I have seen whenever we solve for forces on pulley by rope we take the force on pulley exactly as the tensions in the rope around it. But , why do we do this ? Exactly how does the rope exerts forces on pulley ?
Left side of the figure shows an arbitrary situation.Right side shows force on pulley from the rope. (For simplification assume rope and pulley(frictionless) to be massless, i have depicted only an arbitrary situation and i am focussing only on rope and pulley there maybe other forces on the pulley .)
Best Answer
Let's assume that the center of the massless pulley is fixed. Consider the force on any small piece of the massless rope. Since the rope is massless, by Newton's Second Law, in order for the rope to not have infinite acceleration the forces on any small piece of the rope must be balanced. Therefore, whatever tension is felt from the left on any given piece of the rope is also felt from the right. In this way, every small piece of rope feels the same forces from the left and the right. By Newton's 3rd Law, any small piece of rope also exerts the exact same forces on the pieces of rope to the left and right. In this way, the tension is uniformly felt throughout the rope.
Now what happens at the pulley, you might ask? The pulley only exerts a normal force on the rope which is perpendicular to the direction of the rope. Therefore, the normal force between the pulley and the rope does not change the uniform tension in the rope, which is parallel to the direction of the rope. In this way, the tension is uniformly felt through the rope as above.
Edit: "Can you give an analysis of the forces between small elements of rope and pulley ?" The most important thing is that, because the rope is massless, the force everywhere on the rope must be balanced. Consider the part of the rope that is curved around the wheel. If we consider any small piece of the rope touching the wheel, we'll see that the forces on the left and right due to the neighboring pieces of rope can't cancel each other out since they are not directed exactly antiparallel to one another due to the curvature of the rope. Therefore, the net force on the small piece of rope is only zero if the normal force between the rope and the wheel balances the forces from the right and left described above. You can probably convince yourself of this by drawing a picture. You might even break the rope into many infinitesimal pieces, find the normal force on each small piece of rope, integrate over all pieces of the rope, and find that magnitude of the vector sum of the normal force between the wheel and rope is equal to $2T\cos \frac{\pi -\beta -\alpha}{2}$. If you want to REALLY understand this problem, that might be a worthwhile exercise.