Parallel axis theorem
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In physics, the parallel axis theorem can be used to determine the moment of inertia of a rigid object about any axis, given the moment of inertia of the object about the parallel axis through the object's center of mass and the perpendicular distance between the axes.
Let ICM denote the moment of inertia of the object about the center of mass, M the object's mass and d the perpendicular distance between the two axes. Then the moment of inertia about the new axis z is given by:
This rule can be applied with the stretch rule and perpendicular axes rule to find moments of inertia for a variety of shapes.
The parallel axes rule also applies to the second moment of area (area moment of inertia);
where Iz is the area moment of inertia through the parallel axis, Ix is the area moment of inertia through the center of mass of the area, A is the surface of the area, and d is the distance from the new axis z to the center of gravity of the area.
The parallel axis theorem is one of several theorems referred to as Steiner's theorem, after Jakob Steiner.
