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\includegraphics[max width=\textwidth, alt={}, center]{334b4bdf-6d9c-4208-9032-572eb7c5f9ee-3_451_432_1434_852} One end of a light inextensible string is attached to the highest point \(A\) of a solid fixed sphere with centre \(O\) and radius 0.6 m . The other end of the string is attached to a particle \(P\) of mass 0.2 kg which rests in contact with the smooth surface of the sphere. The angle \(A O P = 60 ^ { \circ }\) (see diagram). The sphere exerts a contact force of magnitude \(R \mathrm {~N}\) on \(P\) and the tension in the string is \(T \mathrm {~N}\).

1. By resolving vertically, show that \(R + ( \sqrt { } 3 ) T = 4\).
   \(P\) is now set in motion, and moves with angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle on the surface of the sphere.
2. Find an equation involving \(R , T\) and \(\omega\).
3. Hence
   (a) calculate \(R\) when \(\omega = 2\),
   (b) find the greatest possible value of \(T\) and the corresponding speed of \(P\).