# How do gears work?

I’ve read the How Stuff Works article on how gears work. It was not very helpful, actually, as it focused on engines and I’m more after the theory. I’m confused as to what exactly makes gears work!.

I’ve managed to grasp the concept that gears behave like levers, in that they can multiply force. HOW they do that is beyond me. It seems to me that (worm gears aside) they’d be ‘pushing back’ just as hard as if you were trying to move the load by hand.

For example: say I have a winch. How is it I can sit there and spin a handle with minimal effort and move a load that, were I pulling on the rope directly, I’d never have a chance of budging? That’s what I don’t get. I understand it works, but I don’t understand why.

Or this: how is it changing gears can produce more force from (presumably) the same input? Doesn’t it matter that you’re not pushing harder? I mean, that’s starting to sound like free energy or something… which I know is bunk, so obviously I’m missing something pretty important here!

Can someone enlighten poor stupid me? (I never grasped why more than one pulley arranged in some mystical fashion can reduce effort, either, but I’m told THAT works as well. Arrgh!)

Gears (and pulleys and winches) create mechanical advantage by trading force for speed. Say you’ve got a 10-tooth gear meshed to a 100-tooth gear. The big gear rotates slower – once for every ten times the small gear rotates. However, the big gear will rotate with ten times the force (minus losses from friction.) So if you rotate the small gear with a force of one pound, the big gear can lift a ten pound object.

This sounds like magic.

No, I’m not being deliberately obtuse. I mean, what’s (for example) to stop someone from hooking up these gears as you’ve stated, throwing a small, cheap motor on the first and latching the second to a big magnet in a coil, and thus creating an electrical generator that produces more than it consumes?

Obviously, something stops that from happening… so what am I missing here?

I guess what I’m saying is, I know this stuff works and I even get the idea that a small gear will turn a large gear with more force than you turn the small one with. But the WHY of it escapes me. I can’t argue that it doesn’t work – there it is, staring me in the face – but I don’t understand why. Again, it seems like something for nothing (or almost nothing), which, if I remember my physics classes, is the big no-no.

Still confused, sorry.

Well first off the big gear will rotate at 1/10th the speed of the small gear. Ten times the force, but 1/10th the speed. You are trading speed for power.
Secondly you have friction and other mechanical losses.
Thirdly the conversion of mechanical energy to electrical energy is not 100% efficient.
Lastly the conversion of electrical energy to mechanical energy is also not 100% efficient.

On the other hand, if you powered the big gear, the small gear would travel at 10X the speed but with 1/10th the force.
This is why you can’t climb a big hill in high gear in either your car, or a bicycle.

It’s not so much the number of teeth on the gears as it is about the radius of the wheels - it just happens to be the case that if you want to make the teeth mesh, they have to be approximately the same size and you can fit less of them on a wheel with a smaller radius than a bigger one - but it’s about the radius of the wheels, not the number of teeth, really.

Forget the wheel for a moment and consider only the teeth that are in contact with each other and a strip of material between them and the hub - you could mentally simplify this to a system of levers - the smaller wheel has a shorter radius, so the lever extending from the hub to the tooth is shorter than the lever extending from the hub to the tooth on a larger wheel.
Given the same input force and distance, a long lever will turn it into long distance at a reduced force, a short lever will turn it into a short distance with an increased force.

I’ll post a picture in a minute that will make more sense of what I’m saying.

…So what that amounts to, Sofaspud, is you could have a small motor turning a small gear turning a big gear turning a big generator slowly… and the output of the generator would be exactly enough to power the motor if you lived in some ideal world where there were absolutely no losses at any stage.

This happens because the power involved is equal to the force multiplied by the speed of rotation - like a big torquey car engine turning slowly, versus a small motorbike engine screaming its nuts off, the car engine is making bigger but fewer power impulses, the bike engine smaller but more, and the practical upshot is that they both make the same power.

Or looking at ropes and pulleys, with a couple of multi-sheave blocks you can lift maybe ten times what you could with just a single fixed pulley, but you also have to haul in ten times as much rope - so you don’t have to work as hard, but you have to work for longer, and it all balances out -

Except that it doesn’t on account of energy being turned into heat or squeak. But that’s life.

A perfect example is lifting a car with a jack. You have to actually move the crank about a thousand times further than the car is raised. You’re applying exactly enough force to lift a car 3 feet off the ground with your bare hands, but the jack allows you to trade less required strength for more motion. You aren’t using any less energy from your body to lift that car than a machine doing the same thing would.

Pulleys do the same thing. You have a lot of extra motion involved, but the amount of force applied is proportional to the extra time and motion it takes to pull it. You’re pulling extra rope and taking extra time, essentially spreading the exertion out over time so that you don’t have to apply it all at once.

edit: I don’t know how much more you have to turn the crank on a car jack than the actual distance you move the car, but it FEELS like a thousand, and that’s good enough for me.

Remember that power is a measure of the speed of delivering force. If you have a certain amount of power (from your little motor) that means it will deliver a certain small amount of force at a certain speed. If you gear it down, you will get a larger amount of force at a lower speed. The power will multiply out to be the same (less efficiency losses) either way.

Careful there, big fella. Speed for torque.

Here is the diagram.

You can see that a simple gear can be reduced to a series of levers. For a given input spindle force and rotation angle, the far end of the longer lever will move further, but weakly, whereas if the same force and rotation is applied to the spindle of the smaller gear, the far end of that lever will move strongly, but over a short distance.

Furthermore, for a given force and distance applied to the end of the long lever, the spindle will turn strongly, but not much, whereas for the same force and distance applied to the end of the short lever, the spindle will turn further, but more weakly.

So driving the larger gear spindle moves the end of the lever a long way, which acts on a short lever to moves the spindle of the smaller gear a long way, but at reduced force.

Driving the smaller gear spindle moves the end of the lever a short way, which then acts on a long lever to move the spindle of the larger gear at increased force, but over a small distance.

It’s the same magic that lets you use a prybar (lever) to raise a 10 lb. object up one inch while pressing the other end down 10 inches with one lb. of force. Essentially, moving 10 lbs. a distance of one inch is the same amount of work as moving one lb. a distance of 10 inches. The lever makes it easier to do the work (as it’s easier to exert one lb. of force than to exert 10 lbs. of force) but doesn’t lessen the amount of work, as you’ve got to exert that one lb. of force for ten times as much distance.

What are you not getting here?

The power of gears is truly awesome. I once built a gearbox (using scrap metal and some regular and worm gears I found) for my graduate school lab work that geared down by a factor of 2000 or so, enabling me to use a cheap, weak electric motor to lift a pretty impressive weight slowly and steadily, after my advisor refused to put down the money to simply buy such a device. Despite my crude handiwork, it worked extremely well.

Mangetout, that picture helped immensely. I understand how levers work; reducing a gear series to a series of levers in my mind made it click. Thank you!

Thanks very much, everyone. I don’t feel so stupid anymore now.

Now, extra-special bonus question: does anyone know when gears were ‘discovered’? Invented would be a better term, I suppose.

Will this help?

I think gears long predate the Antikythera mechanism. What makes the mechanism so apparently anachronistic is that the gears are small and spoked rather than solid.