5.6 Calculating centers of mass and moments of inertia  (Page 5/8)

We conclude this section with an example of finding moments of inertia $I_x,I_y,$ and $I_z.$

Key concepts

Finding the mass, center of mass, moments, and moments of inertia in double integrals:

• For a lamina $R$ with a density function $\rho \left(x,y\right)$ at any point $\left(x,y\right)$ in the plane, the mass is $m={\displaystyle \underset{R}{\iint }\rho \left(x,y\right)dA}.$
• The moments about the $x\text{-axis}$ and $y\text{-axis}$ are
  
  $M_x={\displaystyle \underset{R}{\iint }y\rho \left(x,y\right)}dA\text{and}{M}_y={\displaystyle \underset{R}{\iint }x\rho \left(x,y\right)}dA.$
• The center of mass is given by $\stackrel{\text{−}}{x}=\frac{M_y}{m},\stackrel{\text{−}}{y}=\frac{M_x}{m}.$
• The center of mass becomes the centroid of the plane when the density is constant.
• The moments of inertia about the $x-\text{axis,}$ $y-\text{axis,}$ and the origin are
  
  $I_x={\displaystyle \underset{R}{\iint }{y}^2\rho \left(x,y\right)dA,}{I}_y={\displaystyle \underset{R}{\iint }{x}^2\rho \left(x,y\right)dA,}\text{and}{I}_0=I_x+I_y={\displaystyle \underset{R}{\iint }\left(x^2+y^2\right)\rho \left(x,y\right)dA.}$

Finding the mass, center of mass, moments, and moments of inertia in triple integrals:

• For a solid object $Q$ with a density function $\rho \left(x,y,z\right)$ at any point $\left(x,y,z\right)$ in space, the mass is $m={\displaystyle \underset{Q}{\iiint }\rho \left(x,y,z\right)dV}.$
• The moments about the $xy\text{-plane,}$ the $xz\text{-plane,}$ and the $yz\text{-plane}$ are
  
  $M_{xy}={\displaystyle \underset{Q}{\iiint }z\rho \left(x,y,z\right)dV},M_{xz}={\displaystyle \underset{Q}{\iiint }y\rho \left(x,y,z\right)dV},M_{yz}={\displaystyle \underset{Q}{\iiint }x\rho \left(x,y,z\right)dV}.$
• The center of mass is given by $\stackrel{\text{−}}{x}=\frac{M_{yz}}{m},\stackrel{\text{−}}{y}=\frac{M_{xz}}{m},\stackrel{\text{−}}{z}=\frac{M_{xy}}{m}.$
• The center of mass becomes the centroid of the solid when the density is constant.
• The moments of inertia about the $yz\text{-plane,}$ the $xz\text{-plane,}$ and the $xy\text{-plane}$ are
  
  $\begin{array}{}\\ I_x={\displaystyle \underset{Q}{\iiint }\left(y^2+z^2\right)\rho \left(x,y,z\right)dV,}{I}_y={\displaystyle \underset{Q}{\iiint }\left(x^2+z^2\right)\rho \left(x,y,z\right)dV,}\hfill \\ I_z={\displaystyle \underset{Q}{\iiint }\left(x^2+y^2\right)\rho \left(x,y,z\right)dV.}\hfill \end{array}$

Key equations

• Mass of a lamina
  $m=\underset{k,l\to \infty }{\text{lim}}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^l{m}_{ij}}}=\underset{k,l\to \infty }{\text{lim}}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^l\rho (x_{ij}^{*},y_{ij}^{*})\text{Δ}A={\displaystyle \underset{R}{\iint }\rho (x,y)dA}}}$
• Moment about the x -axis
  $M_x=\underset{k,l\to \infty }{\text{lim}}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^l\left(y_{ij}^{*}\right)m_{ij}}}=\underset{k,l\to \infty }{\text{lim}}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^l\left(y_{ij}^{*}\right)\rho (x_{ij}^{*},y_{ij}^{*})\text{Δ}A={\displaystyle \underset{R}{\iint }y\rho (x,y)dA}}}$
• Moment about the y -axis
  $M_y=\underset{k,l\to \infty }{\text{lim}}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^l\left(x_{ij}^{*}\right)m_{ij}}}=\underset{k,l\to \infty }{\text{lim}}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^l\left(x_{ij}^{*}\right)\rho (x_{ij}^{*},y_{ij}^{*})\text{Δ}A={\displaystyle \underset{R}{\iint }x\rho (x,y)dA}}}$
• Center of mass of a lamina
  $\stackrel{\text{−}}{x}=\frac{M_y}{m}=\frac{{\displaystyle \underset{R}{\iint }x\rho \left(x,y\right)dA}}{{\displaystyle \underset{R}{\iint }\rho \left(x,y\right)dA}}$ and $\stackrel{\text{−}}{y}=\frac{M_x}{m}=\frac{{\displaystyle \underset{R}{\iint }y\rho \left(x,y\right)dA}}{{\displaystyle \underset{R}{\iint }\rho \left(x,y\right)dA}}$

In the following exercises, the region $R$ occupied by a lamina is shown in a graph. Find the mass of $R$ with the density function $\rho .$