Sunday, 9 October 2011

Climbing a hill on a bicycle

Deep down everybody fears hill climbing: the beginners, the veterans and the pro's. It is one of those activities which many people - often including cyclists themselves - find unexplainably masochistic. Why do you climb a hill, when you will come back down again? And if you really insist in climbing, why don't you do it in a car?
I won't go deeper into philosophic essays regarding this subject, although, if you care to think enough about it, you may come to some profound quintessential matter like Universe, Big Bang, Meaning of Life, etc. I will only state some physical facts that may help you to climb the hill easier.

When you cycle, and in particular when you cycle up hill, the nature is working against you. It is trying to stop you from cycling by imposing various forces that act against your movement. There are three major forces acting against you: the force of gravity (G), the force of rolling resistance (R) and the force of air resistance (A). To move against these forces you have to do some work (W). In mathematical terms the work is a (scalar) product of the force(s) and the path (L) of your movement. When you climb a hill of road length L and of grade N (in %), the work against the gravity is:


where g is the acceleration of gravity (9.81 m/s^2) and M is a mass of yourself, bicycle and everything on you and on it. You are doing also the work against the rolling and air resistance, but when climbing steeper hills these two represent only about 5% to 10% of the total work, so, for now, we will disregard them for the sake of simplicity.
The energy to do the work is provided by your body, by your muscles. To climb a hill you must produce at least as much energy as the work W. In reality you are producing much more energy, because some of it (some say even 70% of it) is lost in the dissipative processes in your body, and, of course, some is required to keep you alive, even when not moving at all. We will denote this additional energy as internal body energy (Eb). The total energy required is thus (Eb+W).
You will input that energy during some period of time (t) - the time that you climb the hill. Energy divided by time in physical terms is called power (P):


When measuring and reporting bicycling power we usually only measure the part that corresponds to the energy W; the part Eb, which is lost due to small muscle efficiency, is usually not measured or reported. So, we are left with:


The ability to produce a certain amount of power is in accordance with everyday conception of physical power: trained or stronger people can produce more power. The pro cyclist can produce around 400 Watt to 500 Watt of power when climbing hard. Ordinary cyclists produce 100 Watt up to 250 Watt of power over a period of few hours. According to the above equation, you need more power if you climb the hill faster (in less time), and if you climb the same hill slower, you need less power. Think of it in exaggerated terms: if you had the whole week to climb the hill, you would still need to input the same amount of work W, but you would need so little power that you would probably not even notice there is a hill. The first point to remember is thus: you can climb practically every hill by climbing it slower.

The speed of cycling (v) can be expressed as:

v = L/t = 2*π*R*c*T

where (R) is the radius of the wheel, (c) is cadence of pedaling and (T=F/B) is the transmission ratio (where F and B are number of teeth on front and back rings). By substituting the above equations, we have:


What can we say about this equation? It connects the physical quantities (P,M,N,R,c and T) in a definite way. If we change one of those quantities, then at least one of the rest of them will change too. For example, with a maximum power (P) that we can produce, we can climb a steeper hill (=higher N), if we have lower mass, lower cadence, lower transmission or smaller wheel. Conversely, if we cannot change M, c, T or R, then the only way to climb a steeper hill is to produce more power. Incidentally, from the equation you can even see another possible solution: lowering the effective gradient (N) by zigzagging the climb. Of course, any self-respecting cyclist would never do such a thing - it would be as if stepping down on a climb.

Now, you probably have the feeling that this is perfectly logical, so you may ask why did you have to go through all this math to come to the conclusions that you already know? Well, first of all, it's the proof that physics and common sense sometimes do agree. Second, you can use the equation to calculate, for example, what gearing you require on a certain hill, or what is your power output, or how much will you gain by buying lighter bike, or by loosing 2 kg of fat, or climbing with a MTB instead of road bike. We can very well say that it is the "mother of all cycling equations". Third - and this may be the most important - you may realize that you can climb a certain hill that you considered too hard, and not even that, you may even plan how to climb it without much effort.

To see that, lets first write the above equation in sipmlified form, like this:


From this simpler form, you can see that you can climb a steeper hill either by loosing weight (difficult), producing more power (impossible) or lowering climbing speed (quite possible, as we already mentioned).
Lowering the speed probably instinctively makes you think of lower gearing. But even when you run out of lower gears, you still have another ace in your sleeve: lower the cadence. If you lower the cadence down to around 35 rpm you will be surprised how easy it is to climb even the hills that you previously considered impossible.

However, there is of course a limit: you can climb just as slow as the lowest speed before toppling over. For me it's about 4.5 km/h. By inserting this minimal speed into the equation you can calculate whether you can climb the hill at all (without zigzagging).

There is another interesting thing that we can conclude from the above equations: that the transmission ratio doesn't really matter. If you could pedal at, say, 10 rpm, you may be able to climb in the big front ring, without the need for any more power than you are usually able to produce. Does it mean that you need only one ring in front? No derailleur, no shifter? It is well worth a try!


  1. Interesting post - when I started cycling to work again I tried to stay in as higher gear as possible at all times - but after a while my knees really started to ache. It would be great to pedal slowly but powerfully, but you'd need pretty bulletproof joints.

    Be great if anyone can do it, but even at 32 my joints are too shot to try it for any length of time!


  2. Hi Liam,

    I suggest a bit of exercise for the knees. Ordinary squats are good.

  3. you're missing the rotational energy part in the equation, i.e. angular momentum of the wheels, angular momentum of the 'pedals + feet/calf' attached ...


  4. Hi Tarzan!

    Thanks for your comment, but I don't think I'm missing anything. Angular momentum of the wheels, legs,etc are internal effects which decrease the efficiency of cycling, i.e. they increase the part Eb, which I already considered, and subsequently disregarded.

  5. I actually like the hill climb. I sometimes notice myself accelerating up hills, the bike meter sometimes reads at 15 - 20 km / hour uphill. The steeper the hill, the faster I try to go...

    Maybe I'm just masochistic.