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# Limitations of simple pulley systems

Tuomas Pöysti 2020

I tend to focus on the theoretical aspects of pulley systems, so no practical tips should be expected. I think, though, that there are many interesting explanations to real life phenomena to be found, if one just scratches the surface of physics and mathematics around pulley systems. If my terms, methods or notation seem unclear, read my other articles like this one on basics of pulley system analysis.

If simple pulley systems have some advantages, simplicity surely is among the strongest candidates. Simply manifolding a rope between two points is easily pulled off in any situation.

Probably the most popular pulley system, the Z-rig, is a simple one. Using the notation of this article, the Z-rig has MA expression:

1 + P2 + P1P2.

Using the notation above, simple 4:1 and simple 5:1 have respective expressions:

1 + P3 + P2P3 + P1P2P3 and

1 + P4 + P3P4 + P2P3P4 + P1P2P3P4

There’s seems to be a pattern here: each degree manifests in one term, only, and there’s a term for every possible degree. If we assume all pulleys are similar, P1=P2=P3=P4, we get

1 + P + PP + PPP + PPPP + …

in other terms

P^0 + P^1 + P^2 + P^3 + P^4 …

This is a geometric series with common ratio P. Because the first term is 1, we know that sum of the series is

1/(1-P)

In other words, this is the calculated MA of a simple pulley system that consists of infinite amount of pulleys with efficiency P. No need to say, this is also the maximum value of MA of a simple system using this kind of pulleys.

If you want a more visual proof, check how the sum accumulates if P = 80%:

It all starts well: the first rope leg surely has 100% of the input force and the next one 80%. But as tension of every subsequent leg is taxed by another 80% pulley, the later legs quickly become useless. The fifth leg only brings 40% of the input force, the tenth leg about 13% and the twentieth adds the sum just by 1,4%.

The proud red line represents the miserably failing assumption of ideal pulleys, where each leg is supposed to bring another 100%.

Let’s see the same figure for 60% “pulleys”, like carabiners:

So, if you only have carabiners for pulleys, don’t expect to get anything over 2:1 out of a simple pulley system. There’s no point in going beyond Z-rig on that path.

The 80% and 60% curves seem to approach 5 and 2.5, respectively. Let’s try the sum of geometric series trick:

1/(1-0.8) = 1/0.2 = 5

1/(1-0.6) = 1/0.4 = 2.5

Seems to work!