Bicycles and their Physics
For one of my final training rides, I loaded up a heavy pack and rolled up a nearby mountain. The climb was much harder than any previous attempt, but I also found myself descending more quickly than expected. Why was this?
It is shockingly difficult to explain how bicycles work. At the high level you turn cranks to turn chain to turn wheels, which sounds good enough. But left uncertain is how this two-wheeled contraption stays suspended in the air while in motion, even if the rider struggles to find balance on two feet, let alone ice skating. And uncertain is not an exaggeration.
Some intuitions are good though. For instance, a heavier bicycle is harder to move, especially on hills. This is why so much thought is given to frame material, and why manufacturers struggle to shave fractions of ounces off their components. International competitions regulate the minimum weight of a bike to provide a standard from which riders can prove individual fitness. If we as a society can agree on anything, it is that lighter bikes are faster bikes. Right?
This is where a little knowledge of physics can be a dangerous thing. Recall from high school the following four facts:
- Acceleration of a bicycle comes from your legs, hills (gravity), air resistance, and rolling resistance.
- Not accounting for air resistance, objects of any mass fall towards the earth with the same acceleration. This is because the mass terms for determining acceleration of an object and determining the force of gravity cancel: F = ma = mg, ergo a = g.
- Air resistance is related to the cross sectional area (widest horizontal slice) of the resisted object, not the mass of the object.
- Rolling resistance is some fraction of gravitational force, multiplied by a coefficient determined by the nature of contact with the road.
But wait a second! If this is the case, why does weight matter at all? When you are on a flat surface, the mass terms should cancel for rolling resistance. When you are on hills, the air resistance should not be different as long as your cross section is roughly the same. What gives?
There is a bit more to this story. Thinking about the flat terrain case first, we have the following equation:
Force = Legs + Rolling Resistance
Acceleration = Legs / Mass + Rolling Resistance / Mass
As rolling resistance is proportional to mass, mass will cancel, but it does not cancel with the force of your legs! The more mass you have (the heavier the bike) the harder it is going to be to accelerate. If we are going uphill, the results are substantially the same, just add a term to account for the force of gravity (from which mass will also cancel). In short, a heavier bike does mean you have to work harder, even if the mass term cancels from rolling resistance and gravity.
But what about the downhill case? Here we have to take into account the effects of gravity and air resistance (and take out the legs).
Force = Air Resistance + Rolling Resistance + Gravity
Acceleration = AR / Mass + RR / Mass + Gravity / Mass
Mass will cancel with rolling resistance and gravity as it does above, but it will not cancel with air resistance, causing this term to shrink as mass increases! This means that heavier bikes move faster when descending, even if their cross sections are the same (assuming you are not pedaling).
It is also interesting that a heavier bike is not just affecting the mass term, but also the rolling resistance coefficient. The weight from the vehicle places additional pressure on the tires, causing them to make more contact with the road. You can mitigate this by pumping the tubes to higher pressures, but past a certain point you risk stressing or popping the tube.
So in short, lighter bikes are easier to move through space but there are circumstances in which they will not descend as quickly as their heavier counterparts.
