Center of Mass in Point Configurations: Explorations with GeoGebra

Abstract

The localization of the center of mass of a finite set of point masses can be achieved by minimizing a polynomial function of degree 2 using the least squares method. Considering a finite set of point masses in the plane, this article explores, in some particular situations, the effect that changes in the configuration of the points cause both on the center of mass of the new configuration and on the minimum value of the polynomial function associated with that configuration. In each case, the mathematical details are presented with illustrations, interactive links from GeoGebra, and dynamic color elements, inserted to emphasize visualization and illustrate the relationship of the center of mass as the minimum of a two-variable function using contour plots combined with color scales. The results encompass the visualization of the particle system with randomness in the positions and masses of the particles, the analysis of some cases of displacements of the center of mass and their associated functions, as well as the influence of the masses on the minimum values of these functions. In general, the visualization of mathematical elements, facilitated by the use of GeoGebra, highlights details and fundamental aspects of the problems interactively, allowing observation and analysis.