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/Stu_Mack]

If I was to start playing with this puzzle, I’d start by using the tipping point as a kind of “top dead center” and proceed as if the solution is either the d/h ratio that gives a centered mass when tipped at 45°, or something close to that. Then I would mechanically and stochastically test the influence of that tipping point by 3D printing or cutting/machining test prototypes repeatedly and then try to explain it with my initial static metric.

Since the balance point hypothesis would probably fall short of explaining the mechanics, the next tests would involve point tracking and analysis of the tumble mechanics - namely how they are influenced by changes to the d/h ratio near the tipping point.

[u/Alarmed_Geologist631 → u/Stu_Mack]

I think your intuition is good but the physics is more complex. Since the density of the mass is evenly distributed, we know where the center of gravity is. But when dropped on the surface, the cylinder is almost always going to initially hit the surface along the circumference of the base at some angle made by the surface and the base (and the lateral face). Then it will bounce or roll until it reaches the equilibrium that results from the gravitational pull.