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As derived in the previous section, the moment of inertia
tensor, in 3D Cartesian coordinates, is a three-by-three matrix
that can be multiplied by any angular-velocity vector to
produce the corresponding angular momentum vector for either a point
mass or a rigid mass distribution. Note that the origin of the
angular-velocity vector
is always fixed at
in the space
(typically located at the center of mass). Therefore, the moment of
inertia tensor
is defined relative to that origin.
The moment of inertia tensor can similarly be used to compute the
mass moment of inertia for any normalized angular velocity
vector
as
 |
(B.22) |
Since rotational energy is defined as
(see
Eq.(B.7)), multiplying Eq.(B.22) by 
gives the following
expression for the rotational kinetic energy in terms of the moment of
inertia tensor:
 |
(B.23) |
We can show Eq.(B.22) starting from Eq.(B.14). For a
point-mass 
located at
, we have
where again
denotes the three-by-three identity matrix, and
 |
(B.24) |
which agrees with Eq.(B.20). Thus we have derived the moment of
inertia 
in terms of the moment of inertia tensor
and the
normalized angular velocity
for a point-mass
at
.
For a collection of 
masses 
located at
, we
simply sum over their masses to add up the moments of inertia:
Finally, for a continuous mass distribution, we integrate as usual:
where
is the total mass.
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![\begin{eqnarray*}
I &=& m \left\Vert\,\underline{x}-(\underline{\tilde{\omega}}^T\underline{x})\underline{\tilde{\omega}}\,\right\Vert^2\\ [5pt]
&=& m \left[\underline{x}^T-(\underline{\tilde{\omega}}^T\underline{x})\underline{\tilde{\omega}}^T\right]
\left[\underline{x}-(\underline{\tilde{\omega}}^T\underline{x})\underline{\tilde{\omega}}\right]\\ [5pt]
&=& m \left[\underline{x}^T\underline{x}- \underline{x}^T(\underline{\tilde{\omega}}^T\underline{x})\underline{\tilde{\omega}}-(\underline{\tilde{\omega}}^T\underline{x})\underline{\tilde{\omega}}^T\underline{x}
+(\underline{\tilde{\omega}}^T\underline{x})^2\underline{\tilde{\omega}}^T\underline{\tilde{\omega}}\right]\\ [5pt]
&=& m \left[\underline{x}^T\underline{x}- (\underline{\tilde{\omega}}^T\underline{x})^2\right]
\eqsp
m \left[\left\Vert\,x\,\right\Vert^2 - \underline{\tilde{\omega}}^T\underline{x}\underline{x}^T\underline{\tilde{\omega}}\right]\\ [5pt]
&\eqsp & \underline{\tilde{\omega}}^T \left[m \left(\left\Vert\,\underline{x}\,\right\Vert^2\mathbf{E}-\underline{x}\underline{x}^T\right)\right]\underline{\tilde{\omega}}\\ [5pt]
&\isdef & \underline{\tilde{\omega}}^T\mathbf{I}\,\underline{\tilde{\omega}}
\end{eqnarray*}](img2888.png)
