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Mass Moment of Inertia

The mass moment of inertia $ I$ (or simply moment of inertia), plays the role of mass in rotational dynamics, as we saw in Eq.$ \,$ (B.7) above.

The mass moment of inertia of a rigid body, relative to a given axis of rotation, is given by a weighted sum over its mass, with each mass-point weighted by the square of its distance from the rotation axis. Compare this with the center of massB.4.1) in which each mass-point is weighted by its vector location in space (and divided by the total mass).

Equation (B.8) above gives the moment of inertia for a single point-mass $ m$ rotating a distance $ R$ from the axis to be $ I=mR^2$ . Therefore, for a rigid collection of point-masses $ m_i$ , $ i=1,\ldots,N$ ,B.14 the moment of inertia about a given axis of rotation is obtained by adding the component moments of inertia:

$\displaystyle I = m_1 R_1^2 + m_2 R_2^2 + \cdots + m_N R_N^2, \protect$ (B.9)

where $ R_i$ is the distance from the axis of rotation to the $ i$ th mass.

For a continuous mass distribution, the moment of inertia is given by integrating the contribution of each differential mass element:

$\displaystyle I \eqsp \int_M R^2 dm,$ (B.10)

where $ R$ is the distance from the axis of rotation to the mass element $ dm$ . In terms of the density $ \rho(\underline{x})$ of a continuous mass distribution, we can write

$\displaystyle I \,\mathrel{\mathop=}\,\int_V R^2(\underline{x})\,\rho(\underline{x})\,dV,
$

where $ \rho(\underline{x})$ denotes the mass density (kg/m$ \null^3$ ) at the point $ \underline{x}$ , and $ dV=dx\,dy\,dz$ denotes a differential volume element located at $ \underline{x}\in{\bf R}^3$ .



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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-06-11 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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