Parallel Axis Theorem

Parallel axis theorem lets you find the moment of inertia about any axis given you know its moment of inertia about a parallel axis passing through the object’s centre of mass.
$I=\bar{I}+m{d}^2$

• $\bar{I}$ $-$ moment of inertia about an axis through the mass center G, parallel to the axis of interest (also notated as $I_G$)
• $d$ $-$ perpendicular distance between the two parallel axes
• $m$ $-$ total mass of the body

Non-Centroidal Axis

This theorem only ever moves you to or from the centroidal axis. To go from one non-centroidal axis to another, you must go through $G$: subtract $m{d}_1^2$ to get $\bar{I}$, then add $m{d}_2^2$.

Geometric vs Mass Centre

$I_O$ is the geometric centre.
$I_G$ is the mass centre.

$I_O\ne I_G$ unless $O$ and $G$ coincide.

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