Someone asked this in an IRC channel this morning, and I am not confident in my solution.
Question : In a 6:1 compound pulley system, there are 3 pulleys. If 2 of the pulleys have an efficiency of 66.6%, and one pulley has an efficiency of 95%, where in the system should the high efficiency pulley go to get the best ratio?
It includes thisvisual aid. The 6:1 is the second one from the bottom.
My possibly wrong solution seems to show it doesn’t matter, which doesn’t make intuitive sense.
If it’s really efficiency, it doesn’t matter where the good pulley goes. You’re going to multiply 0.666 X 0.666 X 0.95, and multiplication does not depend on order.
If each pulley has a constant amount of friction (which seems more likely) you want the pulley that turns the most easily to be in the location that does the most turning, so closest to the moving end of the rope.
The linked diagram in the OP shows a non-intuitive arrangement of pulleys. For a more typical multi-purchase pulley rig with one fixed multi-sheaved pulley at the fixed (“dead”) end and one at the load (“live”) end I agree 100% with our esteemed Napier’s explanation of the typical logic.
I’m not sure that typical logic applies in the OP’s weird case though.
There’s no difference in principle between using a multi-sheaved pulley and separate single-sheaved pulleys.
True. But did you look at the diagram? Lots of pulleys pulling on the middle of ropes, etc. Not at all the typical arrangement.
Agreed. But I don’t think that changes the validity of Napier’s analysis.
Damn, this is just too nice – I want to frame it!
I don’t think it matters that we are pulling on the middles of ropes, because we could obviously move the attachment points quite close to the ends without changing the forces and motions. It’s just a question of where the system runs out of traveling range first.
I also think the system is unusual, at least I don’t remember seeing most of these arrangements before.
But I am getting troubled by a thought. Couldn’t we add a pulley someplace in some trivial but pathological role, such that its efficiency was zero or negative? And wouldn’t I then have to argue that the efficiency of the whole system is zero or negative, because I want to multiply them all? Or, otherwise, what are the criteria for deciding whether to include a pulley efficiency in the multiplication?
I also think the diagram is implausible. It looks like a small boat has come fast upon a rock, and our line hero is trying to rescue or salvage it. By them time he has tried all these arrangements, hasn’t he already made too big an effort for such a small boat, which is probably damaged as well? Maybe the efficiency of the whole system actually is zero or negative.
The pictured arrangements certainly aren’t what one would see in a physics textbook, but one can see the appeal: They have a higher mechanical advantage than a textbook arrangement with the same number of pulleys.
I think the advantage is proportional to the number of pulleys a given rope passes through sequentially, and an exponential function of the number of internally sequential subgroups. Or some such.
And they’re fairly easy to improvise from stuff (e.g. rope, carabiners) that might be on hand when your canoe has wrapped itself around a rock.
One bigger racing sailboats of the past before big powerful hand powered winches came along, a big fore sail, even on the boats I have had, there were one multiple turning blocks on the lines so one person could handle a big jib with a small winch. It did not work well for fast tacking without a lot of deck apes but a family could work some big boats with little kids and the cost is way way down from big & heavy 4 sheave blocks.
