7. One end of a light rod of length \(\frac { 5 } { 7 } \mathrm {~m}\) is attached to a fixed point \(O\) and the other end is attached to a particle \(P\), of mass \(m \mathrm {~kg}\). The particle \(P\) is projected from the point \(A\), which is vertically below \(O\), with a horizontal speed of \(u \mathrm {~ms} ^ { - 1 }\) so that it moves in a vertical circle with centre \(O\). When the rod \(O P\) is inclined at an angle \(\theta\) to the downward vertical, the speed of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\) and the tension in the rod is \(T \mathrm {~N}\).
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1. Show that $$v ^ { 2 } = u ^ { 2 } - 14 + 14 \cos \theta$$
2. Hence determine the least possible value of \(u ^ { 2 }\) for the particle to reach the highest point of the circle.
3. Given that \(u ^ { 2 } = 32 \cdot 2\),
   1. find, in terms of \(m\) and \(\theta\), an expression for \(T\),
   2. calculate the range of values of \(\theta\) such that the rod is exerting a thrust.
      State whether your answer to (c)(ii) would be different if the mass of the particle was reduced. Give a reason for your answer. Additional page, if required. Write the question number(s) in the left-hand margin. only