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Name: Ryan
Status: educator
Grade: 9-12
Location: MN
Country: N/A
Date: 4/1/2005


Question:
I am getting ready to do a unit on simple machines and remember a lab I did as a student. We had to lift 150 lbs using 4 pulleys, one of which can change direction. The 150 lbs has to be lifted using as little weight as possible. I do not remember what the best way to set this up is. I have come up with some ways myself but would like to know the very best set-up to share at the end. Do you have any suggestions?


Replies:
Simply put, your best answer will be the one that has the rope doubling back on itself as many times as possible. One pulley on the ceiling does nothing but change the direction. A pulley on the load, with a fixed point on the ceiling gives you a 2:1 mechanical advantage. (Rope goes down, Rope goes up)

If you are using the first one on the ceiling to change direction, you will still have only a 2: 1 advantage. take the rope you're pulling though, put it through a second pulley on the load, and you get a 4:1 advantage. Rope goes down, Rope goes up, rope goes down (again), Rope goes up (again). Your 4th pulley reverses direction, allowing you to pull the rope more efficiently.

Ryan Belscamper


Ryan-

At first all I could think of was this:
.___2_____1_2_4____.
|   |     | | |    |
|  /O\    | | |    |
| |   |1  | | |    |
| |   |  | | |    |
| m1  |   | | |    |
|     |   | | |    |
|     |   | | |    |
|      \O/  | |    |
|     2 |   | |    |
|        \O/  |    |
|       4 |   |    |
|          \O/     |
|         8 |      |
|__________M8______|

which I would describe, working from load (M8) to worker (m1), as "3 cascaded 2:1's and a direction-changer".

The mechanical advantage is 2^3 = 8:1, which felt a little disappointing for 4 pulleys.

Finally I realized that the direction changer could come first:
.__________16______.
|           |      |
|          /O\     |
|       8 |   | 8  |
|        /O\  |    |
|     4 |   | |    |
|      /O\  | |    |
|   2 |   | | |    |
|    /O\  | | |    |
| 1 /   | | | |    |
|  m1   | | | |    |
|       | | | |    |
|       | | | |    |
|       | | | |    |
|_______|_|_|_M8___|
         1 2 4

and that it helps a lot to do so, because all those floor-anchors can be detached from the floor and re-attached to the load. Then the ideal advantage approaches 15:1:
._______16_______.
|        |       |
|       /O\      |
|     8 |  | 8   |
|      /O\ |     |
|    4 |  \|     |
|     /O\  | 12  |
|   2 |  \ |     |
|    /O\  \|     |
| 1 /   \  | 14  |
|  m1    \ |     |
|         \|     |
|          | 15  |
|          |     |
|_________M15____|

A pulley can do 2:1. 4 pulley's can do 2^4 = 16:1. Take away 1 for having a direction reverser (the one at the top). After all, your question specified downwards pull, which cannot be directly used to help lift the load. That is the "1 unit" of force which is lost. So I suspect this is about the best you can do with the ropes all roughly vertical.

In a cascading sequence of pulleys, often the counter-balancing ropes of each pulley are anchored to ceiling or floor. You can detach each of these and re-attach to the axle of a later pulley (or to the load), as long as you are displacing the attachment towards the "lower-gear" end of the pulley sequence. Displacing an attachment to a "higher-gear" place creates "fool's tackle", a mechanical short-circuit, or at least it makes the mechanical advantage smaller instead of larger. (By "low-gear" I mean high-force / low-displacement, and the opposite for "high-gear".)

This 15:1 cascade can only lift the load about 1/8 of the distance from floor to ceiling. Then the pulley nearest the worker reaches the floor. Conventional tackle only gets 4:1 using 4 pulleys because they want to retain 100% range of travel. It can get shorter to "zero" length, or longer indefinitely, as long as the worker pulls or feeds more rope.

Jim Swenson



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