1.
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\caption{Figure 1}
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A light elastic spring, of natural length \(L\) and modulus of elasticity \(\lambda\), has a particle \(P\) of mass \(m\) attached to one end. The other end of the spring is fixed to a point \(O\) on the closed end of a fixed smooth hollow tube of length \(L\).
The tube is placed horizontally and \(P\) is held inside the tube with \(O P = \frac { 1 } { 2 } L\), as shown
in Figure 1. The particle \(P\) is released and passes through the open end of the tube with speed \(\sqrt { } ( 2 g L )\).
- Show that \(\lambda = 8 \mathrm { mg }\).
The tube is now fixed vertically and \(P\) is held inside the tube with \(O P = \frac { 1 } { 2 } L\) and \(P\) above \(O\). The particle \(P\) is released and passes through the open top of the tube with speed \(u\).
- Find \(u\).