A relationship between the moment of inertia of a rigid body about an axis passing through the body's center of mass and the moment of inertia about any parallel axis.
Statement of the Theorem
The parallel axis theorem states that if the moment of inertia of a rigid body about an axis passing through the body's center of mass is Icm then the moment of inertia of the body about any parallel axis can be found by evaluating the sum:
\begin
[ I_
= I_
+ Md^
] \end
where d is the (perpendicular) distance between the original center of mass axis and the new parallel axis.
Motivation for the Theorem
We know that the angular momentum about a single axis of a rigid body that is translating and rotating with respect to a (non-accelerating) axis can be written:
\begin
[ L = m\vec
_
\times\vec
_
+ I_
\omega_
]\end
Now suppose that the rigid body is executing pure rotation about the axis. In that case, the velocity of the center of mass will be perpendicular to the displacement vector from the axis of rotation to the center of mass. Calling the magnitude of that displacement d, to make contact with the form of the theorem, we then have:
\begin
[ L \mbox
= mv_
d + I_
\omega_
]\end
If the body is purely rotating, we can also define an angular speed for rotation about the new parallel axis. The angular speed must satisfy (consider that the center of mass is describing a circle of radius d about the axis):
\begin
[ \omega_
= \frac{v_{cm}}
] \end
Futher, the rotation rate of the object about its center of mass must equal the rotation rate about the parallel axis, since when the object has completed a revolution about the parallel, its oritentation must be the same if it is executing pure rotation. Thus, we can write:
\begin
[ L \mbox
= m\omega_
d^
+ I_
\omega_
]\end
which implies the parallel axis theorem holds.
Derivation of the Theorem
From the definition of the moment of inertia:
\begin
[ I = \int r^
dm ] \end
The center of mass is at a position rcm with respect to the desired axis of rotation. We define new coordinates:
\begin
[ \vec
= \vec
\:' + \vec
_
]\end
where r' measures the positions relative to the object's center of mass. Substituting into the moment of inertia formula:
\begin
[ I = \int\: (r\:'^
+ 2\vec
\:'\cdot\vec
_
+ r_
^
)\: dm ]\end
The rcm is a constant within the integral over the body's mass elements. Thus, the middle term can be written:
\begin
[ \int\:\vec
\:'\cdot \vec
_
\:dm = r_
\cdot \int\:\vec
\:'\:dm ]\end
Since the r' measure deviations from the center of mass position, the integral r'dm must give zero (the position of the center of mass in the r' system). Thus, we are left with:
\begin
[ I = \int\: r\:'^
\:dm + r_
^
\int\:dm = I_
+ Mr_
^
]\end
Which is the parallel axis theorem.