Towards a Completion of Archimedes' Treatise on Floating Bodies

TL;DR

This paper extends Archimedes' work on floating bodies by providing a mathematical model for equilibrium positions of a paraboloid segment when the basis is partially submerged, offering tools for finding and classifying all equilibria.

Contribution

It introduces a new mathematical model and methods to analyze equilibrium positions of floating paraboloid segments in previously unaddressed cases.

Findings

Closed-form conditions for equilibrium positions

Tools for reliable equilibrium detection

Classification framework for equilibria

Abstract

In his treatise on floating bodies Archimedes determines the equilibrium positions of a floating paraboloid segment, but only in the case when the basis of the segment is either completely outside of the fluid or completely submerged. Here we give a mathematical model for the remaining case, i.e., two simple conditions which describe the equilibria in closed form. We provide tools for finding all equilibria in a reliable way and for the classification of these equilibria. This paper can be considered as a continuation of a recent article of Rorres.