6 One end of a light elastic string of natural length 0.4 m and modulus of elasticity 8 N is attached to a fixed point \(O\) on a smooth horizontal plane. The other end of the string is attached to a particle \(P\) of mass 0.2 kg which moves on the plane in a circular path with centre \(O\). The speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the extension of the string is \(x \mathrm {~m}\).

1. Given that \(v = 2.5\), find \(x\).
   It is given instead that the kinetic energy of \(P\) is twice the elastic potential energy stored in the string.
2. Form two simultaneous equations and hence find \(x\) and \(v\).