5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f06704e5-454c-41c1-9577-b1210f60480d-12_693_515_210_781}
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\caption{Figure 3}
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A shop sign is modelled as a uniform rectangular lamina \(A B C D\) with a semicircular lamina removed.
The semicircle has radius \(a , B C = 4 a\) and \(C D = 2 a\).
The centre of the semicircle is at the point \(E\) on \(A D\) such that \(A E = d\), as shown in Figure 3.
- Show that the centre of mass of the sign is \(\frac { 44 a } { 3 ( 16 - \pi ) }\) from \(A D\).
The sign is suspended using vertical ropes attached to the sign at \(A\) and at \(B\) and hangs in equilibrium with \(A B\) horizontal.
The weight of the sign is \(W\) and the ropes are modelled as light inextensible strings.
- Find, in terms of \(W\) and \(\pi\), the tension in the rope attached at \(B\).
The rope attached at \(B\) breaks and the sign hangs freely in equilibrium suspended from \(A\), with \(A D\) at an angle \(\alpha\) to the downward vertical.
Given that \(\tan \alpha = \frac { 11 } { 18 }\)
- find \(d\) in terms of \(a\) and \(\pi\).