Four sets of pulleys demonstrate mechanical advantages of 1:1, 2:1, 4:1 and 6:1.

The photograph above shows four sets of pulleys, arranged from left to right in order of increasing complexity. Starting at left, we have a single pulley, with a mass hanger at either end of the string. This pulley is a fixed pulley; it hangs from the hook collar, and it does not move during operation. The hanger with masses on the right exerts a tension on the string equal to its weight, mg, where m is the mass of the hanger and masses, and g is the acceleration of gravity. We can call this the “load.” Because the pulley can rotate, this tension is equal throughout the length of the string, and if the weight of the left-hand mass hanger and masses equals this tension, the two weights balance, and the hangers stay in position. (This is identical to an Atwood machine; see demonstration 12.33 -- Vertical Atwood’s machine.) Thus, to lift the mass on the right we must apply a force, our “effort”, of slightly more than mg to the left side of the string, and as we do this, we lift the mass on the right by the same distance that we pull down the left end of the string. If we call this distance h, the work we do is mgh. Since the effort equals the load, the mechanical advantage afforded by this arrangement is 1:1. That is, we must apply the same force to lift the load as we would without the pulley. The pulley merely changes the direction in which we apply that force to do the lifting. Note that while the tension in the string is mg, the tension on the hanger that supports the pulley is twice this (plus, of course, the weight of the pulley itself).

Next is a pair of pulleys, one fixed and one movable, with the load applied to the movable pulley. Again, since the pulleys can rotate, the tension throughout the string is the same, but now there are two lengths of string between the pulleys, and the free end coming out. So to raise the load a distance h, we must pull the free end down by 2h, and the total force balancing the load is twice the tension we pull on the free end of the string. We can lift the load by pulling with just over half its weight, but must pull twice as far as we lift it. The mechanical advantage is thus 2:1, but the work we do is the same as it would be without the pulleys (mgh = (1/2)mg2h). For simplicity, we have ignored the weight of the bottom pulley, which adds slightly to the load. We will do this for the last two examples, as well.

Third is a pair of double pulleys. Here we have four sections of string between the pulleys, and the free end. Pulling the free end down thus raises the lower set of pulleys by one-quarter the distance through which we pull, and the tension supporting the load is four times the tension we apply. This represents a mechanical advantage of 4:1. Again, the effort we must apply to lift the load is slightly more than one-fourth the load, but the work we must do to lift it through a particular height is the same as it would be without the pulleys.

Last is a pair of triple pulleys, arranged vertically instead of side by side. There are now six sections of string between the pulleys, plus the free end. So the load rises by one-sixth the distance through which we pull the free end, and we must apply just over one-sixth its weight to lift it. The mechanical advantage is thus 6:1, and, again, the work we do to lift the load is the same as it would be without the pulleys.