The moment of inertia with respect to a given axis of a solid body with density is defined by the volume integral
(1)
|
where
is the perpendicular distance from the axis of rotation. This can be broken into
components as
(2)
|
for a discrete distribution of mass, where is the distance to a point (not the perpendicular distance)
and
is the Kronecker delta, or
(3)
|
for a continuous mass distribution. Depending on the context, may be viewed either as a tensor or a matrix. Expanding (3) in terms of Cartesian axes gives the equation
(4)
|
The moment of inertia of a region can be computed in the Wolfram Language using MomentOfInertia[reg].
The moment of inertia tensor is symmetric, and is related to the angular momentum vector
by
(5)
|
where
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) ,
, and
in order of decreasing magnitude. In the principal axes frame,
the moments are also sometimes denoted
,
, and
. The principal axes of a rotating body are defined by
finding values of
such that
(6)
|
(7)
|
which is an eigenvalue problem.
The following table summarizes the moments of inertia of some common solids around some of their principal axes.
solid/axis | moment of inertia |
cylinder about symmetry axis | |
cylinder about central diameter | |
ellipsoid about principal axis | |
elliptical slab about major axis | |
elliptical slab about vertical | |
cuboid about major axis | |
ring about perpendicular axis | |
ring about diameter | |
rod about end | |
rod about center | |
sphere about diameter | |
spherical shell | |
torus about diameter | |
torus about symmetry axis |