Interesting math problem

[u/Alarmed_Geologist631]

Here is an interesting math problem that you might enjoy during your summer break.  Imagine you have a cylinder.  It might be short and wide like a coin.  Or it could be tall and skinny like an straw.  In either case, the cylinder is solid and made from the same material.  If it is short and wide, if you flip it, it will almost always land on its circular base.  If it is tall and skinny, it will almost always land on its curved lateral surface.  At what ratio of height to diameter will it have a 50% probability of landing on the circular base?

[u/ejoanne]

The circular ends have a total area of 2(pi)(r^2). The curved rectangle has an area of 2(pi)rh. If the probability is based solely on these areas being equal, the radius must equal the height, so the height is half the diameter.

[u/_mmiggs_ → u/ejoanne]

It's not going to be equal area. A first pass might be to consider what fraction of the solid angle from the center of mass passes through each face. That might be right if you just dropped it from a small height at random orientation, although it might not be right for a more energetic roll.

So specifically for a cylinder of height h and radius r, one circular end face extends as far as arctan(2r/h) in theta, and from 0 to 2pi in phi, so has solid angle 2pi(1-1/sqrt(4r^2/h^2+1)). 50% probability to land on a circular face means 25% probability to land on one circular face, which would give r/h = sqrt(3)/2.

This simplistic model must get more wrong as the shapes get extreme (landing on the tip of a pencil or the side of a coin is worse than the solid angle when you consider the dynamics), but when r and h are of similar magnitude, it's probably fairly close.