Pulley system calculator

How do you calculate the mechanical advantage of a pulley system?

Count the rope segments that pull directly on the moving block (and the load). With no friction, that number is the ideal mechanical advantage: two segments give about 2 : 1, three give 3 : 1, four give 4 : 1. OpenStax College Physics says the number of cables pulling directly upward on the system of interest is approximately the mechanical advantage.

Does a single fixed pulley help lift a load?

No, not in force. An ordinary fixed pulley has a mechanical advantage of 1; it only changes the direction of the pull. With friction it is slightly worse than 1: at 91 percent efficiency, holding a 100 lb load takes 109.9 lb.

What is the catch with a pulley system?

Distance. You pull n times as much rope as the load rises. With four supporting ropes and a 10 ft lift, you pull 40 ft of rope. The work, force times distance, stays the same in the ideal case; friction makes you do more.

Which ropes count as supporting ropes?

Only the segments that end on the moving block (or the load itself). A rope that runs over a fixed pulley at the top and then down to your hands is a redirect: it changes the direction of your pull but adds nothing to the advantage. If you pull up on the end that comes off the moving block, that segment does count, and the system has one more supporting rope than if you had to pull down through a redirect.

How much does friction cost?

It depends on the pulleys. Petzl, a maker of rescue and climbing pulleys, lists efficiencies of about 91 to 95 percent for models with sheaves on sealed ball bearings; those are single-pulley figures from the manufacturer’s pages, and a plain bushing, a carabiner or a wet or icy rope will be worse. This page compounds a per-pulley figure along the rope. For four supporting ropes at 91 percent a 200 lb load takes 57.3 lb instead of 50 lb, a real advantage of 3.49 : 1 rather than 4 : 1, and 62.9 lb if you also pull through one redirect.

Is a system with more pulleys always better?

Each extra pulley adds friction, so the real advantage grows more slowly than the ideal one, and the extra rope you must pull grows linearly. The table shows how the real advantage falls short of the number of ropes as you add them.

Can I use this for rigging or lifting people?

No. This is a physics estimate for a simple, continuous-rope system. It does not check the rope, the pulleys, the anchors or the load on them, which are all higher than the weight you lift. For rigging, rescue and anything overhead, use rated equipment and a qualified person.

The ideal case

If friction is zero, the tension is the same everywhere in one continuous rope. The moving block hangs on n segments, each pulling up with tension T, so the load W = n × T and the pull is T = W ÷ n. A 200 lb load on four supporting ropes needs 50 lb. The rope you pull is n times the rise, so lifting the load 10 ft means 40 ft of rope through your hands.

Adding friction: the model used here

Each time the rope passes over a pulley, the tension on the side farther from your hands is lower by the factor e, the efficiency of that pulley. For n supporting ropes and r redirect pulleys before the first supporting rope, the load is W = T × e^r × (1 + e + e² + … + e^(n−1)), so the pull is T = W ÷ [e^r × (1 + e + … + e^(n−1))]. The real mechanical advantage is the bracket. This reduces to W ÷ n when e = 1 and to W ÷ e for one redirect pulley alone. It assumes one continuous rope with a pulley between each pair of neighbouring supporting ropes; separate ropes in series (compound systems) are not covered.

Worked example

A 200 lb load, four supporting ropes, pulleys at 91 percent. The ladder of tensions is T, 0.91 T, 0.83 T, 0.75 T; their sum is 3.49 T, so the pull is 200 ÷ 3.49 = 57.3 lb (the ideal 50 lb). The efficiency of the whole system is 3.49 ÷ 4 = 87.3%. If the rope then runs over one redirect pulley to your hands, the pull rises to 62.9 lb. A 100 lb load on a two-rope system at 90 percent needs 52.6 lb, not 50.

Limits of this method

The figures for pulley efficiency come from manufacturers and apply to one pulley under a stated load; they change with the rope diameter, its stiffness, wetness and the load. The model also ignores the weight of the rope, the friction of the rope on its anchor point if it is tied and not on a sheave, rope stretch, and the force on the anchors, which depends on the arrangement and can be larger than the load. A system lifting at speed has to accelerate the load too. None of this is covered, so treat the result as an estimate for planning, not as a rating.

Common mistakes

The most common is counting the redirect rope as a supporting rope, or the reverse: forgetting that the rope you pull upward off the moving block counts. Entering a percentage as a fraction (0.91 instead of 91) is next. Others are to use the efficiency of one pulley for the whole system rather than per pulley, and to forget that a lift of one foot needs n feet of rope, which matters when a rope is short.

Sources: OpenStax College Physics 2e, section 9.5 “Simple Machines” (mechanical advantage; the number of cables pulling up on the load is approximately the mechanical advantage; Figure 9.24); pulley efficiency figures of about 91 to 95 percent for sealed ball-bearing models are published by Petzl on its product pages (manufacturer data, seen through search results; the pages were not opened directly). The friction model is the sum of a geometric series along one continuous rope. Related: gear ratio calculator, inclined plane calculator, torque calculator, work calculator.