The Gravitational Attraction between Hemispherical Masses

Abstract

This paper is a study of the gravitational attraction between two uniform hemispherical masses aligned such that the pair is cylindrically symmetric. Three variations are considered: flat side to flat side, curved side to curved side, and flat side to curved side. Expressions for the second and third variation are derived from the first, with the use of superposition and the well-known gravitational behavior of a spherical mass as equivalent to a point mass at its center. The study covers two masses of equal diameter and of different diameters, such that one is four times that of the other. Calculations are done for separations from zero to fifty times the radius of the larger of the two, which is effectively the asymptotic limit. It is demonstrated that at any separation, the force can be expressed as if the two hemispheres were point masses separated by a certain distance. Expressions for that distance and the location of the (fictitious) point masses within each hemisphere are presented. Unlike the case of two spherical masses, the location within their respective hemisphere is not necessarily the same for each point and both are dependent upon the separation between the two hemispheres. The calculation for the first variation is done in two ways. The first is a “brute force” multi-dimensional integral with the help of Wolfram Mathematica . The second is an axial expansion for the potential modified for off-axis locations by Legendre polynomials. With only a few terms in the expansion, the results of the second method are in extremely good agreement with those of the first. Finally, an interesting application to a split earth is presented.