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\includegraphics[max width=\textwidth, alt={}, center]{0270d51a-e252-46d3-8c97-7f71ba91fa65-12_618_835_255_616} A particle \(P\) of mass \(m\) is attached to one end of a light rod of length \(3 a\). The other end of the rod is able to pivot smoothly about the fixed point \(A\). The particle is also attached to one end of a light spring of natural length \(a\) and modulus of elasticity \(k m g\). The other end of the spring is attached to a fixed point \(B\). The points \(A\) and \(B\) are in a horizontal line, a distance \(5 a\) apart, and these two points and the rod are in a vertical plane. Initially, \(P\) is held in equilibrium by a vertical force \(F\) with the stretched length of the spring equal to \(4 a\) (see diagram). The particle is released from rest in this position and has a speed of \(\frac { 6 } { 5 } \sqrt { 2 a g }\) when the rod becomes horizontal.

1. Find the value of \(k\).
2. Find \(F\) in terms of \(m\) and \(g\).
3. Find, in terms of \(m\) and \(g\), the tension in the rod immediately before it is released.
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