Tuomas Pöysti 2020

If you find my pulley system articles even a bit interesting, you probably are familiar with a basic rule of thumb regarding pulley systems (or “rule of fist”, as we appropriately say in Finnish): the most efficient pulley should be placed closest to the pulling hand.

This has been proven many times using the T-method modification that factors in pulley efficiencies – or at least shown that it works in some single cases. How about trying a bit more systematic approach, though?

## The specimens

Once again, here are the pulley systems we are talking about. For pulley system analysis, it is very important to carefully define the pulley labels.

The Z-rig represents all simple pulley systems. Just add pulleys in a logical numerical order. The same goes with the “V on V” family.

## Sensitivity to a pulley

Using the pulley system analysis method, the MA expressions of the pulley systems were obtained. See this article for details. Here they are in a table form, including a note about what pulley is “closest to hand”:

Now, if we count how many times each pulley manifests in terms of each degree:

The more times a pulley appears in the MA expression, the more impact it has on total MA. The degree of the term also matters, since a higher degree term involves other pulleys as well, and generally has lower value to start with. Changing a single first degree term from 0.8 to 0.3 actually decreases the total MA by 0.5, whereas a third degree term would have an initial value of 0.8^3=0.51.

That is, the term occurrences tell something about the **sensitivity of total MA to efficiency of a given pulley**. Let’s study the numbers.

In this table, the clear winners of the sensitivity contest are highlighted:

## Interpreting the numbers

The simple systems are a clear case. The closest pulleys rule the first degree, the other degrees are a tie.

But already the Z on Z is an unsolved case between P2 and P4. This might sound a bit surprising, but it is almost self-evident if you think it this way: The total MA of a compound system is the product of the subsystems MA’s, and thus the order (which 3:1 pulls which) does not matter.

The 11:1 is again a clear victory to the closest pulley.

A is a tie between P2 and P4, so the rule applies here, too. Likewise, D does not differentiate between P2 and P3.

B is interesting: P4 is clearly the most critical, but not closest to the pulling hand! Same goes with E, although with a minuscule margin.

C and “crevasse” behave nicely and let the closest win.

F and “double mariner” are so close to the theoretical maximum (almost all boxes ticked) that they barely make a difference between any of the pulleys. All we can say is P1 is not the critical one.

Reversed or upside down “V on V …” systems are even closer to having all terms, they do not make any difference between pulleys. This goes with all “V on V on V” pulley systems, no matter how many V’s there are and if the pulley system is upside down or not. These systems have all possible terms in their expression, so there cannot be a difference.

One more note: I tend to label pulleys so that P1 is the most obvious place for a PCD. Even if all pulley systems actually are not the most sensitive to the closest pulley to hand, none of them seems to be the most sensitive to P1. This is great news to all who like using descenders as PCD’s!

To wrap it up:

- Simple systems: the rule works
- Z on Z: the rule works, but P2 is as critical
- There are pulley systems (B and E here), for which the rule does not apply
- There are also a lot of pulley systems which have several equally important pulleys
- Luckily, the PCD location is usually not critical
- Remember, this is mainly theory and the differences may be small. Calculate, assess and field test!