The points \(A\) and \(B\) are a distance 1.2 m apart on a smooth horizontal surface. A particle \(P\) of mass \(\frac { 2 } { 3 } \mathrm {~kg}\) is attached to one end of a light spring of natural length 0.6 m and modulus of elasticity 10 N . The other end of the spring is attached to the point \(A\). A second light spring, of natural length 0.4 m and modulus of elasticity 20 N , has one end attached to \(P\) and the other end attached to \(B\).

1. Show that when \(P\) is in equilibrium \(A P = 0.75 \mathrm {~m}\).
   The particle \(P\) is displaced by 0.05 m from the equilibrium position towards \(A\) and then released from rest.
2. Show that \(P\) performs simple harmonic motion and state the period of the motion.
3. Find the speed of \(P\) when it passes through the equilibrium position.
4. Find the speed of \(P\) when its acceleration is equal to half of its maximum value.