7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2c0bb9ea-31a6-42f1-9e2e-d792eee8fd10-11_517_254_278_845}
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\caption{Figure 4}
\end{figure}
A smooth hollow narrow tube of length \(l\) has one open end and one closed end. The tube is fixed in a vertical position with the closed end at the bottom. A light elastic spring has natural length \(l\) and modulus of elasticity \(8 m g\). The spring is inside the tube and has one end attached to a fixed point \(O\) on the closed end of the tube. The other end of the spring is attached to a particle \(P\) of mass \(m\). The particle rests in equilibrium at a distance \(e\) below the top of the tube, as shown in Figure 4.
- Find \(e\) in terms of \(l\).
The particle \(P\) is now held inside the tube at a distance \(\frac { 1 } { 2 } l\) below the top of the tube and released from rest at time \(t = 0\)
- Prove that \(P\) moves with simple harmonic motion of period \(2 \pi \sqrt { \left( \frac { l } { 8 g } \right) }\).
The particle \(P\) passes through the open top of the tube with speed \(u\).
- Find \(u\) in terms of \(g\) and \(l\).
- Find the time taken for \(P\) to first attain a speed of \(\sqrt { \left( \frac { 9 g l } { 32 } \right) }\).