Summary of Moments and Centers of Mass

Essential Concepts

  • Mathematically, the center of mass of a system is the point at which the total mass of the system could be concentrated without changing the moment. Loosely speaking, the center of mass can be thought of as the balancing point of the system.
  • For point masses distributed along a number line, the moment of the system with respect to the origin is [latex]M=\displaystyle\sum_{i=1}^{n} {m}_{i}{x}_{i}.[/latex] For point masses distributed in a plane, the moments of the system with respect to the [latex]x[/latex]– and [latex]y[/latex]-axes, respectively, are [latex]{M}_{x}=\displaystyle\sum_{i=1}{n} {m}_{i}{y}_{i}[/latex] and [latex]{M}_{y}=\displaystyle\sum_{i=1}{n} {m}_{i}{x}_{i},[/latex] respectively.
  • For a lamina bounded above by a function [latex]f(x),[/latex] the moments of the system with respect to the [latex]x[/latex]– and [latex]y[/latex]-axes, respectively, are [latex]{M}_{x}=\rho {\displaystyle\int }_{a}^{b}\frac{{\left[f(x)\right]}^{2}}{2}dx[/latex] and [latex]{M}_{y}=\rho {\displaystyle\int }_{a}^{b}xf(x)dx.[/latex]
  • The [latex]x[/latex]– and [latex]y[/latex]-coordinates of the center of mass can be found by dividing the moments around the [latex]y[/latex]-axis and around the [latex]x[/latex]-axis, respectively, by the total mass. The symmetry principle says that if a region is symmetric with respect to a line, then the centroid of the region lies on the line.
  • The theorem of Pappus for volume says that if a region is revolved around an external axis, the volume of the resulting solid is equal to the area of the region multiplied by the distance traveled by the centroid of the region.

Key Equations

  • Mass of a lamina
    [latex]m=\rho {\displaystyle\int }_{a}^{b}f(x)dx[/latex]
  • Moments of a lamina
    [latex]{M}_{x}=\rho {\displaystyle\int }_{a}^{b}\frac{{\left[f(x)\right]}^{2}}{2}dx\text{ and }{M}_{y}=\rho {\displaystyle\int }_{a}^{b}xf(x)dx[/latex]
  • Center of mass of a lamina
    [latex]\overline{x}=\frac{{M}_{y}}{m}\text{ and }\overline{y}=\frac{{M}_{x}}{m}[/latex]

Glossary

center of mass
the point at which the total mass of the system could be concentrated without changing the moment
centroid
the centroid of a region is the geometric center of the region; laminas are often represented by regions in the plane; if the lamina has a constant density, the center of mass of the lamina depends only on the shape of the corresponding planar region; in this case, the center of mass of the lamina corresponds to the centroid of the representative region
lamina
a thin sheet of material; laminas are thin enough that, for mathematical purposes, they can be treated as if they are two-dimensional
moment
if [latex]n[/latex] masses are arranged on a number line, the moment of the system with respect to the origin is given by [latex]M=\displaystyle\sum_{i=1}{n} {m}_{i}{x}_{i};[/latex] if, instead, we consider a region in the plane, bounded above by a function [latex]f(x)[/latex] over an interval [latex]\left[a,b\right],[/latex] then the moments of the region with respect to the [latex]x[/latex]– and [latex]y[/latex]-axes are given by [latex]{M}_{x}=\rho {\displaystyle\int }_{a}^{b}\frac{{\left[f(x)\right]}^{2}}{2}dx[/latex] and [latex]{M}_{y}=\rho {\displaystyle\int }_{a}^{b}xf(x)dx,[/latex] respectively
symmetry principle
the symmetry principle states that if a region R is symmetric about a line [latex]l[/latex], then the centroid of R lies on [latex]l[/latex]
theorem of Pappus for volume
this theorem states that the volume of a solid of revolution formed by revolving a region around an external axis is equal to the area of the region multiplied by the distance traveled by the centroid of the region