Cords Hanging Between Pulleys and the Pull of the Sag
The statics of a cord slung over pulleys: how a shallower sag draws the pulleys together
This page, with three diagrams of cords slung over pulleys, works out the statics of a hanging cord. In the right column Leonardo considers a cord hanging in an arc between two pulleys: if the arc is a semicircle its straight chord passes through the pulley centres, but a deeper angle adds to the cord's natural weight. The left column states that an arc equal to or greater than a semicircle never pulls one pulley toward the other, while a shallower arc pulls the more strongly the smaller its sagitta. A lower figure shows cords a b and c d wound over the pulleys b c so that increasing the weight f e straightens the cord.
On this page
A cord hanging in an arc between two pulleys
If the arc of the hanging cord is a semicircle, the cord passes with its straightness through the centre of its pulleys and joins them at a right, spherical angle. But if the arc drops its right angle below the diameter of the two pulleys, it gains as much of the cord's own natural gravity as it hangs below those diameters.
The shallower the sag, the stronger the pull
An arc equal to or greater than a semicircle never pulls one pole of a pulley toward the other. But if the arc is smaller than a semicircle, its power to draw the poles together grows the smaller its sagitta; the force drawing the pulleys together increases as the hanging arc has a smaller sagitta.
Cord over four pulleys straightened by a weight
The cords a b and c d are wound around the pulleys b c in contrary motion to the cord b c, and the more the weight f e increases, the more the cord b c straightens.