Gears and Mechanical Advantage

There are three things that really set mankind apart from the remainder of the animal kingdom: 1) opposable thumbs, 2) the collected works of William Shatner, and 3) tools.* The ability to use tools has allowed man to build, create, and otherwise accomplish feats no human being could do alone, largely through a phenomenon called mechanical advantage.

Mechanical advantage is kind of a blanket term for any sort of force amplification created by a simple machine—think crowbars, levers, gears—that outputs much more force than a person could create on his or her own. We've seen this already with levers, where a light object can balance a heavier one by increasing its distance from the fulcrum (Archimedes' Law of the Lever). Gears can accomplish the same thing based on their size or number of teeth.

When gears mesh together, their teeth interlock and their motion is inextricably connected:

This interlocking motion means the tangential velocity of any point on either of two interlocking gears must be the same. (If this wasn't true, the tip of the gears' teeth would move at different speeds and the gears wouldn't mesh at all.) Identical tangential velocities but different radii means that two interlocking gears must have two different angular velocities, since . The larger gear must rotate slower in order to mesh with the smaller gear.

Imagine you start turning the smaller gear (maybe you're pedaling a bike, or cranking a winch, or maybe you just like gears). You're applying some torque τ_sm to the smaller gear, which has radius r_sm and turns at speed ω_sm. Conservation of energy tells us that the amount of energy per second you put into the gear—that is, the power you supply—must be the same as the amount of energy per second output by the larger gear, with radius r_lg and angular velocity ω_lg. The power you input to the small gear is P = τ_smω_sm, while the power output by the larger gear is exactly the same: P = τ_lgω_lg. Then we have τ_smω_sm = τ_lgω_lg, or:

We know ω_lg < ω_sm, so this tells us τ_lg will always be bigger than τ_sm—in other words, the larger gear will always output more torque than input to the smaller gear, but will rotate more slowly.

So suddenly, just by turning a gear, you can make more torque than you ever could on your own. Shatner would be proud.

*Language, too, maybe. But mostly thumbs, Shatner, and tools.