4. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of length \(a\) and modulus of elasticity \(\frac { 1 } { 2 } m g\). The other end of the string is fixed at the point \(A\) which is at a height \(2 a\) above a smooth horizontal table. The particle is held on the table with the string making an angle \(\beta\) with the horizontal, where \(\tan \beta = \frac { 3 } { 4 }\).

1. Find the elastic energy stored in the string in this position. The particle is now released. Assuming that \(P\) remains on the table,
2. find the speed of \(P\) when the string is vertical. By finding the vertical component of the tension in the string when \(P\) is on the table and \(A P\) makes an angle \(\theta\) with the horizontal,
3. show that the assumption that \(P\) remains in contact with the table is justified.