State and prove perpendicular axes theorem.

Answer:

Statement: The moment of inertia of a plane lamina about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two mutually perpendicular axes lying in its plane and intersecting at the point where the perpendicular axis passes through the lamina: \(I_z=I_x+I_y\).

Proof: [Figure: a plane lamina in the XY plane with origin O; OX and OY in its plane, OZ perpendicular to it; a particle of mass \(dm\) at point P(\(x\), \(y\)) at distance \(r\) from O.]

Consider a small element of mass \(dm\) at P. Its distance from the Z axis is \(r\), from the X axis is \(y\) and from the Y axis is \(x\), with \(r^2=x^2+y^2\).

MI of the lamina about the Z axis: \(I_z=\int r^2\,dm=\int(x^2+y^2)\,dm=\int x^2\,dm+\int y^2\,dm\).

But \(\int y^2\,dm=I_x\) (MI about X axis) and \(\int x^2\,dm=I_y\) (MI about Y axis).

\(\therefore I_z=I_x+I_y\). The theorem applies only to plane (two-dimensional) laminae.