8 A particle P of mass \(3 m \mathrm {~kg}\) is attached to one end of a light elastic string of modulus of elasticity \(4 m g \mathrm {~N}\) and natural length 0.4 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.
- Find the distance OA .
At time \(t = 0\) seconds, P is given a speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downwards from A .
- Show that P initially performs simple harmonic motion with amplitude \(a \mathrm {~m}\), where \(a\) is to be determined correct to \(\mathbf { 3 }\) significant figures.
- Determine the smallest distance between P and O in the subsequent motion.