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]
\includegraphics[alt={},max width=\textwidth]{831ba5da-df19-43bb-b163-02bbddb4e2b8-4_659_540_397_244}
\captionsetup{labelformat=empty}
\caption{Fig. 5.1}
\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 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]
\includegraphics[alt={},max width=\textwidth]{831ba5da-df19-43bb-b163-02bbddb4e2b8-4_444_899_1889_246}
\captionsetup{labelformat=empty}
\caption{Fig. 5.2}
\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.