## Moment of inertia

Moment
of inertia is the rotational analogue to mass. The mass moment of
inertia about a fixed axis is the property of a body that measures the
body's resistance to rotational acceleration. The greater its value,
the greater the moment required to provide a given acceleration about a
fixed pivot.

The moment
of inertia must be specified with respect to a
chosen axis of rotation.

The symbols Ixx, Iyy and Izz are frequently used
to express the moments
of inertia
of a 3D rigid body

about
its three axis.

(A)

Products
of
Inertia are
given by Ixy, Ixz and Iyz where

(B)

## Inertia
Matrix

The

moment of momentum,

can be
expressed as

(C) (

See PDF
for an explanation of how this is obtained)

Where

is the Inertia Matrix

Problems
where the moment of momentum vector,

h
is parallel to

are easier to
solve, so the moment of momentum can be expressed as

If
this expression for is substituted into equation

(C)
then the following expression is obtained.

This
can be
seen to be an eigenvalue problem, the three
eigenvalues

of

define the axis about which the body can spin maintaining h
parallel to

The three eigenvalues are the

principle
moments of inertia and are known
as

A B and

C

The three eigenvectors are the

principle axis of inertia
and are orthogonal

.
When the axis are aligned with the principle axis Ip can be
expressed as

Therefore
axis aligned with principle are useful in solving
practical
problems

### Moments of Inertia of a gyroscope

A gyroscope is an axisymmetric body

Due
to the
axisymmetry of a gyroscope all axis in the i-j
plane are
principle. A gyroscope can be thought of as an

AAC body. It
has
principle moments of inertia

A A and

C