7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c9d14ac-0757-4cdd-9534-337e6b3acee0-12_195_922_237_511}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
\(A\) and \(B\) are two points on a smooth horizontal floor, where \(A B = 5 \mathrm {~m}\).
A particle \(P\) has mass 0.5 kg . One end of a light elastic spring, of natural length 2 m and modulus of elasticity 16 N , is attached to \(P\) and the other end is attached to \(A\). The ends of another light elastic spring, of natural length 1 m and modulus of elasticity 12 N , are attached to \(P\) and \(B\), as shown in Figure 4.
- Find the extensions in the two springs when the particle is at rest in equilibrium.
Initially \(P\) is at rest in equilibrium. It is then set in motion and starts to move towards \(B\). In the subsequent motion \(P\) does not reach \(A\) or \(B\).
- Show that \(P\) oscillates with simple harmonic motion about the equilibrium position.
- Given that the initial speed of \(P\) is \(\sqrt { } 10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the proportion of time in each complete oscillation for which \(P\) stays within 0.25 m of the equilibrium position.