3 A particle \(P\) of mass 0.3 kg moves in a straight line on a smooth horizontal surface. \(P\) passes through a fixed point \(O\) of the line with velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A force of magnitude \(2 x \mathrm {~N}\) acts on \(P\) in the direction \(P O\), where \(x \mathrm {~m}\) is the displacement of \(P\) from \(O\).

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = k x\) and state the value of the constant \(k\).
2. Find the value of \(x\) at the instant when \(P\) comes to instantaneous rest.
   \includegraphics[max width=\textwidth, alt={}, center]{8d128de7-f6ca-4187-ae8b-cbd8dfcdfc93-3_921_672_258_733} One end of a light inextensible string is attached to a fixed point \(A\). The string passes through a smooth bead \(B\) of mass 0.3 kg and the other end of the string is attached to a fixed point \(C\) vertically below \(A\). The bead \(B\) moves with constant speed in a horizontal circle of radius 0.6 m which has its centre between \(A\) and \(C\). The string makes an angle of \(30 ^ { \circ }\) with the vertical at \(A\) and an angle of \(45 ^ { \circ }\) with the vertical at \(C\) (see diagram).
3. Calculate the speed of \(B\). The lower end of the string is detached from \(C\), and \(B\) is now attached to this end of the string. The other end of the string remains attached to \(A\). The bead is set in motion so that it moves with angular speed \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle which has its centre vertically below \(A\).
4. Calculate the tension in the string.