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\includegraphics[max width=\textwidth, alt={}, center]{f6887893-66c5-40df-ba8d-9439a5c268eb-2_582_798_616_671} A particle of mass \(m\) is attached to the end \(B\) of a light inextensible string. The other end of the string is attached to a fixed point \(A\) which is at a distance \(a\) above the vertex \(V\) of a circular cone of semi-vertical angle \(60 ^ { \circ }\). The axis of the cone is vertical. The particle moves with constant speed \(u\) in a horizontal circle on the smooth surface of the cone. The string makes a constant angle of \(30 ^ { \circ }\) with the vertical (see diagram). The tension in the string and the magnitude of the normal force acting on the particle are denoted by \(T\) and \(R\) respectively. Show that $$T = \frac { m } { \sqrt { } 3 } \left( g + \frac { 2 u ^ { 2 } } { a } \right) ,$$ and find a similar expression for \(R\). Deduce that \(u ^ { 2 } \leqslant \frac { 1 } { 2 } g a\).