Center of Mass Technique and Affine Geometry

Abstract

The notion of center of mass, which is very useful in kinematics, proves to be very handy in geometry (see [1]-[2]). Countless applications of center of mass to geometry go back to Archimedes. Unfortunately, the center of mass cannot be defined for sets whose total mass equals zero. In the paper we improve this disadvantage and assign to an n-dimensional affine space L over any field k the (n+1)-dimensional vector space over the field k of weighty points and mass dipoles in L. In this space, the sum of weighted points with nonzero total mass is equal to the center of mass of these points equipped with their total mass. We present several interpretations of the space of weighty points and mass dipoles in L, and a couple of its applications to geometry. The paper is self-contained and is accessible for undergraduate students.

Figures (6)

• Figure 1: If the mass at $B$ is twice as big as the mass at $A$, then the center of mass $C$ will be two times closer to $B$ than to $A$
• Figure 2: If $AC=2BC$, then equilibrium holds if $m_B=-2m_A$
• Figure 3: Force proportional to $m_1+m_2$ must be placed at $C$ to keep the table in equilibrium
• Figure 4: Center of 3 unit masses and 3 medians
• Figure 5: The center of mass of a weighted set $\{(A,2),(B,-1),(C,1)\}$ is the point $O$, where $2AO=BC$

Theorems & Definitions (40)

• proof
• proof : Proof of Theorem \ref{['induction']}
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof