6 A particle \(P\) of mass 0.15 kg is attached to one end of a light elastic string of natural length 0.4 m and modulus of elasticity 12 N . The other end of the string is attached to a fixed point \(A\). The particle \(P\) moves in a horizontal circle which has its centre vertically below \(A\), with the string inclined at \(\theta ^ { \circ }\) to the vertical and \(A P = 0.5 \mathrm {~m}\).

1. Find the angular speed of \(P\) and the value of \(\theta\).
2. Calculate the difference between the elastic potential energy stored in the string and the kinetic energy of \(P\).
   \(7 \quad\) A particle \(P\) of mass 0.5 kg is at rest at a point \(O\) on a rough horizontal surface. At time \(t = 0\), where \(t\) is in seconds, a horizontal force acting in a fixed direction is applied to \(P\). At time \(t \mathrm {~s}\) the magnitude of the force is \(0.6 t ^ { 2 } \mathrm {~N}\) and the velocity of \(P\) away from \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It is given that \(P\) remains at rest at \(O\) until \(t = 0.5\).
3. Calculate the coefficient of friction between \(P\) and the surface, and show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = 1.2 t ^ { 2 } - 0.3 \quad \text { for } t > 0.5$$
4. Express \(v\) in terms of \(t\) for \(t > 0.5\).
5. Find the displacement of \(P\) from \(O\) when \(t = 1.2\).