### Mass Moment of Inertia Tensor

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

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:

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

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:

#### Simple Example

Consider a mass at . Then the mass moment of inertia tensor is

*should*be zero about that axis. On the other hand, if we look at , we get

#### Example with Coupled Rotations

Now let the mass be located at so that

We expect to yield zero for the moment of inertia, and sure enough . Similarly, the vector angular momentum is zero, since .

For , the result is

#### Off-Diagonal Terms in Moment of Inertia Tensor

This all makes sense, but what about those off-diagonal terms in ? Consider the vector angular momentum (§B.4.14):

*coupling*of rotation about with rotation about . That is, there is a component of moment-of-inertia that is contributed (or subtracted, as we saw above for ) when

*both*and are nonzero. These cross-terms can be eliminated by

*diagonalizing*the matrix [449],

^{B.25}as discussed further in the next section.

**Next Section:**

Principal Axes of Rotation

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Angular Momentum Vector