Standard +0.8 This is a multi-part centre of mass problem requiring composite lamina calculations using standard formulas (distance 4r/3π for semicircles), coordinate geometry setup, and equilibrium with moments. While it involves several steps and careful bookkeeping of two semicircular regions with different radii, the techniques are standard for Further Maths mechanics. The 'show that' part guides students through the harder calculation, and part (b) is a straightforward moments application once the centre of mass is found.
\(A C B\) is the diameter of a semi-circular lamina of radius \(2 a\) and centre \(C\). Another semi-circular lamina, having \(A C\) as its diameter, is added to form a uniform lamina, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-10_755_521_520_772}
(i) Show that the distance of the centre of mass of the lamina from \(A B\) is \(\frac { 28 } { 15 \pi } a\).
(ii) Calculate the distance of the centre of mass of the lamina from a line drawn through \(A\) that is perpendicular to \(A B\).
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Suppose that the lamina is suspended in equilibrium by means of two vertical wires attached at \(A\) and \(B\) so that \(A B\) is horizontal. Find the fraction of the lamina's weight that is supported by the wire attached at \(B\).
\begin{enumerate}
\item $A C B$ is the diameter of a semi-circular lamina of radius $2 a$ and centre $C$. Another semi-circular lamina, having $A C$ as its diameter, is added to form a uniform lamina, as shown in the diagram below.\\
\includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-10_755_521_520_772}\\
(a) (i) Show that the distance of the centre of mass of the lamina from $A B$ is $\frac { 28 } { 15 \pi } a$.\\
(ii) Calculate the distance of the centre of mass of the lamina from a line drawn through $A$ that is perpendicular to $A B$.\\
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(b) Suppose that the lamina is suspended in equilibrium by means of two vertical wires attached at $A$ and $B$ so that $A B$ is horizontal. Find the fraction of the lamina's weight that is supported by the wire attached at $B$.\\
\end{enumerate}
\section*{PLEASE DO NOT WRITE ON THIS PAGE}
\hfill \mbox{\textit{WJEC Further Unit 6 2024 Q3}}