6 A uniform rod AB has length \(2 a\) and weight \(W\). The rod is in equilibrium in a horizontal position. The end A rests on a smooth plane which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The force exerted on AB by the plane is \(R\). The end B is attached to a light inextensible string inclined at an angle of \(\theta\) to AB as shown in Fig. 6. The rod and string are in the same vertical plane, which also contains the line of greatest slope of the plane on which A lies. The tension in the string is \(T\).
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\caption{Fig. 6}
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- Add the forces \(R\) and \(T\) to the copy of Fig. 6 in the Printed Answer Booklet.
- By taking moments about B , find an expression for \(R\) in terms of \(W\).
- By resolving horizontally, show that \(6 T \cos \theta = W \sqrt { 3 }\).
- By finding a second equation connecting \(T\) and \(\theta\), determine
- the value of \(\theta\),
- an expression for \(T\) in terms of \(W\).