Capillarity and Archimedes’ Principle of Floatation

We consider some of the complications that arise in attempting to generalize a version of Archimedes’ principle of floatation to account for capillary effects. The main result provides a means to relate the floating position (depth in the liquid) of a symmetrically floating sphere in terms of readily observable geometric quantities. A similar result is obtained for an idealized case corresponding to a symmetrically floating infinite cylinder. Certain possibilities are also outlined in the event symmetry is relaxed in this latter problem. Central to all of these results is a specialized variational formula for floating bodies which was derived in a special case earlier [Pac. J. Math. 231 (2007) pp.167–191] and is here generalized to account for gravitational forces.