pulley system. (In fact, this is true in any situation, not just for pulleys.)