7 A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light elastic string of modulus of elasticity 24 mgN and natural length 0.6 m . The other end of the string is attached to a fixed point \(O\); the particle \(P\) rests in equilibrium at a point \(A\) with the string vertical.

1. Show that the distance \(O A\) is 0.625 m . Another particle \(Q\), of mass \(3 m \mathrm {~kg}\), is released from rest from a point 0.4 m above \(P\) and falls onto \(P\). The two particles coalesce.
2. Show that the combined particle initially moves with speed \(2.1 \mathrm {~ms} ^ { - 1 }\).
3. Show that the combined particle initially performs simple harmonic motion, and find the centre of this motion and its amplitude.
4. Find the time that elapses between \(Q\) being released from rest and the combined particle first reaching the highest point of its subsequent motion. \section*{END OF QUESTION PAPER}