What are the conditions for stable equilibrium?
What it shows:
Place different rectangular solids (of various aspect ratios) on a cylinder to demonstrate which ones balance in stable equilibrium.
How it works:
If the thickness of the board (or whatever rectangular solid is used) is less than the diameter of the cylinder, it can be balanced in stable equilibrium—a light tap on the board will make it tip and oscillate, but not fall off. The width and length do not affect the equilibrium condition. If the thickness is greater than the diameter, it balances unstably (if you can manage to balance it at all).
Setting it up:
The cylinder should be supported in such a way that it will not roll. Provide a selection of boards and blocks whose thickness is either less than, equal to, or greater than the diameter of the cylinder. Also have various lengths available to show how that affects the frequency of oscillation, but not the ability to balance.
This is a realization of problem 8.4 in David Morin's book, Introduction to CLASSICAL MECHANICS with Problems and Solutions, (Cambridge University Press, 2007), p 333. Morin provides extensive and alternative solutions to the problem on pp 352-353.