2.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{99e1d643-7408-4793-9ebc-b33c91bc5fab-06_554_547_246_758}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Uniform wire is used to form the framework shown in Figure 2.
In the framework
- \(A B C D\) is a rectangle with \(A D = 2 a\) and \(D C = a\)
- \(B E C\) is a semicircular arc of radius \(a\) and centre \(O\), where \(O\) lies on \(B C\)
The diameter of the semicircle is \(B C\) and the point \(E\) is such that \(O E\) is perpendicular to \(B C\).
The points \(A , B , C , D\) and \(E\) all lie in the same plane.
- Show that the distance of the centre of mass of the framework from \(B C\) is
$$\frac { a } { 6 + \pi }$$
The framework is freely suspended from \(A\) and hangs in equilibrium with \(A E\) at an angle \(\theta ^ { \circ }\) to the downward vertical.
- Find the value of \(\theta\).
The mass of the framework is \(M\).
A particle of mass \(k M\) is attached to the framework at \(B\).
The centre of mass of the loaded framework lies on \(O A\). - Find the value of \(k\).