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\includegraphics[max width=\textwidth, alt={}, center]{1dba0ab0-f3a4-4e7e-a67a-00fd37223cc7-10_445_735_264_696} A particle of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The point \(A\) is such that \(O A = a\) and \(O A\) makes an angle \(\alpha\) with the upward vertical through \(O\). The particle is held at \(A\) and then projected downwards with speed \(\sqrt { } ( a g )\) so that it begins to move in a vertical circle with centre \(O\). There is a small smooth peg at the point \(B\) which is at the same horizontal level as \(O\) and at a distance \(\frac { 1 } { 3 } a\) from \(O\) on the opposite side of \(O\) to \(A\) (see diagram).

1. Show that, when the string first makes contact with the peg, the speed of the particle is \(\sqrt { } ( \operatorname { ag } ( 1 + 2 \cos \alpha ) )\).
   The particle now begins to move in a vertical circle with centre \(B\). When the particle is at the point \(C\) where angle \(C B O = 150 ^ { \circ }\), the tension in the string is the same as it was when the particle was at the point \(A\).
2. Find the value of \(\cos \alpha\).