Calculus Small Changes, Big Ideas

Section 15.5 Center of mass

• Mass in 2D (where \(\rho(x,y)\) is density):

  \begin{equation*} M = \iint_{D}\rho(x,y)\dd{A} \end{equation*}

  Moments in 2D:

  🔗
  \begin{align*} M_{x=0} \amp = \iint_{D} x\rho(x,y)\dd{A}\\ M_{y=0} \amp = \iint_{D} y\rho(x,y)\dd{A} \end{align*}
  Center of mass:

  \begin{equation*} (\bar{x},\bar{y})=\qty(\frac{M_{x=0}}{M},\frac{M_{y=0}}{M}) \end{equation*}

  🔗
  🔗
• Mass in 3D (where \(\rho(x,y,z)\) is density):

  \begin{equation*} M = \iiint_{D}\rho(x,y,z)\dd{V} \end{equation*}

  Moments in 3D:

  🔗
  \begin{align*} M_{x=0} \amp = \iiint_{D} x\rho(x,y,z)\dd{V}\\ M_{y=0} \amp = \iiint_{D} y\rho(x,y,z)\dd{V}\\ M_{z=0} \amp = \iiint_{D} z\rho(x,y,z)\dd{V} \end{align*}
  Center of mass:

  \begin{equation*} (\bar{x},\bar{y},\bar{z})=\qty(\frac{M_{x=0}}{M},\frac{M_{y=0}}{M},\frac{M_{z=0}}{M}) \end{equation*}

  🔗
  🔗
• If we assume uniform density, we get the centroid of the region.

  🔗
  🔗
• Start with density along a rod first?

  🔗
  🔗
• Moment of inertia:

  🔗
  \begin{align*} I_{x=0} \amp = \iint_{D} x^{2}\rho(x,y)\dd{A}\\ I_{y=0} \amp = \iint_{D} y^{2}\rho(x,y)\dd{A}\\ I_{z=0} \amp = \iint_{D} (x^{2}+y^{2})\rho(x,y)\dd{A} \end{align*}
  These will be done WITHOUT polar coordinates; I believe they can instead be done with symmetry arguments and trigonometric substitutions. (Never a bad idea to revisit those techniques!)

  🔗
  🔗