# Balance a coin on a floating lemon

There is a challenge involving a lemon floating in a jug of water which seems impossible to beat. I've noticed it in several pubs of Edinburgh.

The challenge is as follows:

• There is a jug half filled with water.
• Floating in the water, there's a lemon. The lemon doesn't touch either the bottom nor the edges of the jug.
• The challenge is to successfully balance a coin on the lemon.
• Modifying, moving, or more generally touching the lemon are not allowed.

Any attempt to balance a coin on the lemon seems to result in the lemon flipping over, and the coin to sink in the water.

Why is it so hard to balance the coin while it's extremely easy to balance a coin on a lemon set on a table? How do you beat the lemon challenge?

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.

• When you are writing about a point I think you mean the metacentre. Metacentric height is the distance of the metacentre above the centre of gravity. Also, the average point through which the water pushes up is the centre of buoyancy, not the metacentre. Aug 20, 2016 at 18:19
• So even if the lemon and coin were perfectly shaped, and the coin was perfectly positioned above the center of buoyancy, the system would be unstable. But could it (theoretically) hold that position, i.e. not flip over? Aug 20, 2016 at 20:29
• @AurélienGasser: Yes theoretically it's an unstable equilibrium but still an equilibrium if everything is perfect. However, reality does not allow perfection so it will flip. Aug 21, 2016 at 3:54
• @sammygerbil fixed. Aug 21, 2016 at 3:55

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.

• Interesting, however in the challenge one is not allowed to touch the lemon. I edited the question to indicate it. Aug 20, 2016 at 0:20
• When you check on the web, there are various ideas about squashing the lemon to alter it's COG before its put in the water, but that doesn't seem to work. I, like you I guess,would prefer to see no cheating involved.
– user108787
Aug 20, 2016 at 11:27

I don't happen to have a lemon at the moment. But here's what I would try. I would try to place the coin so that it's position is as close as possible to the water plane. So then near either tip of the lemon rather than the center. My guess is the lemon is somewhat more stable in pitch than in roll, so it may not pitch up. And if the coin is closer to the water plane the disturbing torque in the roll axis may not be enough to turn the lemon.

• I tried it a hundred (OK, 20 really) times, the %@*&\$#@ lemon kept rolling over.
– user108787
Aug 20, 2016 at 11:29