Mass and Center of Mass

Integrals in curvilinear coordinates are just like any other integrals: chop and add. The most important difference is that you chop along curves for which one of the coordinates is constant; these curves are no longer straight lines. It is useful to know how to convert between various coordinate systems, but in practice one often constructs the integral directly in the appropriate coordinate system. Round problems should be done in round coordinates!

The area element is \begin{eqnarray} dA &=& dx\,dy \\ &=& r\,dr\,d\phi \end{eqnarray} in rectangular and polar coordinates, respectively, and the volume element is \begin{eqnarray} dV &=& dx\,dy\,dz \\ &=& r\,dr\,d\phi\,dz \\ &=& r^2\,\sin\theta\,dr\,d\theta\,d\phi \end{eqnarray} in rectangular, cylindrical, and spherical coordinates, respectively.

Thus, the volume of a region $R$ is given by \begin{equation} V = \int\limits_R dV \end{equation} and it only remains to choose the appropriate form of $dV$, then express this triple integral as three iterated single integrals, choosing limits based on the order of integration. It is a good idea to sketch the region, and to explicitly indicate on the sketch which way you are chopping the region; the limits can then be read off from the sketch.

A simple application is to find the mass of such a region, given the (volume) mass density $\rho$, which is given by \begin{equation} m = \int\limits_R \rho\,dV \label{massvol} \end{equation} Think of this as adding up the mass $\rho\,dV$ of small regions. The process is exactly the same as before, except that $\rho$ could be a function. The symmetry of $\rho$ often suggests the best choice of coordinates to use.

The mass of a 2-dimensional region is similar, except that it is customary to use $\sigma$ for the (surface) mass density, resulting in \begin{equation} m = \int\limits_R \sigma\,dA \end{equation}

Averages can also be computed using integrals. For example, the average mass density would be simply the total mass divided by the total volume (or area), and is thus a fraction whose numerator and denominator are both integrals.

The center of mass of an object is defined as the weighted average (pun intended!) of the coordinates. In three dimensions, the coordinates of the center of mass are typically denoted by ($\bar{x}$,$\bar{y}$,$\bar{z}$), where for example \begin{equation} \bar{x} = \frac{1}{m} \int\limits_R x \rho \,dV \end{equation} and where the mass $m$ would of course also be computed by integration, using (\ref{massvol}).

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