\includegraphics{figure_7}

The diagram shows the cross-section $OABCDE$ through the centre of mass of a uniform prism on a rough inclined plane. The portion $ADEO$ is a rectangle in which $AD = OE = 0.6$ m and $DE = AO = 0.8$ m; the portion $BCD$ is an isosceles triangle in which angle $BCD$ is a right angle, and $A$ is the mid-point of $BD$. The plane is inclined at $45°$ to the horizontal, $BC$ lies along a line of greatest slope of the plane and $DE$ is horizontal.
\begin{enumerate}[label=(\roman*)]
\item Calculate the distance of the centre of mass of the prism from $BD$. [3]
\end{enumerate}

The weight of the prism is $21$ N, and it is held in equilibrium by a horizontal force of magnitude $P$ N acting along $ED$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item \begin{enumerate}[label=(\alph*)]
\item Find the smallest value of $P$ for which the prism does not topple. [2]
\item It is given that the prism is about to slip for this smallest value of $P$. Calculate the coefficient of friction between the prism and the plane. [3]
\end{enumerate}
\end{enumerate}

The value of $P$ is gradually increased until the prism ceases to be in equilibrium.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Show that the prism topples before it begins to slide, stating the value of $P$ at which equilibrium is broken. [5]
\end{enumerate}

\hfill \mbox{\textit{CAIE M2 2015 Q7 [13]}}