# How does the masless pulley gets the force from rope?

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 .)

• Is the center of the pulley constrained? – Ian Jun 16 '15 at 13:39
• @ Ian It may or may not be .Alpha and beta may change with time but the above diagrams are only meant for an instant. – Robin Hood Jun 16 '15 at 13:42
• I would assume that it is. The pulley is hinged and hence the weight of the pulley is really irrelevant in a static situation. The vector sum of the tension + weight of the pulley is being balanced by the hinge reaction. Rope transmits tension (think pulling) evenly due to its very nature – chilljeet Jun 16 '15 at 13:44
• @RobinHood in that case the weight becomes relavant and the vector sum of the tensions + weight of the pulley accelerates the pulley in the direction of the vector at the instance in consideration – chilljeet Jun 16 '15 at 13:46
• @chilljeet can you please elaborate , and exactly how does the rope exerts force on the pulley i mean the rope is in contact with the pulley but how do we know that the force it exerts on pulley is as shown in the diagram ? – Robin Hood Jun 16 '15 at 13:50

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.