7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d93ae982-9395-4311-9972-be727b3ce954-22_197_945_251_497}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{figure}
The fixed points \(A\) and \(B\) are 4.2 m apart on a smooth horizontal floor. One end of a light elastic spring, of natural length 1.8 m and modulus of elasticity 20 N , is attached to a particle \(P\) and the other end is attached to \(A\). One end of another light elastic spring, of natural length 0.9 m and modulus of elasticity 15 N , is attached to \(P\) and the other end is attached to \(B\). The particle \(P\) rests in equilibrium at the point \(O\), where \(A O B\) is a straight line, as shown in Figure 5.
- Show that \(A O = 2.7 \mathrm {~m}\).
The particle \(P\) now receives an impulse acting in the direction \(O B\) and moves away from \(O\) towards \(B\). In the subsequent motion \(P\) does not reach \(B\).
- Show that \(P\) moves with simple harmonic motion about centre \(O\).
The mass of \(P\) is 10 kg and the magnitude of the impulse is \(J \mathrm { Ns }\).
Given that \(P\) first comes to instantaneous rest at the point \(C\) where \(A C = 2.9 \mathrm {~m}\),
- find the value of \(J\),
- find the time taken by \(P\) to travel a total distance of 0.5 m from when it first leaves \(O\).