Perpendicular axis theorem

In physics, the perpendicular axis theorem (or plane figure theorem) can be used to determine the moment of inertia of a rigid object that lies entirely within a plane, about an axis at right angles to the plane, given the moments of inertia of the object about two perpendicular axes lying within the plane. The axes must all pass through a single point in the plane.

Define perpendicular axes <math>x,</math>, <math>y,</math>, and <math>z,</math> (which meet at origin <math>O,</math>) so that the body lies in the <math>xy,</math> plane, and the <math>z,</math> axis is perpendicular to the plane of the body. Let I_x, I_y and I_z be moments of inertia about axis x, y, z respectively, the perpendicular axis theorem states that

<math>I_z = I_x + I_y,</math>

This rule can be applied with the parallel axis theorem and the stretch rule to find moments of inertia for a variety of shapes.

If a planar object (or prism, by the stretch rule) has rotational symmetry such that <math>I_x,</math> and <math>I_y,</math> are equal, then the perpendicular axes theorem provides the useful relationship:

<math>I_z = 2I_x = 2I_y,</math>

Derivation

Working in Cartesian co-ordinates, the moment of inertia of the planar body about the <math>z,</math> axis is given...
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