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Moment of Inertia Tensor
Consider a rigid body rotating with fixed angular velocity about
an axis which passes through the originsee Figure 28.
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 Equation (A.1309)
that

(457) 
Thus, the above equation specifies the velocity,
,
of each mass element as the body rotates with fixed angular velocity about
an axis passing through the origin.
Figure 28:
A rigid rotating body.

The total angular momentum of the body (about the origin) is written

(458) 
where use has been made of Equation (457), and some standard vector
identities (see Section A.10). The above formula can be written as a matrix equation
of the form

(459) 
where



(460) 



(461) 



(462) 



(463) 



(464) 



(465) 
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, etc. The matrix of the values is
known as the moment of inertia tensor.^{} Note that 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 (459) can be written more succinctly as

(466) 
Here, it is understood that and are
both column vectors, and is the matrix of the values.
Note that is a real symmetric matrix:
i.e.,
and
.
In general,
the angular momentum vector, , obtained from Equation (466),
points in a different direction to the angular velocity vector, . In other words,
is generally not parallel to .
Finally, although the above 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
20110331