Moment of Inertia

The moment of inertia with respect to a given axis of a solid body with density rho(r) is defined by the volume integral

where r__|_ is the perpendicular distance from the axis of rotation. This can be broken into components as

for a discrete distribution of mass, where r is the distance to a point (not the perpendicular distance) and delta_(jk) is the Kronecker delta, or

for a continuous mass distribution. Depending on the context, I may be viewed either as a tensor or a matrix. Expanding (3) in terms of Cartesian axes gives the equation

The moment of inertia of a region can be computed in the Wolfram Language using MomentOfInertia [reg].

The moment of inertia tensor I is symmetric, and is related to the angular momentum vector L by

where omega is the angular velocity vector.

The principal moments of inertia are given by the entries in the diagonalized moment of inertia matrix, and are denoted (for a solid) A, B, and C in order of decreasing magnitude. In the principal axes frame, the moments are also sometimes denoted I_(xx), I_(yy), and I_(zz). The principal axes of a rotating body are defined by finding values of I such that

which is an eigenvalue problem.

The following table summarizes the moments of inertia of some common solids around some of their principal axes.

See also

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References

Referenced on Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Moment of Inertia." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MomentofInertia.html

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