6.
\begin{figure}[h]
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\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{c68c85a1-9d80-4ced-bfb6-c7b5347e9bb8-4_244_1264_1314_382}
\end{figure}
A particle \(P\) of mass 2 kg is attached to the mid-point of a light elastic spring of natural length 2 m and modulus of elasticity 4 N . One end \(A\) of the elastic spring is attached to a fixed point on a smooth horizontal table. The spring is then stretched until its length is 4 m and its other end \(B\) is held at a point on the table where \(A B = 4 \mathrm {~m}\). At time \(t = 0 , P\) is at rest on the table at the point \(O\) where \(A O = 2 \mathrm {~m}\), as shown in Fig. 3. The end \(B\) is now moved on the table in such a way that \(A O B\) remains a straight line. At time \(t\) seconds, \(A B = \left( 4 + \frac { 1 } { 2 } \sin 4 t \right) \mathrm { m }\) and \(A P = ( 2 + x ) \mathrm { m }\).
- Show that
$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 x = \sin 4 t$$
- Hence find the time when \(P\) first comes to instantaneous rest.
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