Past exam of the physics 1 paper of the University of Oxford / 2014 resit / 1 / b / Solution

Use two central perpendicular axes in the plate. The perpendicular axis theorem and the preceding moments of inertia giveAlternatively, axes along adjacent edges have moments of inertia $M{a}^2/3$ each. The normal axis through their shared corner therefore has $2M{a}^2/3$. It lies a distance $a/\sqrt{2}$ from the parallel normal axis through the centre, so the parallel axis theorem yields $I_{z,G}=2M{a}^2/3-M(a^2/2)=M{a}^2/6$. This normal central result is independent of the in-plane axis-location ambiguity.