On the buoyancy force and the metacentre

TL;DR

This paper investigates the point of application of buoyancy force and the metacentre in floating bodies, providing analytical expressions and exploring their dependence on motion type, shape, and stability considerations.

Contribution

It introduces a unified analytical approach to determine the metacentre and buoyancy point for various shapes and motions, clarifying their relationships and implications for stability.

Findings

Location of the buoyancy point varies with motion type.

Analytical expressions for the metacentre are derived for general shapes.

The height of the metacentre relates to the stability of the floating body.

Abstract

We address the point of application A of the buoyancy force (also known as the Archimedes force) by using two different definitions of the point of application of a force, derived one from the work-energy relation and another one from the equation of motion. We present a quantitative approach to this issue based on the concept of the hydrostatic energy, considered for a general shape of the immersed cross-section of the floating body. We show that the location of A depends on the type of motion experienced by the body. In particular, in vertical translation, from the work-energy viewpoint, this point is fixed with respect to the centre of gravity G of the body. In contrast, in rolling/pitching motion there is duality in the location of A ; indeed, the work-energy relation implies A to be fixed with respect to the centre of buoyancy C, while from considerations involving the rotational moment it follows that A is located at the metacentre M. We obtain analytical expressions of the location of M for a general shape of the immersed cross-section of the floating body and for an arbitrary angle of heel. We show that three different definitions of M viz., the ?geometrical? one, as the centre of curvature of the buoyancy curve, the Bouguer's one, involving the moment of inertia of the plane of flotation, and the ?dynamical? one, involving the second derivative of the hydrostatic energy, refer to one and the same special point, and we demonstrate a close relation between the height of M above the line of flotation and the stability of the floating body. Finally, we provide analytical expressions and graphs of the buoyancy, flotation and metacentric curves as functions of the angle of heel, for some particular shapes of the floating bodies.