3
\includegraphics[max width=\textwidth, alt={}, center]{08760a55-da6c-41f2-a88a-289ecc227f69-3_812_773_260_685} Two uniform rods \(A B\) and \(B C\), each of length \(2 a\), have weights \(2 W\) and \(W\) respectively. The rods are freely jointed to each other at \(B\), and \(B C\) is freely jointed to a fixed point at \(C\). The rods are held in equilibrium in a vertical plane by a light string attached to \(A\) and perpendicular to \(A B\). The rods \(A B\) and \(B C\) make angles \(45 ^ { \circ }\) and \(\alpha\), respectively, with the horizontal. The tension in the string is \(T\) (see diagram).

1. By taking moments about \(B\) for \(A B\), show that \(W = \sqrt { 2 } T\).
2. Find the value of \(\tan \alpha\).