Proof of Perpendicular Axis Theorem

The perpendicular axis theorem is also referred to as plane figure theorem. In physics, perpendicular axis theorem is used to determine the moment of inertia of a rigid object that lies entirely within a plane, about an axis which is perpendicular to the plane. This theorem page holds the proof of perpendicular axis theorem. This plane figure theorem is applicable only to a plane lamina with a particle of mass 'm' at point 'p' and distance 'r'.

Perpendicular Axis Theorem

Definition

The moment of inertia of a plane lamina about an axis perpendicular to its plane is equal to the sum of the moments of inertia of the lamina about any two mutually perpendicular axes, passing in its own plane, intersecting each other at the point through which the perpendicular axis passes.

Proof

Let us consider a plane lamina lying in the XOY plane. It has large particles where a particle of mass 'm' present at point ‘P’ and its positions are at the distance ‘r’ . From point ‘P’, PN and PNl are drawn perpendicular to X-axis and Y-axis.

Now we say, PNl= x,
PN = y The moment of inertia of the particle of mass ‘m’ about X-axis = my2 The moment of inertia of the whole of lamina about X-axis is, Ix = ∑ my2 ---------------> 1 The moment of inertia of the whole of lamina about Y-axis is, Iy = ∑ mx2 ---------------> 2 The moment of inertia of the whole of lamina about Z-axis, is Iz = ∑ mr2 ----------------> 3 For simplicity, let a particle of mass m, present at point P (x, y) and its position is denoted by r. r2 = x2 + y2 Adding 1 & 2 in 3, we get Iz = ∑ m (x2 + y2) Therefore, ∑ mx2 + ∑my2 or Iz = Iy + Ix Hence, Perpendicular Axis Theorem is proved.


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