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\includegraphics[max width=\textwidth, alt={}, center]{84a2b2eb-a750-4047-864b-4a165fc66b2a-4_558_857_260_644} One end of a light elastic string with modulus of elasticity 15 N is attached to a fixed point \(A\) which is 2 m vertically above a fixed small smooth ring \(R\). The string has natural length 2 m and it passes through \(R\). The other end of the string is attached to a particle \(P\) of mass \(m \mathrm {~kg}\) which moves with constant angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle which has its centre 0.4 m vertically below the ring. \(P R\) makes an acute angle \(\theta\) with the vertical (see diagram).

1. Show that the tension in the string is \(\frac { 3 } { \cos \theta } \mathrm {~N}\) and hence find the value of \(m\).
2. Show that the value of \(\omega\) does not depend on \(\theta\). It is given that for one value of \(\theta\) the elastic potential energy stored in the string is twice the kinetic energy of \(P\).
3. Find this value of \(\theta\).