A <gear> is a toothed mechanical element that transmits motion and <torque> through contact with another <gear> or a <rack and pinion>. The teeth maintain a prescribed relation between the motions: unlike a smooth friction drive, the ideal <gear ratio> is set by geometry and tooth counts. The <gear pitch circle> supplies an imaginary rolling geometry for describing that relation; it is not the circle through the tooth tips.<KHK Gear Nomenclature>[“Intermeshing Pitch Circle”; The ABCs of Gears, A, §7, printed p. 18: reference circle and tip circle]{ref}

For two external <gears> on fixed parallel axes, let $N_1,N_2$ be their tooth counts and let signed <angular velocities> $\omega_1,\omega_2$ use the same orientation. Each tooth on one <gear> must pass a tooth space on the other, so counting tooth passages gives
$$
N_1\omega_1+N_2\omega_2=0,
\qquad \frac{\omega_2}{\omega_1}=-\frac{N_1}{N_2}.
$$
Thus a 20-tooth input driving a 40-tooth output makes the output turn half as fast in the opposite direction. An input of $1200$ revolutions per minute gives an output of $600$ revolutions per minute in magnitude. These signs describe an external pair; a <rack and pinion> instead converts rotation into translation.

The useful trade is between <angular speed> and <torque>. In an ideal steady transmission, <conservation of energy> equates input and output <power>, $|\tau_1\omega_1|=|\tau_2\omega_2|$. The same 20-tooth/40-tooth pair therefore turns an input <torque> of $2\,\mathrm{N\,m}$ into an output <torque> of $4\,\mathrm{N\,m}$ while halving <angular speed>. This does not create <energy>: more <torque> accompanies less <angular displacement> per input revolution. Real <gears> dissipate <energy> through <friction>, and <backlash> permits a small lost motion on reversal; tooth strength and bearings, rather than the <gear ratio> alone, determine allowable load.


 Gear (source code)