5 The cover of a children's book is modelled as being a uniform lamina $L . L$ occupies the region bounded by the $x$-axis, the curve $\mathrm { y } = 6 + \sin \mathrm { x }$ and the lines $x = 0$ and $x = 5$ (see Fig. 5.1). The centre of mass of $L$ is at the point ( $\mathrm { x } , \mathrm { y }$ ).

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{831ba5da-df19-43bb-b163-02bbddb4e2b8-4_659_540_397_244}
\captionsetup{labelformat=empty}
\caption{Fig. 5.1}
\end{center}
\end{figure}
\begin{enumerate}[label=(\alph*)]
\item Show that $\bar { X } = 2.36$, correct to 3 significant figures.
\item Find $\bar { y }$, giving your answer correct to 3 significant figures.

The cover of the book weighs 6 N . $A$ is the point on the cover with coordinates $( 3 , \bar { y } )$ and $B$ is the point on the cover with coordinates $( 5 , \bar { y } )$. A small badge of weight 2 N is attached to the cover at $A$.

The side of $L$ along the $y$-axis is attached to the rest of the book and the book is placed on a rough horizontal plane. The attachment of the cover to the book is modelled as a hinge.

The cover is held in equilibrium at an angle of $\frac { 1 } { 3 } \pi$ radians to the horizontal by a force of magnitude $P N$ acting at $B$ perpendicular to the cover (see Fig. 5.2).

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{831ba5da-df19-43bb-b163-02bbddb4e2b8-4_444_899_1889_246}
\captionsetup{labelformat=empty}
\caption{Fig. 5.2}
\end{center}
\end{figure}
\item State two additional modelling assumptions, one about the attachment of the cover and one about the badge, which are necessary to allow the value of $P$ to be determined.
\item Using the modelling assumptions, determine the value of $P$ giving your answer correct to 3 significant figures.
\end{enumerate}

\hfill \mbox{\textit{OCR Further Mechanics 2020 Q5 [9]}}