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\includegraphics[max width=\textwidth, alt={}, center]{c7844913-5c2e-47b4-87b6-f822f4d4bf22-2_426_862_1553_644} A hemispherical bowl of radius \(r\) is fixed with its rim horizontal. A thin uniform rod rests in equilibrium on the rim of the bowl with one end resting on the inner surface of the bowl at \(A\), as shown in the diagram. The rod has length \(2 a\) and weight \(W\). The point of contact between the rod and the rim is \(B\), and the rim has centre \(C\). The rod is in a vertical plane containing \(C\). The rod is inclined at \(\theta\) to the horizontal and the line \(A C\) is inclined at \(2 \theta\) to the horizontal. The contacts at \(A\) and \(B\) are smooth. In any order, show that

1. the contact force acting on the rod at \(A\) has magnitude \(W \tan \theta\),
2. the contact force acting on the rod at \(B\) has magnitude \(\frac { W \cos 2 \theta } { \cos \theta }\),
3. \(2 r \cos 2 \theta = a \cos \theta\).