The Physics of Cycling

Random notes


1 horsepower = 746 Watts.


A professional cyclist can steadilly output 400 to 450 Watts of power.

A fit cyclist can steadilly output 200 to 250 Watts of power.

A casual cyclist can steadilly output 100 to 150 Watts of power.


Q: How to determine the amount of power you can steadilly output?


A:In my case, I went to a gym that has an indoor exercise bicycle with a display of speed, power, distance, etc...  I set the display to indicate the power output and started gradually increasing load to the bicycle until I reached my comfort limit. At that point I was able to steadilly output around 150 W. By steadilly, I mean for at least 1/2 hour. The more you cycle, the fitter you'll get and that limit will increase up to a certain point. You will most likely do better with a real bicycle as the riding position, saddle and handlebars will give you a better fit.


A more accurate approach would be to try the hardest steep hill that we can comfortably climb (we need to know the slope of the hill that can be obtained in websites with a map that also show the elevation, like ) and record the speed (the climb should be done at a constant speed). 

The power can then be obtained from the following formula:

Power [Watts] = m [Kg] x Speed [Km/h] x Sqrt (S^2/(1+S^2)) x 2.71


m=mass of the whole set: bicycle + rider + cargo (expressed in Kilograms)


Speed =the recorded speed in Km/h


S=the slope of the hill  (5%  S=0.05, 8%  S=0.08, 10%  S=0.1, 12%  S=0.12, etc...)



Gear ratio in gear inches:


G = (Tire Diameter [in inches] x  Nr. of chainring teeth / Nr. of rear sprocket teeth


ex: A single speed bicycle with 26" tires, 42 teeth on the chainring and 28 teeth on the rear sprocket will have


26" x 42 / 28 = 39 gear inches


A triple chainring bicycle with 28" tires, a 46-36-26 chainring and a cassette with sprockets 11 to 32 teeth will have a range of:


lowest:  28" x 26 / 32 = 22.75 gear inches


highest: 28" x 46 / 11 = 117.09 gear inches




The following article on the Copenhagen Cycle Chic blog has some good cycling tips, unfortunatelly the point 5 (gears) can be misleading since it tries to apply flat city arguments to hilly cities. And even though I agree that the bicycle's weight is not signifficant up to a certain point, it is not totally irrelevant either. The whole weight (bike+cargo+cyclist) will be something to consider when chosing the gearing range of a bicycle, except when that bicycle is to be used only in flat areas. 

 Not many hilly cities like Trondheim, Norway, have this kind of cycling facilities:


To overcome some steep hills when this type of infrastructure is not present, one can either walk the bike up the hill or have a lowest gear that is low enough to make it within the rider's available power. It's not enough to say that "if 3 speeds don't make it get 5 speeds". You could even have 18 speeds and still not be able to climb that hill. In this case what matters is how low is your lowest speed and how much weight will you have to carry uphill.


Lets start by weight. The following 2 graphs show the necessary power for a bicycle with a gear of 39 gear inches (lowest gear) to carry different weights at a cadence of 60 rpm (1 full crank rotation per second):

This weight will be the total weight of the set bicycle+rider+cargo, so, lets see what happens if we have a 70 Kg person riding a 13 Kg bicycle carrying 7 Kg of cargo (total weight  90 Kg) and we add 5 Kgs to the bicycle's weight.

These 5 Kilos can mean 1 set of fenders, chainguard, mudflaps, rack, basket, saddlebag and set of tools which would turn an inadequate commuting bicycle into something much more practical.


For a 1% slope, which is almost flat, the rider would need to output:

27 Watts to carry 90 Kg

29 Watts to carry 95 Kg

a difference of 2 Watts which is not even noticeable by the rider. Also, the power needed to carry the load is small to begin with because it's an almost flat surface.

As you can see on the graph above, even if the total weight raises to 160 Kg the power needed to carry it up the 1% slope will be under 50 Watts. In these conditions anybody can ride even a fully loaded cargo bike.


Now lets do the same analysis for the 8% slope: 



 For an 8% slope the rider would need to output:

218 Watts to carry 90 Kg 

230 Watts to carry 95 Kg

One would have to be a fit cyclist to carry this load with a 39 gear inch ratio. But that being the case,

the 12 Watts difference could barelly be noticed by the rider, and would it be worth saving weight on a set of accessories that make all the difference when using the bicycle for transportation purposes?

I still don't think so. But what we see is that the steeper the hill, the bigger the difference even a small 5 Kg load will make, because now, the necessary amount of power to carry the load up the hill is aproaching the power we can steadilly output.

If this difference would pose a problem for climbing hills, two things could be done:

1-Lowering the gear ratio: replacing the rear sprocket by a bigger one or the chainring by a smaller one.

2-Lowering the cadence: cycling physics.xls 

by using this sheet, you will see that reducing the cadence will translate in a decrease in the necessary amount of power to go up the slope. This happens because you're lowering your speed and Power = Force x Speed. However, the force you will have to apply to the pedals (Avge. crank force) will be the same and if this force goes beyond your comfort level, reducing the cadence will make it even more painfull on your knees. In my case, I feel comfortable pushing the pedals with a force up to 20% of my weight because I never get up from the saddle to push the pedals.

If you usually stand to push the pedals, then maybe you can use your whole weight for a while. If it's a long climb you will need to do something to the gear ratio.



 The following graph shows that for an 80 Kg set at a cadence of 60 rpm with 39 gear inches most people would be able to climb hills up to a 6% slope (under 150 Watts of power required).