State and prove the theorem of perpendicular axes.

The theorem of perpendicular axes applies to flat, two-dimensional objects, known as planar laminae. It states that the moment of inertia of a planar object about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two mutually perpendicular axes that lie within the plane and intersect at the point where the perpendicular axis passes through the lamina.

To understand this, consider a planar object lying in a horizontal plane. Imagine three axes: one perpendicular to the plane, and two that lie in the plane and that intersect at the vertical axis. The moment of inertia about the vertical axis is a measure that accounts for all the rotational resistances of all mass elements of the object in terms of how far they lie from this axis. Similarly, the moments of inertia about the two horizontal axes account for the resistance of the same mass elements relative to these axes.

The perpendicular axes theorem makes the computation easier by correlating the moment of inertia of the vertical axis to the sum of moments of inertia of the two horizontal axes. The theorem is more beneficial in the context of engineering and physics as it is applicable for analyzing rotational dynamics of flat objects, facilitating designs of systems with rotation as well as for planar structures.

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