2 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 \(y = 6 + \sin x\) and the lines \(x = 0\) and \(x = 5\) (see Fig. 2.1). The centre of mass of \(L\) is at the point \(( \bar { x } , \bar { y } )\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d6bf2fa5-2f29-4632-b27d-ed8c5a0379cf-02_650_534_1030_255}
\captionsetup{labelformat=empty}
\caption{Fig. 2.1}
\end{figure}
- Show that \(\bar { x } = 2.36\), correct to 3 significant figures.
- Find \(\bar { y }\), giving your answer correct to 3 significant figures.
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 \(P \mathrm {~N}\) acting at \(B\) perpendicular to the cover (see Fig. 2.2).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d6bf2fa5-2f29-4632-b27d-ed8c5a0379cf-03_412_213_402_525}
\captionsetup{labelformat=empty}
\caption{Fig. 2.2}
\end{figure} - 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.
- Using the modelling assumptions, determine the value of \(P\) giving your answer correct to 3 significant figures.