gear and gear ratio
A gear is a toothed wheel that meshes with another toothed wheel so that when one turns, it forces the other to turn too — the teeth lock together so neither can slip. The gear ratio is just the bargain those two wheels strike: it compares how fast and how forcefully one shaft turns against the other. The core idea is a trade. You can have lots of speed with little force, or lots of force with little speed, but you do not get both for free — gears let you swap one for the other to suit the job.
The ratio comes straight from counting teeth. If the driving wheel has 10 teeth and it meshes with a wheel of 30, the big wheel turns only once for every three turns of the small one — a ratio of 3 to 1. The slow-down is exactly threefold, and in the same bargain the turning force at the output is multiplied by roughly the same factor (a little is always lost to friction). This is why a small, fast, weak motor can move a heavy load: route it through gearing and you cash in speed you did not need to buy the strength you did. Run the same gears the other way and you trade strength back for speed.
Gearing is everywhere in robots because raw motors are almost always too fast and too weak for a joint that has to lift, push, or hold steady. The right ratio lets a tiny motor swing a heavy arm or grip with real force. The trade has costs, though: more reduction usually means a stiffer joint that is harder to push backward (less backdrivable), and cheap gears suffer backlash — a sloppy gap where the teeth rattle without driving — which shows up as a small dead zone before the joint responds. Picking the gear ratio is one of the first and most consequential decisions in designing any powered joint.
A hobby servo motor spins fast but can barely budge a load on its own. Behind its output sits a stack of gears, often around a 200-to-1 reduction: the motor whirs about 200 times for each slow turn of the horn, and in exchange that horn comes out roughly 200 times stronger — strong enough to swing a small robot leg.
A 200:1 gear stack converts a servo's fast feeble spin into the slow, strong turn a robot leg needs.
A ratio above one (e.g. 3:1) is a reduction: it trades speed for force. A ratio below one speeds things up at the expense of force. Force is multiplied by roughly the same factor speed is divided — the trade is symmetric apart from small friction losses.