3.23 Gears

“It was a 2000 year old computer designed by Archimedes” claimed Helen to an incredulous response in a science discussion one evening.  She was talking about the ‘Antikythera mechanism’, found in 1901 amongst the remains of an ancient ship lying on the seabed off the eponymous Greek island. Classical civilisations were a source of fascination for her. She had read that this caked and corroded discovery was in fact an extraordinary block of gears.

Figure 1 Antikythera mechanism. Image credit: Tilemahos Efthimiadis

Research revealed that it was capable of showing the apparent positions of the Moon and Sun and of predicting eclipses. “It was an ancient computer!” she repeated enthusiastically. By using X rays and scanning technology scientists were able show that it consisted of at least 30 separate gears. This was itself remarkable given that gear mechanisms of such complexity had not hitherto been found in ancient civilisations – they were assumed to have arrived with the medieval clockmakers.

Unexpectedly, discussion on this occasion did not dwell on ancient civilisations or early astronomy or marine archaeology. Instead it was the gears that attracted attention and became the issue “What exactly are gears – they’ve always mystified me, even though I use them almost every day!” ventured Jean. “What do they do; how do they work”

It was bicycle gears that sprung most readily to mind in the discussion that ensued. Unlike car gears, they are at least visible; yet, for Helen, they were still baffling. Discussion opened up about features of gears that can be clearly seen and understood. “There are  small cogs and big cogs” Sarah stated. “The small cogs go round more quickly, the big ones more slowly. The effect depends on the number of teeth” Jean added. “There’s a gadget that puts the chain up or down a level. It looks as though different gears make the chain shorter or longer, but it can’t actually do this, can it?” queried Mary.

What are gears?

These simple observations captured the essence of bike gears; and, as Mary concluded, the chain does indeed remain the same length, whatever the gear. Its trajectory, however, alters as the gear is  changed so that it continues to fit round the cogs, whatever their sizes (figure 2).

Figure 2 Bicycle chain fitting around derailleur gears

The important question, of course, is: what are gears for – what is it they actually do for a bike or car? The cogwheels in Figure 2 give a clue. On the right-hand side the chain is wrapped around a larger cogwheel with 38 teeth (the chainwheel). When the rider turns the pedal by one revolution, the chain will move along by 38 links. On the back wheel of the bike, the left hand side of the diagram, the same chain is wrapped around only 26 teeth of the smaller, rear cogwheel (the freewheel). As result one turn of the chainwheel causes more than one turn of the freewheel at the back – nearly one and half turns in fact (38/26 or 1.46 turns to be precise). This rear cogwheel is  firmly attached to the rear wheel of the whole bike and this drives the bike and rider along their path.  As a result, one turn of the chainwheel by the rider rotates the back wheel approximately one and half times.

A closer look at figure 2 shows a cluster of different size cogwheels at the rear (known as a cassette). The smallest cogwheel has just 11 teeth, compared to the 26 of the largest in the cassette. If the chain were wrapped around this smaller cogwheel instead, a single turn of the chainwheel, with its 38 teeth, would cause the rear wheel to rotate more than three times (38/11 or 3.45 to be precise). The action of the rider, in turning the pedals just once, would make the bike run much further in this case – over twice as far. This is known as a higher gear. The more turns of the freewheel from a single turn of the chainwheel, the higher the gear.

At first sight, a higher gear would seem to give an unqualified advantage – you travel further for a single turn of the pedals. If you’ve ever ridden a bike in too high a gear, however, you would know that it doesn’t work out that way. You soon realise that you may go further when you turn the chainwheel around once in a higher gear, but  it takes more effort to do so. This is simply because the effort you are expending is being used to overcome greater resistance: more friction with road, more friction in the wheel bearings and a longer spell of air resistance, especially on a windy day. In the case of an uphill climb, additional effort would also be needed as you rise up, to work against gravity. So, there’s a trade-off: the higher the gear, the further you travel with a single turn of the driving wheel, but the great the effort needed to cover that distance.

