Summary of Calculating Centers of Mass and Moments of Inertia

Essential Concepts

  • Finding the mass, center of mass, moments, and moments of inertia in double integrals:
    • For a lamina [latex]R[/latex] with a density function [latex]\rho(x,y)[/latex] at any point [latex](x,y)[/latex] in the plane, the mass is [latex]m=\underset{R}{\displaystyle\iint} \rho(x,y) dA[/latex]
    • The moments about the [latex]x[/latex]-axis and [latex]y[/latex]-axis are [latex]M_{x}=\underset{R}{\displaystyle\iint} y\rho(x,y) dA[/latex] and [latex]M_{y}=\underset{R}{\displaystyle\iint} x\rho(x,y) dA[/latex]
    • The center of mass is given by [latex]\overline{x}=\frac{M_{y}}{m}[/latex], [latex]\overline{y}=\frac{M_{x}}{m}[/latex]
    • The center of mass becomes the centroid of the plane when the density is constant.
    • The moments of inertia about the [latex]x[/latex]-axis, [latex]y[/latex]-axis, and the origin are [latex]I_{x}=\displaystyle\iint_{R} y^{2}\rho(x,y) dA[/latex], [latex]I_{y}=\underset{R}{\displaystyle\iint} x^{2}\rho(x,y) dA[/latex], and [latex]I_{0}=I_{x}+I_{y}=\underset{R}{\displaystyle\iint} (x^{2}+y^{2})\rho(x,y) dA[/latex]
  • Finding the mass, center of mass, moments, and moments of inertia in triple integrals:
    • For a solid object [latex]Q[/latex] with a density function [latex]\rho(x,y,z)[/latex] at any point [latex](x,y,z)[/latex] in space, the mass is [latex]m=\displaystyle\iiint_{Q} \rho(x,y,z) dV[/latex]
    • The moments about the [latex]xy[/latex]-plane, the [latex]xz[/latex]-plane, and the [latex]yz[/latex]-plane are [latex]M_{xy}=\displaystyle\iiint_{Q} z\rho(x,y,z) dV[/latex], [latex]M_{xz}=\underset{Q}{\displaystyle\iiint} y\rho(x,y,z) dV[/latex], [latex]M_{yz}=\underset{Q}{\displaystyle\iiint} x\rho(x,y,z) dV[/latex]
    • The center of mass is given by [latex]\overline{x}=\frac{M_{yz}}{m}[/latex], [latex]\overline{x}=\frac{M_{xz}}{m}[/latex], [latex]\overline{x}=\frac{M_{xy}}{m}[/latex]
    • The center of mass becomes the centroid of the solid when the density is constant.
    • The moments of inertia about the [latex]yz[/latex]-plane, the [latex]xz[/latex]-plane, and the [latex]xy[/latex]-plane are [latex]I_{x}=\underset{Q}{\displaystyle\iiint} (y^{2}+z^{2})\rho(x,y,z) dV[/latex], [latex]I_{y}=\underset{Q}{\displaystyle\iiint} (x^{2}+z^{2})\rho(x,y,z) dV[/latex], [latex]I_{z}=\underset{Q}{\displaystyle\iiint} (x^{2}+y^{2})\rho(x,y,z) dV[/latex]

Key Equations

  • Mass of a lamina
    [latex]m=\underset{k,l\to\infty}{\lim}\displaystyle\sum_{i=1}^{k}\displaystyle\sum_{j=1}^{l}m_{ij}=\underset{k,l\to\infty}{\lim}\displaystyle\sum_{i=1}^{k}\displaystyle\sum_{j=1}^{l}\rho(x_{ij}^{\ast},y_{ij}^{\ast})\Delta{A}=\underset{R}{\displaystyle\iint} \rho(x,y) dA[/latex]
  • Moment about the [latex]x[/latex]-axis
    [latex]M_{x}=\underset{k,l\to\infty}{\lim}\displaystyle\sum_{i=1}^{k}\displaystyle\sum_{j=1}^{l}(y_{ij}^{\ast})m_{ij}=\underset{k,l\to\infty}{\lim}\displaystyle\sum_{i=1}^{k}\displaystyle\sum_{j=1}^{l}(y_{ij}^{\ast})\rho(x_{ij}^{\ast},y_{ij}^{\ast})\Delta{A}=\underset{R}{\displaystyle\iint} y\rho(x,y) dA[/latex]
  • Moment about the [latex]y[/latex]-axis
    [latex]M_{y}=\underset{k,l\to\infty}{\lim}\displaystyle\sum_{i=1}^{k}\displaystyle\sum_{j=1}^{l}(x_{ij}^{\ast})m_{ij}=\underset{k,l\to\infty}{\lim}\displaystyle\sum_{i=1}^{k}\displaystyle\sum_{j=1}^{l}(x_{ij}^{\ast})\rho(x_{ij}^{\ast},y_{ij}^{\ast})\Delta{A}=\underset{R}{\displaystyle\iint} x\rho(x,y) dA[/latex]
  • Center of mass of a lamina
    [latex]\overline{x}=\frac{M_{y}}{m}=\dfrac{\underset{R}{\displaystyle\iint} x\rho(x,y) dA}{\underset{R}{\displaystyle\iint} \rho(x,y) dA}[/latex] and [latex]\overline{y}=\frac{M_{x}}{m}=\dfrac{\underset{R}{\displaystyle\iint} y\rho(x,y) dA}{\underset{R}{\displaystyle\iint} \rho(x,y) dA}[/latex]

Glossary

radius of gyration
the distance from an object’s center of mass to its axis of rotation