3 A light elastic rope has natural length 15 m . One end of the rope is attached to a fixed point O and the other end is attached to a small rock of mass 12 kg .
When the rock is hanging in equilibrium vertically below O , the length of the rope is 15.8 m .
- Show that the modulus of elasticity of the rope is 2205 N .
The rock is pulled down to the point 20 m vertically below O , and is released from rest in this position. It moves upwards, and comes to rest instantaneously, with the rope slack, at the point A .
- Find the acceleration of the rock immediately after it is released.
- Use an energy method to find the distance OA.
At time \(t\) seconds after release, the rope is still taut and the displacement of the rock below the equilibrium position is \(x\) metres.
- Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 12.25 x\).
- Write down an expression for \(x\) in terms of \(t\), and hence find the time between releasing the rock and the rope becoming slack.