5 Fig. 5 shows the curve with equation \(y = - x ^ { 2 } + 4 x + 2\).
The curve intersects the \(x\)-axis at P and Q . The region bounded by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 4\) is occupied by a uniform lamina L . The horizontal base of L is OA , where A is the point \(( 4,0 )\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4acb019b-e630-4766-9d7f-39bc0e174ba1-4_533_930_466_242}
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\caption{Fig. 5}
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- Explain why the centre of mass of L lies on the line \(x = 2\).
- In this question you must show detailed reasoning.
Find the \(y\)-coordinate of the centre of mass of \(L\).
- L is freely suspended from A . Find the angle AO makes with the vertical.
The region bounded by the curve and the \(x\)-axis is now occupied by a uniform lamina M . The horizontal base of M is PQ.
- Explain how the position of the centre of mass of M differs from the position of the centre of mass of \(L\).