Coin spirals (such as the one here or here) are often claimed to mimic the physics of an orbiting planet, including by people who should know better. I vaguely remember being told this as an undergraduate as well. This is exactly half true.
- Since the funnel drops towards the center, the potential energy of a coin rolling on the surface decreases; since energy is conserved, kinetic energy must increase and the coin speeds up. This happens in planetary orbits as well, and – assuming the coin funnel shape is similar to the gravitational potential, i.e., proportional to 1/r2 – the coin funnel is a good representation of this part of celestial mechanics.
- In Newton’s law of gravitation, the force of gravity is purely central and therefore produces no torque, so angular momentum is conserved. This is not the case for the coin funnel, since frictional forces between coin and funnel do produce a torque on the coin; in fact, if you measure the angular momentum of a coin in one of the funnel videos above, you will find that it increases with time. This is a clear violation of conservation of angular momentum and is dramatically different from planetary motion. It’s not a huge effect but it is definitely measurable. I don’t see any way to fix this without making the funnel frictionless.