All geared up to go for a ride

Q. I used to teach from a textbook which had a question about ratio and bicycle gears, but I avoided getting pupils to answer it as I didn't understand it. Can you help?

A. There are three parts in determining the gearing on a bicycle: the number of teeth on the chainring(s); the number of teeth on the rear cluster; and the wheel, as shown in the diagram.

The chainrings are attached to the pedals and revolve with them: one revolution of the pedals gives one revolution of the chainrings.

The first thing to do is to work out how many times the rear cogs go round for a single rotation of the pedals. Consider the largest chainring and largest cog. On my bike, I have 42 teeth on the largest chainring and 32 on the largest cog.

The 42 teeth of the chainring means that for one revolution of the pedals the chain is pulled forward 42 links. The chain then rotates the largest cog by 42 teeth. As there are 32 teeth on the largest cog, it will rotate more than once. This means the ratio will be greater than unity.

To work out the ratio, it is easier to consider how many links of the chain mesh with each sprocket (chainring or cog) in one rotation of the pedals.

For a 42-tooth chainring we know that 42 links of the chain will be pulled forward in each rotation of the pedals. The 32-tooth cog will use 32 of these links in its first rotation. As the 10 remaining links move forward they will rotate the cog some more (about a third more, as 10 is roughly a third of 32). We can write the ratio as fractions of the number of links through which each sprocket moves for one revolution of the pedals: Note the numerator is the same on both sides (links of the chain used) and the denominator depends on the size of the sprocket. Also, as the number of chain links involved depends on the chainring being used, the chainring value is always unity and, therefore, the ratio will be 1 : teeth on chainringteeth on cog.

The ratios of other chainringcog combinations can be worked out in a similar way. The highest gear is that with the largest ratio (ie largest chainringsmallest cog) and lowest gear is that with the smallest ratio.

For my bike these ratios are 1 : 4211 (1:3.8) for the highest, and 1:2232 (1:0.69) for the lowest. But what does that mean for me when I cycle?

The gear system means that as I rotate the pedals once, the bicycle will travel different distances for different gears. That is, the effective size of the wheel changes.

I worked out the effective sizes of my bicycle's rear wheel for the highest and lowest gearing ratios and for 1 : 1 (unity) and I didn't believe the results so I measured them.

The effective diameter (size) of the wheel is found by multiplying the actual diameter of the wheel by that gear's ratio. For the lowest gear, the effective diameter is 2232 x 27 = 19 inches, to the nearest inch (ratio = 1:0.69; wheel diameter (including tyre27 inches). Therefore, one revolution of the pedals should carry me 9 x 2232 x 27 = 58 inches (4 feet, 10 inches).

For the highest gear, the effective diameter of the wheel and distance travelled should be 103 inches and 27 feet. I didn't believe these results so I went outside with the bike and measured them, and they turned out to be remarkably accurate.

I also discovered that as mountain bikes have diameters given in inches (their American origins), this exercise helped give pupils an appreciation of imperial units.

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