Mass Moment of Inertia

The mass moment of inertia (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 mass (§B.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 rotating a distance from the axis to be . Therefore, for a rigid collection of point-masses , ,^B.14 the moment of inertia about a given axis of rotation is obtained by adding the component moments of inertia:

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

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

where is the distance from the axis of rotation to the mass element . In terms of the density of a continuous mass distribution, we can write

where denotes the mass density (kg/m) at the point , and denotes a differential volume element located at .

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Subsections

• Circular Disk Rotating in Its Own Plane
• Circular Disk Rotating About Its Diameter

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Next: Circular Disk Rotating in Its Own Plane

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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.