Why does a pulley rotates? I have beening dealing with problems involving pulleys. It seems like a pulley is something that changes the direction of a string, and it rotates when the string moves on it.
Why do we need something like that? Why can't we use something like a smooth circular tag and pass a string over it?
 A: In introductory problems the working is often simplified by taking the mass of the pulley to be zero. This means it doesn't take any force to make the pulley accelerate so the tension in the string is the same either side of the pulley. In this case there is no difference between a pulley and a fixed frictionless disk over which the string slides.
However in more advanced problems (as well as in real life!) the pulleys have a mass greater than zero, and in that case it takes a non-zero torque to make them accelerate:
$$ \tau = I \alpha \tag{1} $$
This means the tension in the string is different in the two sides of the pulley and you need to take this account in the calculation. Now we do have a difference between a pulley and a fixed disk with friction as the frictional force obeys Amonton's law rather than equation (1) above.
A: There are two types of pulley problems.
In the first time it is specified that the pulley is light. Rotating a light pulley does not require any Torque (or force). So tension in both sides will be same. Such questions are usefull for building intuition for physics in students just starting with Newton's laws. In practice the pulley is going to have some mass.
We can take a 'smooth' peg and pass string over it. The system will work in theory. Of course the peg is not going to be smooth in practice.
In second type of problem pulley does have some mass. To rotate such pulley Torque is needed as per relation $\tau = I\alpha$.
If tangential tension on two sides is $T_1$ and $T_2$ with $T_1 > T_2$; then pulley will move in sense of $T_1$ with equation $$T_1R — T_2R = I\alpha$$
