Center of Gravity of Polygonal Regions and Conic Sections
Tilak de Alwis talwis@selu.edu
Department of Mathematics Southeastern Louisiana University United States
Abstract
In
this paper, we will consider the center of gravity, or the center of mass,
of several types of polygonal regions in the XYplane. In order to calculate
the center of gravity of such regions, instead of techniques from calculus,
one can conveniently use geometric ideas. Our regions are of variable
nature, so one can consider the locus of their center of gravity in the
XYplane. The locus of the center of gravity of one region we have considered
turns out to be a hyperbola with axes parallel to the coordinate axes.
This observation can be reversed to come up with a new definition for
a hyperbola without involving eccentricity. We will use this new definition
along with the dynamic geometry software Geometer's Sketchpad to present
a novel construction of a hyperbola. The locus of the center of gravity
of other polygonal regions we have considered yield some other interesting
curves as well. In addition to Geometer's Sketchpad, we have also used
the computer algebra system Mathematica to facilitate the center of gravity
calculations. The paper amply demonstrates the usage of different types
of software handinhand to experiment and conjecture mathematical results.
