Bike Physics 1: The Basics

This is the first part of a series of blog posts that explore the physics that govern cycling, striving for simplicity and clarity; here we explore the basics.

It must have been a few years ago when I found myself driving my car in the hills around Limassol, in moderate weather, windows rolled down, listening to some chatter on the radio. There were no other vehicles in sight. The road was twisty and rolling, when I saw two cyclists riding abreast talking to each other, negotiating gently the bendy road. It was no unique sight, but fairly uncommon still, to have people powering up a road and looking so serene, so at ease with what they were doing. I have been doing sports as an amatuer pretty much all my life (mainly playing basketball), but I thought that this was completely beyond me. I couldn’t see myself pedalling in the countryside and overcoming the gradient so effortlessly on a bike - it felt impossible. Up until then hills were a necessary evil, something you had to do to cover the distance, or enjoy the exhilaration of going down the hill on the other side. And looking at these cyclists - damn was I envious.

Fast forward a few years and not only did I tolerate, but I rather learned to enjoy this pain. I then decided to do away with the romanticism and think about the mechanics and physics of cycling, which boil down to a few simple rules: Rider and bike need to be light, evade wind as best as possible, and do it all efficiently, minimising all other losses. There’s not much more to it.

So there are basically three forces acting on a cyclist at any time: Gravity, aerodynamics and losses in various parts of the bike, mainly where the tyres meet the road.  These are laws of physics, and apply with equal measure to everyone, Tour de France racers and leisurely commuters alike. Let’s try and unpack one by one.

Gravity is of course the work that you have to put in to overcome the pull from planet Earth. The force that pulls you down is simply the combined mass of yourself, your bike and whatever you happen to be carrying, multiplied by the gravitation constant, ‘g’.

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Your mass (m) is measured in kilograms (kg), the SI unit; other units like stone or lbs can be used of course depending on what system one is accustomed to, but using standardised units does away with unnecessary conversions. ‘g’ is measured in m/s2, is equal roughly to 9.8 m/s2 and it’s constant for people riding on this planet.

Note that the above is a linear equation, which means that an increase in weight of say, 20%, (the only variable that can change, remember that ‘g’ is constant) will result in the increase of the pulling force by that same 20%. Simples.

A private battle with gravity.

A private battle with gravity.

Aerodynamics is a little trickier to calculate, not because it’s governed by some dark, arcane rules, but because there are more variables involved, and their relationship is not linear. Here’s the equation:

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So let’s take them one by one: ‘ρ’ is the air density, and measures how thick (or viscous, as is the technical term) the fluid - in this case air - one goes through is. It’s convenient to keep its value constant, but the truth is that it varies with altitude, ambient temperature and relative humidity: Hotter, more humid locations in thin mountain air decrease the value of ρ considerably. This property of aerodynamics is precisely the reason that indoor world record cycling attempts are set at stifling, sweltering conditions. At sea level and at 15 °C, air has a density of approximately 1.225 kg/m3.

Cd is the coefficient of drag. It is unique to every material object, and is a dimensionless number (as in it’s not measured in any units and it’s there only as a multiplier). Cd is rather less straightforward to calculate and it’s not naturally intuitive which shapes offer less Cd than others - testing is needed. No, there won’t be any testing done in this article, but for the moment bear in mind that a cyclist is an inherently un-aero object, due to the shape of the human body and the way one sits on a bike. Its value varies, but as a base assumption, Cd for a road cyclist should be between 0.70 and 1.3, a relatively wide range of values.

Aerodynamic efficiency will take you down this pretty fast

Aerodynamic efficiency will take you down this pretty fast

A is the frontal area, measured in m2. A measures the projected frontal area, i.e. how much space a cyclist + bike occupy when viewed from the front. Here larger riders have an inherent disadvantage, and this is why getting low is so important.

V is the air velocity that the rider experiences. For the purposes of simplicity we’ll assume here that there is no wind, and all the air apparent to the moving cyclist is caused by the movement of the bike only. What makes this term quirkier than the others is that in the above equation V is squared; hence the aerodynamic force acting against the cyclist accelerates quickly at higher speeds. V is measured in m/s (the SI unit).

Rolling resistance is the most significant loss of a moving bicycle, and represents the energy lost where the tyres meet the road. This in fact consists of two elements: The energy lost due to the bumpiness of the road, and the quality of your tyres and tubes. Here’s some more math:

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The fancy trigonometric term in the end is just there to calculate the normal force, which is the one perpendicular to the surface on which the bike is travelling. m is of course the combined mass of rider, bike and equipment, and g is the gravitation constant, the same ones discussed above.

So what is Crr? It’s another dimensionless term that summarises the efficiency of the tyre-road system. Remember that a bicycle is a relatively simple mechanical contraption that converts leg revolutions to forward motion, and all principles discussed here have been laid down ages ago; despite this, only recently has it become clear what's best for a road cyclist. Out is the idea that skinny, rock hard tyres are faster, in comes the notion that wider tyres mounted in wider rim beds produce quicker times. If you want to geek things out and are good in reading German, the Tour Magazin is a wealthy source of engineering tests, including rolling resistance comparisons of different tyre brands. A future piece will discuss these in more detail, including a discussion on hysteresis losses etc.

So to see what forces act against a cyclist is a simple matter of adding all the above together:

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It’s simple and elegant. In part II of this article series I’ll define what power is and relates to the above, and how the above equations relevant - perhaps also where we should expect to see gains by equipment upgrades. In part III I’ll examine a typical ride in Cyprus and investigate using a simple example how important a few changes to power output, equipment or riding style can be to our performance. Stay tuned.

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