Question

Prove the theorem of perpendicular axes about the moment of inertia.

Answer 1

Theorem of perpendicular axes

Proof: Let Oz be an axis perpendicular to the lamina's plane, and let Ox and Oy be two perpendicular axes in the lamina's plane. At point P(x, y), consider an infinitesimal mass element (dm) of the lamina. MI centred on the z-axis in the lamina

I_z = ∫ OP² dm`

The element is perpendicular to the x - and y - axes, respectively, at a distance of y and x. Because of this, the lamina's moments of inertia about the x - and y -axes are, respectively,

I_x = ∫ y² dm and I_y = ∫ x² dm 

Since OP² = y² + x²,

I_z = ∫ OP² dm

= ∫ (y² + x²) dm

= ∫ y² dm + ∫ x² dm

= ∴ I_z = I_x + I_y

This proves the theorem of perpendicular axes.