Gears
An engine that produces plenty of torque at low speed isn't automatically useful for a task that needs high speed and low torque — or vice versa. Gears are the mechanical trick that lets one rotating source be reshaped into whatever combination of speed and torque a given job actually needs.
What a gear pair does
A gear is a toothed wheel used to transmit rotational motion and torque from one shaft to another. When two gears mesh together, their teeth interlock so that rotation of one (the driving gear) forces rotation of the other (the driven gear) at a speed and torque set by the ratio of their sizes. This connects directly to the rotational dynamics and rotational energy covered earlier in this module: gears are simply a mechanical device for taking a rotating shaft's angular speed $\omega$ and torque $T$, and converting them into a different combination of $\omega$ and $T$ on a second shaft, in a controlled and entirely predictable ratio.
The gear ratio relationship
For a simple pair of meshing gears, gear 1 (with $N_1$ teeth and radius $r_1$) driving gear 2 (with $N_2$ teeth and radius $r_2$), the teeth on both gears must be the same size for them to mesh correctly — so the number of teeth on each gear is directly proportional to its radius. Since both gears' teeth move past the meshing point at the same linear speed, the ratio of their angular speeds is the inverse of the ratio of their tooth counts (or radii):
A gear with more teeth (a larger radius) turns proportionally more slowly than a gear with fewer teeth (a smaller radius) meshed with it — exactly the same geometric relationship as two circles of different radii rolling against each other without slipping.
Now consider power. Assuming an ideal, frictionless gear pair with no losses at the mesh, power is conserved as it passes from one shaft to the other:
Recall from the rotational energy topic that power delivered by a rotating shaft is $P = T\omega$. Combining this with power conservation gives
So torque and angular speed trade off inversely across an ideal gear pair: whatever factor a gear pair uses to reduce speed, it increases torque by that same factor, and whatever factor it uses to increase speed, it reduces torque by that same factor. A gear pair can never increase both speed and torque simultaneously without an external power input — that would mean creating power from nothing.
Why gearboxes exist
An engine or electric motor typically has some speed and torque combination at which it runs most efficiently, or a maximum torque it can safely deliver — but the machine it's driving often needs a very different combination at different times. A vehicle pulling away from rest needs a large torque to accelerate a stationary mass, but only a modest rotational speed; the same vehicle cruising at a steady speed needs much less torque, but the wheels must turn much faster. A gearbox with multiple selectable gear ratios lets a single engine, running efficiently within its own preferred speed range, be matched to both of these very different demands by simply changing the ratio between the engine's output shaft and the wheels.
This is the everyday, practical reason gearboxes exist in vehicles, machine tools, and countless other rotating machinery applications: they decouple the speed/torque characteristics of the power source from the speed/torque characteristics actually needed by the load.
The relationships above assume an ideal, frictionless mesh. In reality, meshing gear teeth experience sliding and rolling contact against each other under load, and some power is inevitably lost to friction at the mesh — exactly the kind of friction loss that adequate lubrication (covered in the previous lesson) is designed to minimise. As a result, real gear pairs always deliver slightly less output power than input power, $P_2 < P_1$, and the ratio $P_2/P_1$ is the gear pair's mechanical efficiency.
Key takeaways
- Gears are toothed wheels that transmit rotational motion and torque between shafts, changing speed and/or torque in a controlled ratio.
- For a meshing gear pair, angular speed ratio is the inverse of the tooth-count (or radius) ratio: $\omega_1/\omega_2 = N_2/N_1 = r_2/r_1$.
- For an ideal, frictionless gear pair, power is conserved: $P_1 = P_2$.
- Since $P = T\omega$, torque and angular speed trade off inversely across an ideal gear mesh — reducing speed by some factor increases torque by that same factor.
- Gearboxes let an engine or motor running efficiently at one speed/torque combination be matched to a task needing a different combination, such as high torque at low speed for pulling away and lower torque at high speed for cruising.
- Real gear pairs lose some power to friction between meshing teeth, so real output power is always slightly less than input power.