Math? Physics? Those who enjoy brief exercises in applied speculation will certainly learn a thing or two from Rhett Allain, who took some time to consider bicycles and hills. At first blush, it seems an easy enough question: What’s the steepest gradient you could possibly ride on a road bike?
I think there are three reasons why a slope would be too steep. For all of these cases, I am going to assume that it is a prolonged slope. This means that you can’t just build up a large speed and zoom up the slope. If this was the case, you could go straight up a wall (which you can for a short time).
Those three reasons are the limitations of human power, center of mass, and friction. If one wishes to point out, “What if you used these tires instead of those?” or, “What if you had a different gear set?” it’s all well and fine to do so, but therein lies the point about the complexity of accounting for all the factors required.
Really, the friction problem might be worse than this. The bike only uses the back wheel for moving forward, so it is the friction on the back wheel that matters. If the biker is leaning forward, the weight distribution might not be even on the two wheels. I will leave this estimation (combining the previous two limits) as an exercise for the reader.
And, of course, one is welcome to pursue such endeavors. (In truth, that might be part of the point.)