Problem

A uniform lamina $OABCD$ consists of a rectangle $OACD$ and a triangle $ABC$. The length of $OA$ is $ka$, the length of $OD$ is $2a$, the height of triangle $ABC$ is $h$ and angle $CAB$ is $45^\circ$ (see diagram). Relative to axes through $O$, parallel and perpendicular to $OA$ as shown, the centre of mass of triangle $ABC$ is $(\bar{x},\bar{y})$.

(a) Show that

$$\bar{x}=\frac13(3ka+h),$$

and find an expression for $\bar{y}$.

The lamina $OABCD$ is placed vertically on its edge $OA$ on a horizontal plane.

(b) Find, in terms of $a$ and $k$, the set of values of $h$ for which the lamina is in equilibrium.

It is now given that $k=\frac{\sqrt3}{3}$ and that the lamina is on the point of toppling.

(c) Find, in terms of $a$, the coordinates of the centre of mass of the triangle $ABC$.