9 A small sphere, 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 sphere is at rest in equilibrium directly below \(O\) when it is struck, giving it a horizontal impulse of magnitude \(m U\) After the impulse, the sphere follows a circular path in a vertical plane containing the point \(O\) until the string becomes slack at the point \(C\) At \(C\) the string makes an angle of \(30 ^ { \circ }\) with the upward vertical through \(O\), as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{86817115-46a1-4702-8a33-8f9b05d69bb9-12_583_331_875_901} 9

1. Show that $$U ^ { 2 } = \frac { a g } { 2 } ( 4 + 3 \sqrt { 3 } )$$ where \(g\) is the acceleration due to gravity.
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2. With reference to any modelling assumptions that you have made, explain why giving your answer as an inequality would be more appropriate, and state this inequality.
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