Problem

A uniform lamina $OABCD$ is in the form of a rectangle, $OBCD$, joined along the edge $OB$ to a quarter circle $OAB$. The length of $DO$ is $ka$ and the length of $OB$ is $a$. The lamina rests in a vertical plane with its edge $CB$ on a horizontal surface (see diagram).

(a) Find, in terms of $k$, $a$ and $\pi$, an expression for the distance of the centre of mass above the horizontal surface.

[You may use without proof the result for the centre of mass of a circular sector in the list of formulae (MF19).]

The lamina is on the point of toppling about $B$.

(b) Find the value of $k$.