8 A particle, of mass 10 kg , is attached to one end of a light elastic string of natural length 0.4 metres and modulus of elasticity 100 N . The other end of the string is fixed to the point \(O\).
- Find the length of the elastic string when the particle hangs in equilibrium directly below \(O\).
- The particle is pulled down and held at a point \(P\), which is 1 metre vertically below \(O\).
Show that the elastic potential energy of the string when the particle is in this position is 45 J .
- The particle is released from rest at the point \(P\). In the subsequent motion, the particle has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is \(x\) metres below \(\boldsymbol { O }\).
- Show that, while the string is taut,
$$v ^ { 2 } = 39.6 x - 25 x ^ { 2 } - 14.6$$
- Find the value of \(x\) when the particle comes to rest for the first time after being released, given that the string is still taut.