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Mass of a thin shell, moments and center of mass.

Consider the curve $\mathcal{S}$ as a very thin shell; if we know at each point what is the density of the material used to build this shell, we can compute the mass, the moments of this object about the coordinate planes, the center of mass of the object, and so on. We denote by $\delta (x,y,z)$ the density at the point (x,y,z).

1.
Mass:

\begin{displaymath}M= \iint_{\mathcal{S}} \delta (x,y,z) \; ds.
\end{displaymath}

2.
First moments about the coordinate planes:
\begin{align*}M_{xy} & = \iint_{\mathcal{S}} z\delta (x,y,z) \; ds\\
M_{yz} & =...
...,y,z) \; ds\\
M_{zx} & = \iint_{\mathcal{S}} y\delta (x,y,z) \; ds
\end{align*}
3.
Coordinates of the center of mass:

\begin{displaymath}\overline{x}=\frac {M_{yz}}{M} \qquad ; \qquad \overline{y}=\frac {M_{xz}}{M}
\qquad ; \qquad \overline{z}=\frac {M_{xy}}{M}
\end{displaymath}

4.
Moments of inertia:
\begin{align*}I_x &= \iint_{\mathcal{S}} (y^2+z^2)\delta (x,y,z) \; ds\\
I_y &=...
...x,y,z) \; ds\\
I_L &= \iint_{\mathcal{S}} r^2 \delta (x,y,z) \; ds
\end{align*}
where L is any line in the space and r is the distance from the point (x,y,z) to the line L.
5.
Radius of gyration about the line L:

\begin{displaymath}R_L= \sqrt{\frac {I_L}{M}}.
\end{displaymath}

Example 8.32   Find the center of mass of a thin shell cut from the unit sphere by the coordinate planes so that the coordinates on $\mathcal{S}$ are non negative, and whose density is $\delta (x,y,z) = xyz$.


  
Figure 9: An 8th of the unit spehere.
\begin{figure}
\mbox{\epsfig{file=An8thunitSphere.eps,height=5cm} }
\end{figure}

1.
The mass: by the same way as in Example  8.27, we have:
\begin{align*}M & = \underset{\begin{matrix}x^2+y^2=1 \\ x \geq 0 \\ y \geq 0 \e...
...ac 14 \int_0^{\frac {\pi}{2}} \sin 2 \theta \; d \theta = \frac 14.
\end{align*}
2.
First moments:
  • about the xy-plane:
    \begin{align*}M_{xy} & = \iint_{\mathcal{S}} xyz^2 \; d \sigma\\
\quad &= \unde...
... \int_0^{\frac {\pi}{2}} \sin 2 \theta \; d \theta = \frac {2}{15}.
\end{align*}
  • about the yz-plane:
    \begin{align*}M_{yz} & = \iint_{\mathcal{S}} x^2yz \; d \sigma \\
\quad &= \und...
...\frac {1}{15} \cos^3 \theta \right] _0^{\frac {\pi}{2}} = \frac 13.
\end{align*}
  • about the xz-plane:
    \begin{align*}M_{zx} & = \iint_{\mathcal{S}} xy^2z \; d \sigma \\
\quad &= \und...
...\frac {1}{15} \sin^3 \theta \right] _0^{\frac {\pi}{2}} = \frac 13.
\end{align*}
    The fact that this last is equal to the previous one result could be foreseen, as the settings are totally symmetric in x and y.
3.
The coordinates of the center of mass:

\begin{displaymath}\overline{x}=\frac {M_{yz}}{M} = \frac {1/3}{1/4} = \frac 43 ...
...line{z}=\frac {M_{xy}}{M} = \frac {2/15}{1/4} = \frac {8}{15}.
\end{displaymath}


next up previous contents
Next: Orientation of a surface. Up: Surface integrals and Surface Previous: Special formulas for Surface
Noah Dana-Picard
2001-05-30