Archimedian Theorems for Composite Solids

Abstract: We consider the Center of Gravity of a solid, partly filled with some homogeneous material, and find its qualitative and quantitative properties. In particular, we prove that the Center of Gravity has its lowest position when it lies on the top surface of the material inside the solid and find a differential equation for the first moments that explains this result in both mathematical and physics terms. We make explicit calculations of this lowest position in a number of cases, such as cylinders, cones, solids of revolution, power solids, spheres and half spheres.