Perpendicular Axis Theorem

The perpendicular axis theorem states that

the moment of inertia of a planar lamina (2-dimensional object) about an axis perpendicular to its plane and passing through the origin of two mutually perpendicular axes, lying in the plane of the object, is equal to the sum of the moments of inertia of the lamina about axes lying in the plane of the object.

Assuming three mutually perpendicular axes x, y, z so that the xy plane is coinciding with the plane of the object as shown in the figure. Let I_x, I_y, and I_z be the moments of inertia about x, y, and z-axis respectively.

Then from the perpendicular axis theorem

I_{z}=I_{x}+I_{y}

If an object has rotational symmetry such that I_{x} and I_{y} are equal then the perpendicular axes theorem provides the useful relationship:

I_{z}=2I_{x}=2I_{y}

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