Say you have a rope looped around a frictionless pulley, and a mass m is attached to each side of the rope. When the system is released, the tension in the rope is T = mg, and the total force exerted on the pulley due to tension in the string is 2T = 2mg. We usually derive this by adding up the downward tension in the rope on either side of the pulley, but I don't buy this. There are only two points for which the rope is in contact with the pulley and the tension is directly downward. These two points can't possibly be the sole origin of force pulling the pulley down. I understand that this issue can be resolved if one considers differential portions of the rope as it loops around the pulley, but does this math always translate directly into adding up the tensions of the two sides of the rope?