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How gears work

What’s great about modern gear design is that it allows you to keep pedalling at the same speed, regardless of terrain, by varying the amount of energy needed to turn the back wheel. Here, I’ll explain it with math. Please ignore this page if it seems too technical – it makes no difference to how much you enjoy your cycling.

Imagine you no longer have a spangly bike with dozens of gears. Instead, you have a model with only two chainrings at the front, which you turn by pedalling, and two sprockets on the back, which push the back wheel around when they are turned. This leaves you with a 10-tooth and 20-tooth sprocket at the back, and a 20-tooth and a 40-tooth sprocket at the front. These combinations aren’t useful for cycling, but they make the math easier.

Start with your chain running between the 40 at the front and the 10 at the back. Begin with one pedal crank pointing upwards, in line with the seat-tube. Turn the cranks round exactly once. Each link of the chain gets picked up in the valley between two teeth. Since there are 40 teeth on the chainring, exactly 40 links of chain get pulled from the back of the bike to the front.

At the back of the bike, exactly the reverse happens. Since each link of the chain picks up one sprocket valley, pulling 40 chain links through will pull 40 sprocket valleys around. But the sprocket you’re using has only 10 teeth, so it will get pulled round four times (4×10 = 40). The sprocket is connected directly to the wheel, so, in this instance, turning the chainring one turn means that the rear wheel will turn four complete turns. To measure how far this is, imagine cutting across an old tyre to make a strip instead of a hoop, then laying it out along the ground. Measure the distance and that’s how far the bike goes if you turn the wheel once. Turn the wheel four times, and the bike goes four times as far.

For comparison, leave the chain on the 40-tooth chainring, but move it to the 20-tooth sprocket at the back. Turn the cranks once and the chainring will still pull the 40 links around, but, at the back, pulling 40 links around a 20-tooth sprocket will only pull the wheel around twice (2×20 = 40), so the bike goes half as far as in the previous example.

Finally, leave the chain on the 20-tooth at the back, and pop it on the 20-tooth at the front. Now, turning the cranks around once only pulls 20 links of chain through, which in turn pulls the 20-tooth sprocket and the wheel around exactly once. Thus, turning the pedals around once moves the bike forward one tyre length – a one-to-one ratio.

In the first example, the bike goes much further, but it is harder work to push the pedals around one turn. In the last example, it is very easy to push the pedals around, but you don’t go far. Sometimes, you need to go as fast as possible and you don’t care how hard you work, so you use a combination of big chainring and small sprocket. You might be charging downhill or trying to catch someone ahead of you or sprinting for the sake of it.

Other times, like when you happen to be climbing a steep hill or start off from rest, it takes all your energy simply to keep the wheels going round, so you need the easiest gear possible. Then you choose something like the last combination – small chainring and big sprocket.

Going back to the original bike, you have up to 10 sprockets at the back and up to three chainrings at the front. These allow subtle variations in how far the bike goes and how easy it is when you turn the cranks. The aim is to maintain a constant cadence, at a level that is most efficient for your body over varying terrain. The two derailleurs (gear mechs – short for mechanisms) are controlled by separate shifters at the handlebars, the front by your left hand and the back by your right. These evolved by mechanical necessity but have proved very convenient.

On the back wheel you have a selection of sprockets on the cassette, as mentioned above, with small gaps between the sizes. Moving up or down from one sprocket to the next might make a gear 10 per cent higher or lower, allowing very subtle changes in pedalling speed. You shift the sprockets (i.e. rear derailleur) most often, so it makes sense for the handlebar shifter to be on the right-hand side, since most people are right-handed.

Your left hand controls the front derailleur. You only have three chainrings on the front, but the differences between the number of teeth on the ring are far greater to allow a radical change to pedalling speed with a single shift. This is useful when the terrain changes unexpectedly – for example, if you turn a corner and the trail suddenly climbs. By moving the chain from the big to the little chainring, you can change into a much lower gear quickly enough to maintain momentum. Although it’s possible to change into all the combinations of chainrings and sprockets, in practice, some should be avoided. Using the largest chainring and the largest sprocket means that the chain has to cut across from one to the other at a steep angle, which makes the chain wear faster and wastes pedalling energy. The same goes for the combination of the smallest sprocket and smallest chainring. Both these gears are duplicates – the same gear ratio can be found by switching into the middle chainring at the front and one of the middle sprockets at the back.