# Balance a coin on a floating lemon

We assume the lemon is rigid, which is reasonably accurate for these small forces.

Stability in buoyancy requires a small rotation to create a net restoring torque. This is conceptualized as the metacenter, which is the "average" point the water pushes upward on. For *small* displacement angles this point remains fixed to the object. If the center of gravity is above the metacenter it's unstable. For the lemon, the metacenter is very close to the center since it's almost cylindrically symmetric. A coin raises the center of mass above the metacenter and makes the system unstable, regardless of exactly where it's positioned.

For a lemon on the table, the bumps and/or flat-regions act like a tiny tripod. As long as the center of mass stays above this "tripod" (above a point inside the triangle defined by it's three feet), it is stable. The center of mass depends on the position of the coin, so we can find a location that is stable for an arbitrarily small tripod.

As to the water case, it may be possible if the lemon is oddly shaped enough.

The real answer is a trick. Sorry

Take a **heavier** coin and squeeze it in sideways underneath, so it's now a lemon with the centre of gravity at the bottom, like the keel on a sailboat.

As long as the top coin is small and light, it should balance.