Perpendicular Axis Theorem

Question

A thin square plate has a moment of inertia I about an axis lying in its plane and passing through its centre parallel to one edge. What is its moment of inertia about an axis perpendicular to the plate through the same centre?

RankGuru · Accepted Answer

The perpendicular axis theorem applies to flat planar laminae and states $I_z = I_x + I_y$, where the $z$-axis is perpendicular to the plate and the $x$- and $y$-axes lie in its plane through the same point. By the symmetry of a square plate, the in-plane moments of inertia about the two central axes parallel to its edges are equal, $I_x = I_y = I$. Therefore $I_z = I + I = 2I$. The value I forgets that two separate in-plane contributions must be summed. The value I/2 inverts the relationship between the perpendicular and in-plane axes. The value 4I wrongly squares the count of contributions instead of adding them. This is the NCERT perpendicular axis theorem, valid only for two-dimensional bodies of negligible thickness. As a consistency check, the perpendicular axis sees mass distributed in both planar directions at once, so its moment of inertia must be larger than either single in-plane value, and $2I > I$ confirms this expectation cleanly. The theorem holds only for thin laminae because it assumes every mass element lies in the plane of the plate; for a thick three-dimensional block the mass spreads along the perpendicular axis itself and the simple sum $I_x + I_y$ no longer applies.

Perpendicular Axis Theorem

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