moment of inertia

Mass weighted by the square of its distance from an axis, $I=\iint_R r^2\rho\,dA$ — a measure of resistance to rotation.

The moment of inertia weights mass by the square of its distance from an axis, $I=\iint_R r^2\rho\,dA$ , rather than linearly the way $M_x$ and $M_y$ do. Squaring falls directly out of physics: kinetic energy of a rotating point mass is $\tfrac12mv^2$ , and for circular motion $v=r\omega$ , so summing over a lamina produces rotational kinetic energy $\tfrac12I\omega^2$ with $I$ playing the role ordinary mass plays in $\tfrac12mv^2$ . The perpendicular axis theorem $I_0=I_x+I_y$ follows immediately since $x^2+y^2=r^2$ splits additively.

Examples

For the unit square plate with $\rho=1$ , $I_0=\iint_R(x^2+y^2)\,dA=\tfrac23$ , and indeed $I_x+I_y=\tfrac13+\tfrac13=\tfrac23$ .