8 A rectangular lamina of mass $M$ has vertices at the origin $O ( 0,0 ) , A ( 24 a , 0 ) , B ( 24 a , 6 a )$ and $C ( 0,6 a )$, where $a$ is a positive constant. A small object $P$ of mass $m$ is attached to the lamina at the point ( $x , y$ ). The centre of mass of the system consisting of the lamina and $P$ is at the point ( $\mathrm { x } , \mathrm { y }$ ). $P$ is modelled as a particle and the lamina is modelled as being uniform.
\begin{enumerate}[label=(\alph*)]
\item Show that $x = \frac { 12 M a + m x } { M + m }$.
\item Find a corresponding expression for $\bar { y }$.

The lamina, with $P$ no longer attached, is placed on a horizontal rectangular table, with its sides parallel to the edges of the table, and partly overhanging the edges of the table, as shown in the diagram. The corner of the table is at the point ( $6 a , 2 a$ ).\\
\includegraphics[max width=\textwidth, alt={}, center]{c6445493-9802-46ca-b7eb-7738a831d9ee-6_538_1431_849_246}

When $P$ is placed on the lamina at $O$, the lamina topples over one of the edges of the table.
\item Show that $\mathrm { m } > \frac { 1 } { 2 } \mathrm { M }$.

The lamina is now put back on the table in the same position as before. $P$ is placed at the point $( 12 a , 6 a )$ on the smooth upper surface of the lamina, and is projected towards $O$. At a subsequent instant during the motion, $P$ is at the point (12ak, 6ak) where $0 < k < 1$.
\item Assuming that the lamina has not yet toppled, find, in terms of $M$ and $m$, the value of $k$ for which the centre of mass of the system lies on the table edge parallel to $O C$.
\item For the case $\mathrm { m } = \frac { 3 } { 2 } \mathrm { M }$, determine which table edge the lamina topples over.
\end{enumerate}

\hfill \mbox{\textit{OCR Further Mechanics 2021 Q8 [13]}}