6 A particle \(P\) of mass \(m\) is attached to one end of a light inextensible rod of length \(3 a\). An identical particle \(Q\) is attached to the other end of the rod. The rod is smoothly pivoted at a point \(O\) on the rod, where \(\mathrm { OQ } = \mathrm { x }\). The system, of rod and particles, rotates about \(O\) in a vertical plane. At an instant when the rod is vertical, with \(P\) above \(Q\), the particle \(P\) is moving horizontally with speed \(u\). When the rod has turned through an angle of \(60 ^ { \circ }\) from the vertical, the speed of \(P\) is \(2 \sqrt { \mathrm { ag } }\), and the tensions in the two parts of the rod, \(O P\) and \(O Q\), have equal magnitudes.

1. Show that the speed of \(Q\) when the rod has turned through an angle of \(60 ^ { \circ }\) from the vertical is \(\frac { 2 x } { 3 a - x } \sqrt { a g }\).
2. Find \(x\) in terms of \(a\).
3. Find \(u\) in terms of \(a\) and \(g\).
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