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Wikipedia, 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 X, Y, and Z (which meet at origin O) so that the body lies in the XY plane, and the Z axis is perpendicular to the plane of the body. Let
The perpendicular axis theorem states that
This rule can be applied with the parallel axis theorem and the stretch rule to find moments of inertia for a variety of shapes. ProofLet p be a plane thin uniform lamina. Let m_{i} be a mass element with perpendicular distance r_{i} from an axis OZ perpendicular to the plane and passing through O in the plane. Let OX and OY be two perpendicular axes lying in the plane. Let a_{i} be the perpendicular distance of m_{i} from OX and b_{i} be the perpendicular distance of m_{i} from OY, both in the plane. Let be the moment of inertia of p about OX and be the moment of inertia of p about OY. The moment of inertia of p about OZ is given by See also
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Published  July 2009
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