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\includegraphics[max width=\textwidth, alt={}, center]{bcd7ee99-e382-4cb6-aa39-d8b385b01319-2_506_623_977_760} Two uniform rods \(A B\) and \(B C\), each of length \(2 a\) and mass \(m\), are smoothly hinged at \(B\). They rest in equilibrium with \(C\) in contact with a smooth vertical wall and \(A\) in contact with a rough horizontal floor. The rods are in a vertical plane perpendicular to the wall. The rods \(A B\) and \(B C\) make angles \(\alpha\) and \(\beta\) respectively with the horizontal (see diagram). Show that

1. the reaction at \(C\) has magnitude \(\frac { 1 } { 2 } m g \cot \beta\),
2. \(\tan \alpha = 3 \tan \beta\). The coefficient of friction at \(A\) is \(\mu\). Given that \(\alpha = 60 ^ { \circ }\), find the least possible value of \(\mu\).