Part 2: How to optimise a pulley / hoist system

Part 1 looked at the way to count a Mechanical Advantage (MA) of a rope system.

Part 2 will now look at the benefits of different systems, and the actual mechanical advantage as opposed to the ideal one.


Any hoist system will have aspects of friction, and of course obey the laws of thermodynamics. 1: As heat comes out via friction, that system loses energy. 2: That energy loss dissipates out, and cannot be totally regained. 3: Doesn't really apply. With this in mind, is there any way to best organise a system to reduce the friction with given equipment (pulley vs carabiner)? Yes. And given the friction co-efficients of different pieces of equipment, can we choose a lower Ideal MA but benefit from higher Practical MA? Yes...


Friction co-efficients:
I haven't done the testing for figures on a pulley, a revolver or a normal carabiner, and can't find any exact figures (but have on fairly good authority that these figures are good for example purposes).
Lets assume a well made pulley has 90% efficiency, a carabiner with a small pulley built in has 50%, and a normal lightweight climbing carabiner has 40%. Now let us assess the different practical MA depending on where you put the pulley.

3 to 1 with pulley at the prussic near the climber
In this instance we count the mechanical advantage in the same way as before, but at each turn in the system we take into account the friction loss. So as the 1 unit of force comes around the pulley, there will only be 0.9 units on the other side.  This 0.9 unit will then pass through a normal carabiner (40%) and come out the other side as 0.36 units.  We then total up the system as before to arrive at the practical mechanical advantage.  In this case, we add the 1.9 from the prussic to the 0.36 remain and get a ratio of 2.26:1

3 to 1 with carabiner at the prussic near the climber
This time the system has a normal carabiner at the prussic by the climber, and a pulley at the belay.  This positioning has big implication for the system. The 1 unit of force drops to 0.4 at the carabiner, then drops to 0.36 after the pulley.  Leaving a PMA of 1.76:1 overall.  This is 22% less efficient than the other way around.

Conclusion:
These findings are probably more suited to a situation such as crevasse rescue, or slacklining, where the organiser can rearrange the belay anchors and place pulleys in the best positions. In a climbing situation, 3:1 is usually the only viable option, unless one finds themself a long way down the rescue scenario and has escaped the system and freed up a rope. But generally speaking, if you put the most efficient type of carabiner or pulley at the moving part of the system, you will achieve the biggest PMA. In that way, the system can be far more efficient than a more complex one with much higher Ideal MA.  Such as this 5:1 IMA, that only achieves  1.65:1 when using all carabiners, showing that it's better to use a 3:1 with a pulley.
Last diagram, showing that a relatively inefficient set up of an 11:1 system (2.658:1 PMA), is only 15% better than a 3:1 with a pulley (2.26:1 PMA) !