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Physical Audio Signal Processing
    Elementary Physics, Mechanics, and Acoustics
       Rigid-Body Dynamics
          Center of Mass


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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{\bf 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(\un... ...p \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{\bf 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.


Subsections
Previous: Rigid-Body Dynamics
Next: Linear Momentum of the Center of Mass

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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