According to Parallel axis theorem, the moment of inertia of any object through its center of mass is the minimum moment of inertia for an axis in that particular direction in the space. The moment of inertia about any axis parallel to that axis through the center of mass is given by

In the given expression, the centre of mass moment of inertia will be recognized as the moment of inertia of a point mass - the moment of inertia about a parallel axis is the centre of mass moment plus the moment of inertia of the entire object treated as a point mass at the centre of mass.




The perpendicular distance between the axes lies along the x-axis and that the centre of mass lies at the origin. The moment of inertia relative to the z-axis, passing through the centre of mass, is:

 I_{cm} = \int{(x^2 + y^2)} dm

The moment of inertia relative to the new axis, perpendicular distance r along the x-axis from the centre of mass, is:

 I_z = \int{((x - r)^2 + y^2)} dm

 I_z = \int{(x^2 + y^2)} dm + r^2 \int dm - 2r\int{x} dm

The first term is Icm, the second term becomes mr2, and the final term is zero since the origin is at the centre of mass. So, this expression becomes:

  I_z = I_{cm} + mr^2\,


In classical mechanics

In classical mechanics, the Parallel axis theorem (also known as Huygens-Steiner theorem) can be generalized to calculate a new inertia tensor Jij from an inertia tensor about a center of mass Iij when the pivot point is a displacement a from the center of mass:

Jij = Iij + M(a2δij  aiaj)



is the displacement vector from the center of mass to the new axis, and


is the Kronecker delta.

We can see that, for diagonal elements (when i = j), displacements perpendicular to the axis of rotation results in the above simplified version of the parallel axis theorem.


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