The mathematics of gearing
Martin Boote asks if I know in which issue of CB I explained how to calculate the sprocket sizes needed to raise gearing, as he wants to make his Morini Kanguro a bit more relaxed on the road but the range of sprockets available is limited.
Well I don’t, but I know I’ve mentioned the ratio of the sprockets – dividing the rear by the front as, for example, a 60 rear and a 15 front has a factor of four, meaning one tooth on the front is the same as four off the rear. That might help work a way round what sprockets are available, but I don’t think I’ve ever thought about how to relate that to road performance before, so here goes.
Looking online, stock Kanguro sprockets are 14 front, 43 rear – a factor of 3.07, or one on the front roughly equals three off the rear. The sports model 3½ sprockets (14/38 – factor 2.7) might raise the softer-tune Kanguro too high, so I think midway between the two – a 15t front or 40t rear might be better, giving a factor of 2.86. It would help to work out the ‘mph per 1000rpm in top gear’ road tests used to quote. Tests say the 3½ does 95mph and peaks at 8500. Assuming a typical 1:1 top gear ratio (actually 0.954:1 on a 3½), 95mph at 8500rpm equals 11.2mph per 1000.
Now I think if we multiply 11.2 mph by the 3½ ’s factor of 2.07 and then divide it by the Kanguro’s factor of 3.07 we get a mph/1000 figure for the Kanguro – let’s see... 11.2 x 2.07 = 30.24, divide that by 3.07 and you get 9.75mph per 1000rpm or 78mph at the Kanguro’s 8000 peak – that’s actually about right.
Applying that formula with a 40t rear sprocket, we get 10.57mph per 1000 – top speed: 85mph. That’s about 10% up on standard gearing, and means that 60mph cruising would be around 5600rpm, not 6100 rpm.
Martin says that would be perfect. Hah – if only my maths teachers could see me now!