As an aside, the complicated apparatus at the rear of the gear system illustrated in figure 2 (called “Derailleur” gears) is now easy to explain. Immediately below the cassette – or cluster of cogwheels – is a device for shifting the chain in and out, so that it can engage with different sized cogwheels. A lever on the frame of the bike makes this happen. Below this cassette, is a spring-loaded extension which pulls constantly towards the back of the bike. This ensures the chain is fully stretched whatever the size of the cogwheels over which it runs.

What’s the point of gears?

Some bikes, particularly those used to race around a flat track, don’t have gears. There’s simply one large cogwheel at the front and a smaller one at the back. They work well on a track, but present a challenge when facing an uphill climb. The rider simply hasn’t sufficient muscle power to both overcome the resistances and work uphill against gravity. Put simply the point of gears is to enable riders to manage inclines, rough surfaces or strong headwinds comfortably.

Engaging a lower gear mean shifting from a smaller to a larger cogwheel at the rear (figure 3, top). As explained above, one spin of the pedals means the back cogwheel rotates fewer times, the bike travels less far, so less work has to be done in overcoming resistance. Less effort is needed. In summary, you need gears to ensure that the effort needed to overcome the friction, air resistance and gravity does not exceed the capacity of the rider’s muscles. The converse is true for a downhill run: with the rear wheel now rotating rapidly, a smaller cog needs to be engaged to ensure the rider isn’t having to pedal madly to keep up.

Figure 3 Low and high gear  on a bike. Image credit: Moebiusuibeom-en

Gear boxes in motor vehicles

“What about the gears in our cars” asked Sarah. “There’s more of them than bikes, but we just don’t get to see how they work”. The principle is exactly the same as for bikes: cogs of different sizes meshing with one another. Unlike a bike, however, the cogs mesh directly with one another in a gearbox, rather than being connected by a chain.

Figure 4 is a photo of a  gearbox , showing cogwheels engaging with each other. In this case the teeth are cut diagonally so that they mesh gradually with one another, giving a smoother and quieter operation

Figure 4 A vehicle gearbox

Figure 5 is a simple animation that helps us see how shifting a gearstick in a vehicle causes different pairs of cogs to engage in the gearbox. The purple cogwheel, linked to the gearstick, engages with largest cogwheel (blue) for low gear (1st) and the smallest (green) for top gear (4th).

Figure 5 Changing gear – animation

Mechanical advantage

The multiplication effect of gears is reminiscent of a similar effect with levers. As we know from common experience, a lever – like a spanner or door handle -enables you to create a larger force from a smaller one.

This illustration of Archimedes statement “Give me the place to stand, and I shall move the earth” makes the point well, though it hasn’t quite captured the difference of scale (figure 6)!

Figure 6 Illustration of a lever to lift the Earth

For levers, a large force can be generated by a smaller force – the ratio is called “mechanical advantage”. There has, however, to be a trade-off: the smaller force must travel though a larger distance. This is simply expressed by the general principle that you can’t get something for nothing – the energy expended in moving a small force though a large distance is more than the energy gained from the large force moving through a small distance.

A more practical example of a lever is a pair of pliers. Here a large force used to grip and bend a piece of metal or wire is generated by a lesser force generated by your hand, thanks to the greater distance the handle moves compared to the jaws of the pliers. The energy you expend in moving the handle though a larger distance with a small force is greater than the energy consumed in moving a large force though a smaller distance in the jaws.

Figure 7 pliers as an example of a lever

Gears work in comparable fashion, though the linear force of a lever is replaced by a turning force or “torque” and the linear distance though which the force moves is replaced by a rotational equivalent. In Figure 8, the smaller cogwheel rotates more rapidly but with less turning force or torque than the larger one.

Figure 8 small and large cogwheel in a gear

Different kinds of gear

The examples of a bicycle and a car discussed above show gears being used to multiply a force to varying degrees. This enables the limited capacity of a rider’s leg muscles or a vehicle’s engine to cope with strenuous conditions, such as a steep hill or rough terrain, while also being able to breeze along fast on the flat or downhill. But gears are also used for other purposes.

