The Stability of Boats, Triangular Configurations: The cos~4 Rule

摘要

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. Let the measure of the apex angle of the triangle be θ, the triangle's two equal legs have length e, and the density be ρ. Then the area of the cross section is A_0 = 1/2e~2 sin θ. For a boat of length ℓ, the volume is V_0 = A_0ℓ and its weight is ρV_0. We measure the density of the boat as a fraction of the (unit) density of the fluid in which the boat is to float, hence ρ ＜ 1. (If the density of the boat is greater than that of the fluid, the boat will surely-"stably"!-sink.) For convenience, we casually refer to water as the medium, even as the mathematics allows other fluids and their densities.