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You Were Lied to About Pulleys

Tags: physics mechanics

In elementary school, we all learn the simple machines. I remember being very disappointed that we were supposed to be learning about machines, but all studied were stupid things that just sat there, like ramps an pulleys. A ramp isn’t a machine. There’s not that much difference between a ramp and a steep hill, but people would look at you like a madman if you called a hill a machine.

Part of the reason that the discussion of simple machines is so disappointing is because they’re usually taught in a misguided way. The lever works like a teeter-totter. The wheel and axle make it easy to move things because they roll. The ramp let’s you get that wheel up the steeps. Then the pulley…. well, it let’s you lift thing by pulling down, which is way easier than pulling up. Far less likely to throw your back out. Oh, and you can tie the rope off to hold it in place. Yeah, that’s what pulleys are for. Now, who wants a juice box.

In actuality, all of the simple machines are based behind a simple idea: I’m willing to apply a small force over a long distance in exchange for moving something heavy for a very short distance.

At some point, we’ve all seen the sitcom scene where the dumb character is in the gym and is told to do 20 reps with 15 pounds. The dumb character decides to save time and just do one rep at 300 pounds. We’re all supposed to laugh at how stupid he’s being, but the joke usually falls flat. As an aside, how come no one ever does the joke the other way around? Just once, I’d like to see the camera cut back to the character pounding through 300 reps with a one pound weight. We could listen to the character rapidly count out the numbers (247,248,249,250) while his arm flails up and down unathletically with a pitifully small dumbbell easily contained in his palm.

Moving back to the point, we all know that there are things too heavy for us to lift. However, when something is easy to lift, we can usually carry it pretty far. The reason this becomes important is one of the great physics relations:

(I’m going to take a moment to apologize here for the fact that I haven’t actually defined energy or force. I’ll save that for a later post, and I’ll link to that one once it’s written. In the mean, you can roughly think of force as how hard something is being pushed and energy as a measure of how much force something contains. They’re much deeper than that and you’ll have to forget this description later, but it will work for the length of this post.)

The key bit about the above relationship is that, if we go a longer distance, we don’t need as much force to do the job. All of the simple machines are ways to provide a small amount of force over a long distance to provide a larger force over a small distance.

We’ll start with the lever. The teeter-totter is a horrible analogy, since the two people sit the same distance away from each other. Instead, just think about a crowbar. You place the crowbar in something that is stuck, push the bar as far as you can, and whatever you were prying on only moved less than an inch. Of course, you couldn’t have moved it that tiny amount by yourself - that’s why you needed the crowbar in the first place. So you moved the crowbar a long distance in exchange for opening whatever was stuck by a tiny distance.

Next comes the ramp. The ramp is about moving a long distance in an easy direction to escape needing to move a short distance in a hard direction. Think about pushing a heavy cart up a ramp. Still seems like hard work, right? What’s being gained from this? Well, try imagining that I took the ramp away and there’s just a three foot wall in your path. Now my hypotheticals are just getting sadistic. Does it seem impossible?

It’s not impossible. You just pick up the entire cart and carry it over the wall. Of course, that’s not what we imagine in these sort of situations, because our intuition tells us that’s it’s stupid. When we have a heavy cart that’s already a pain to move, we’re certainly not going to pick it up and carry it over the wall, because we can’t. The cart is just too heavy. However, if there’s a ramp, then we can very slowly lift the cart over the wall. It’s to heavy for us to lift the cart three feet, but we can push it upward very slightly over the course of twenty feet and we’ll be just fine.

Now comes the wheel and axle. This one might seem like it doesn’t fit, because wheels aren’t about multiplying force. The catch is that the “wheel and axle” of a simple machine have nothing to do with the wheel and axle that we push our carts around one. Instead, when the Ancient Greeks first cataloged the simple machines, they were thinking more of a bucket entering a well.

        /   \
       /     \
      /       \
      I       I
   H  I       I
   H  I  |    I
   H  I  |    I
      O  |    O
      O  |    0
      0  |    O
      O  A    O
      O  U    O

In case my text are diagram isn’t perfectly clear, there’s a bucket (U) lowered into a stone well by a rope (|). The rope is tied around an axle (_) that connects to a big wheel (H). When you make a full rotation of the wheel, you also make a full rotation of the axle. Since the wheel is large, it’s a long way around to make a full rotation. Conversely, the axle is narrow and only pulls up a tiny bit of rope with each rotation. Again, since we’ve shrunk the distance, we’ve increased the force, so the heavy water jug comes up much more easily than if we tried to just turn the axle by hand.

Finally, we come to the pulley. The great lie of elementary school science. The reason it’s a great lies is because the way it’s usually presented is completely wrong. The diagram often looks something like this:

   | |
   | |
   | |
   A |
 XXX |
 XXX \O/
 XXX  |
 XXX / \

There’s some poor fool trying to lift a heavy object and using a pulley on the ceiling to make their job /easier/. The great lie is that this pulley doesn’t do anything. It takes just as much force to lift the block with the pulley as without it. Nothing is being accomplished here. This isn’t a simple machine - it’s a waste of time.

A better use of pulleys can be seen here:

 /O\ /O\
 | | | |
 | | | |
 | | | |
 A \O/ |
 XXXXX \0/
 XXXXX / \

At first glance, this is the same situation, but worse. We now have three pulleys. We already know that one pulley does nothing. Three pulleys might seem like an even bigger waste. The key is to look at the ropes. The rope now touches the block three ways: It’s tied to the block at the (A), it comes down to the pulley at the (O), and it comes back off the pulley at the (O).

If the block is raised one foot into the air, each of these three rope section is shortened by one foot. However, since it’s all really one big rope running through some pulleys, that means the person pulling the rope needs to have pulled it three feet. See where this is going? The block moved one foot, but the person pulled the rope three feet, so the block was lifted with three times as much force as the person pulled the rope. We now have a real, useful machine that lets us lift things that we never could before. The beauty of the pulley isn’t that it let’s us pull something in a different direction, but rather that each pulley let’s us multiply the amount of force we can provide.

This is why we’re supposed to be learning about simple machines in school. We get an understanding of how we can trade some things that are plentiful, like distance, for things that we don’t have, like force.