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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

$\displaystyle \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

$\displaystyle \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

$\displaystyle 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.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2021-11-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University