Center of Mass

The center of mass (or centroid) of a rigid body is found by averaging the spatial points of the body $\underline{x}_i\in\mathbb{R}^3$ weighted by the mass $m_i$ of those points:^B.12

$$\underline{x}_c \isdefs \left. \sum_{i=1}^N m_i \underline{x}_i \right/ \sum_{i=1}^N m_i$$

Thus, the center of mass is the mass-weighted average location of the object. For a continuous mass distribution totaling up to $M$ , we can write

$$\underline{x}_c \isdefs \frac{1}{M}\int_V \underline{x}\, dm(\underline{x}) \eqsp \int_V \underline{x}\, \rho(\underline{x})\, dV \eqsp \iiint_{\underline{x}\in V} \underline{x}\, \rho(\underline{x})\, dx\,dy\,dz$$

where the volume integral is taken over a volume $V$ of 3D space that includes the rigid body, and $dm(\underline{x}) = m(\underline{x})dV = m(\underline{x})\, dx\,dy\,dz$ denotes the mass contained within the differential volume element $dV$ located at the point $\underline{x}\in\mathbb{R}^3$ , with $\rho(\underline{x})$ denoting the mass density at the point $\underline {x}$ . The total mass is

$$M \eqsp \int_V dm(\underline{x}) \eqsp \int_V \rho(\underline{x})\, dV.$$

A nice property of the center of mass is that gravity acts on a far-away object as if all its mass were concentrated at its center of mass. For this reason, the center of mass is often called the center of gravity.