The bevel cogwheels shown in figure 9, for example, are simply used to convert a horizontal rotation to a vertical one in this case for use inside a Dutch grain mill.

Figure 9 bevel gears in a grain mill

Another kind of cogwheel (figure 10), known as a worm gear, is used to convert a relatively fast rotation in one direction to a much slower one at right angles to it.

Figure 10 Worm gear

For some purposes gearing is used to slow down rotation to an extreme degree, the classic example being clockwork.

An example of a church clock in Marlborough, Wiltshire is shown in figure 11. A number of very small cogs can be seen engaging with much larger ones. When the smaller one has rotated just once, the larger one may have only turned though a tenth or less of its cycle.

Figure 11 Church clock mechanism

A succession of such gear changes enable the hand of the clock to turn very slowly indeed (figure 12). Each tick of the clog, moves the first wheel on by just one tooth; then this slow rotation is reduced even further as the small cog at the centre, labelled “c” engages with a big cog labelled “B” and  is slowed again at the small cog “b”.

Figure 12 clock mechanism showing the escapement

The slow speed of the hands on the clockface bring us back to the mechanism of the ancient Antikythera mechanism that launched this story. Somewhere, buried in that crusted, clogged-up machine are pairs of dissimilar size cogwheels, designed to slow down rotation around their axes, in order to mimic the stately motion of the heavenly bodies. Ancient wisdom, embodied in a marvel of sophisticated engineering.

© Andrew Morris 3rd November 2022

Further reading

Everyday words such as “effort” and “capacity” have been used in the foregoing text in describing the effect of gears. As they are rather vague, we introduce here some of the relevant scientific terms and their relationships in the field of mechanics.


Force is a push or pull of one object on another. In some contexts, such as building structures and bridges, forces do not move. They balance one another, statically. In other contexts, such as transport and ocean currents, forces are on the move.


When a force moves an object – a leg pushing a bike pedal or a rocket pushing out hot gases, for example – energy gets used up. The amount of energy used depends on the distance through which the force moves. A force is said to “do work” as it moves and the amount of work it does is the product of the force x the distance through which it moves. This work done by a force gets stored in the object upon which it operates, in one form of energy or another. For example the muscles in a cyclist’s legs do work on the pedals. This is stored as energy in the rotation of the cogwheels and bike wheels of the moving bicycle. Some will also appear as heat energy in the parts that rub together and warm up thanks to friction.


When the force of leg muscles on bicycle pedals or the driving force of a car engine conveys the vehicle through a given distance, a given quantity of energy will be consumed. But car drivers and cyclists are concerned about more than covering a distance; they need to know how quickly that distance will be covered – the speed. In other words they are interested in how quickly the required energy will be consumed and therefore how quickly it needs to be supplied. The issue of greatest interest is the rate at which energy gets transferred. This is defined in science as the “power”. A powerful muscle or engine is one that delivers a high quantity of energy every second.

As an aside, understanding these terms and their units can be useful in everyday matters. For example the energy content in a carton of yoghurt is measured in Joules or kilo Joules (kJ) or more commonly in the related unit, Calories. This helps choices to be made about which foods are most likely to help you limit your weight.

Power, on the other hand, refers to the rate at which energy is consumed or supplied. It is therefore measured in Joules per second. This unit is called a Watt. Thus, a 40 Watt light bulb is consuming 40 Joules of energy every second. The power is what determines the brightness of a your light bulb, but what you get billed for at the end of the month is the energy you’ve consumed that month. Just to complicate things, energy used in the home is not measured in the usual units of Joules or kJoules – these units are too small (equivalent to farthings and halfpennies). Instead they are measured in an alternative unit: the kilowatt-hour (kWh).This is the amount of energy consumed in one hour by a 1000 Watt device (or 10 hours by a 100 Watt device).