2 A particle \(P\) of mass 0.4 kg is attached to one end of a light inextensible string of length 1.8 m . The other end of the string is attached to a fixed point, \(O\), on a smooth horizontal plane. Initially, \(P\) is moving with a constant speed of \(12 \mathrm {~ms} ^ { - 1 }\) in a horizontal circle with \(O\) as its centre.

1. 1. Find the magnitude of the acceleration of \(P\).
   2. State the direction of the acceleration of \(P\). A force is now applied to \(P\) in such a way that its angular velocity increases. At the instant that the angular velocity reaches \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\), the string breaks.
2. 1. Find the speed with which \(P\) is moving at the instant that the string breaks.
   2. Find the tension in the string at the instant that the string breaks. After the string has broken \(P\) starts to move directly up a smooth slope which is fixed to the plane and inclined at an angle \(\theta ^ { \circ }\) above the horizontal. Particle \(P\) moves a distance of 20 m up the slope before coming to instantaneous rest.
3. Use an energy method to determine the value of \(\theta\).