For a given rotation axis direction, the moment of inertia will always be minimized when the axis of rotation passes through the object's center-of-mass. The moment of inertia increases as the rotation axis is moved further from the center-of-mass.

Parallel-axis Theorem

For an object of mass M, the parallel-axis theorem states:

I = I_com + Mh²

where h is the distance from the center-of-mass to the current axis of rotation, and I_com is the moment of inertia for the object rotating about the axis through the center of mass that is parallel to the current axis.

Example

The parallel-axis theorem is usually used to calculate the moment of inertia about a second axis when I_com is known. Let's use it to go the other way, using the moment of inertia we just calculated for a rod rotating about one end.

What is the moment of inertia I_com for a uniform rod of length L and mass M rotating about an axis through the center, perpendicular to the rod?

I = I_com + Mh²

I_com = I - Mh²

We found that the moment of inertia when the rod rotates about a parallel axis passing through the end of the rod is:

I

=

1 3

ML2

The distance from the end of the rod to the center is h = L/2. Therefore:

Icom

=

1 3

ML2

-

M

(

L 2

)

2

=

1 3

ML2

-

1 4

ML2

=

1 12

ML2