5. \begin{figure}[h] \end{figure} A particle \(P\) of mass 1.5 kg is attached to the midpoint of a light elastic spring \(A B\), of natural length 2 m and modulus of elasticity 12 N . The end \(A\) of the spring is attached to a fixed point on a smooth horizontal floor. The end \(B\) is held at a point on the floor where \(A B = 6 \mathrm {~m}\). At time \(t = 0 , P\) is at rest on the floor at the point \(O\), where \(A O = 3 \mathrm {~m}\), as shown in Figure 2. The end \(B\) is now moved along the floor in such a way that \(A O B\) remains a straight line and at time \(t\) seconds, \(t \geqslant 0\), $$A B = \left( 6 + \frac { 1 } { 4 } \sin 2 t \right) \mathrm { m }$$ At time \(t\) seconds, \(A P = ( 3 + x ) \mathrm { m }\).

1. Show that, for \(t \geqslant 0\), $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 16 x = 2 \sin 2 t$$ The general solution of this differential equation is $$x = C \cos 4 t + D \sin 4 t + \frac { 1 } { 6 } \sin 2 t$$ where \(C\) and \(D\) are constants.
2. Find the time at which \(P\) first comes to instantaneous rest. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{44066c44-e366-4f87-b1b2-c5a894e407fa-20_705_1104_116_420} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure}