Next: Rotational kinetic energy
Up: Rigid body rotation
Previous: Fundamental equations
Moment of inertia tensor
Consider a rigid body rotating with fixed angular velocity
about
an axis that passes through the origin. (See Figure 8.1.)
Let
be the position vector of the
th mass element, whose mass
is
.
We expect this position vector to precess about the axis of rotation
(which is parallel to
)
with angular velocity
. It, therefore, follows from Section A.7
that
Thus, Equation (8.4) specifies the velocity,
,
of each mass element as the body rotates with fixed angular velocity
about
an axis passing through the origin.
Figure 8.1:
A rigid rotating body.

The total angular momentum of the body (about the origin) is written
where use has been made of Equation (8.4), and some standard vector
identities. (See Section A.4.) The preceding formula can be written as a matrix equation
of the form

(8.6) 
where
Here,
is called the moment of inertia about the
axis,
the moment of inertia about the
axis,
the
product of inertia,
the
product of
inertia, and so on. The matrix of the
values is
known as the moment of inertia tensor. Each component
of the moment of inertia tensor can be written as either a sum over separate
mass elements, or as an integral over infinitesimal mass elements.
In the integrals,
, where
is the mass density, and
a volume element.
Equation (8.6) can be written more succinctly as
Here, it is understood that
and
are
both column vectors, and
is the matrix of the
values.
Note that
is a real symmetric matrix; that is,
and
.
In general,
the angular momentum vector,
, obtained from Equation (8.13),
points in a different direction to the angular velocity vector,
. In other words,
is generally not parallel to
.
Finally, although the preceding results were obtained assuming a
fixed angular velocity they remain valid, at each instant in time, if the angular velocity varies.
Next: Rotational kinetic energy
Up: Rigid body rotation
Previous: Fundamental equations
Richard Fitzpatrick
20160331