Section Summary

In this section we worked out the formula for the center of mass \(\bar{x}\) of a system of point masses \(m_1,\,m_2,\,...,\,m_n\) located at points \(x_1,\,x_2,\,...,\,x_n\) along an \(x\)-axis:

We then used this to find the center of mass of a uniform tapered rod. If the rod lies along the \(x\)-axis between \(a\) and \(b\), the cross-sectional radius \(r(x)\) varies continuously with \(x\), and the material composing the rod has density \(\delta\), then the center of mass coordinate \(\bar{x}\) is

Here we may think of \(\pi [r(x)]^2\) as an infinitesimal volume located at the point with coordinate \(x\). When we multiply by \(\delta\), we obtain the infinitesimal mass \(dm\) given by \(\delta \pi [r(x)]^2 dx\). Under this identification, our formula for \(\bar{x}\) becomes

Note, however, that \(m\) is not our variable of integration. To use this formula for computation, we also need a formula for the infinitesimal mass.