The Stability of Boats, Triangular Configurations: The cos4 Rule

In Rublein [2015], we considered the stability of long boats with a square cross section. Here we study a long solid boat of uniform weight density whose cross section has an isosceles triangular shape.

We are interested in geometric configurations in which such a boat will float stably. Stability begins with equilibrium. According to Archimedes:

A floating body is in an equilibrium configuration if its center of gravity and the center of gravity of the fluid displaced by it lie on a vertical line.

The latter center of gravity is known as the center of buoyancy. Of course, the center of buoyancy is a phantom quantity. Nevertheless, it can be shown with the aid of surface integrals that the net upward force of the (ideal, incompressible) fluid pressure on the immersed hull of the boat produces the mechanical equivalent of a single upward force acting at precisely the center of gravity of the displaced fluid.