9231 P21 - Jun 2017 - Q5 - 12 marks
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).
(i) Show that, when the string first makes contact with the peg, the speed of the particle is
\[\sqrt{ }(a g(1+2 \cos \alpha)) .\]
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 \(A\).
(ii) Find the value of \(\cos \alpha\).
