Parallel Axis Theorem

Let $ I_0$ denote the moment of inertia for a rotation axis passing through the center of mass, and let $ I_d$ denote the moment of inertia for a rotation axis parallel to the first but a distance $ d$ away from it. Then the parallel axis theorem says that

$\displaystyle I_d = I_0 + Md^2
$

where $ M$ denotes the total mass. Thus, the added inertia due to displacement by $ d$ meters away from the centroidal axis is equal to that of a point mass $ M$ rotating a distance $ d$ from the center of rotation.


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Perpendicular Axis Theorem