- A particle \(P\) of mass \(m\) lies on a smooth plane inclined at an angle \(30 ^ { \circ }\) to the horizontal. The particle is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(2 m g\). The other end of the string is attached to a fixed point \(O\) on the plane. The particle \(P\) is in equilibrium at the point \(A\) on the plane and the extension of the string is \(\frac { 1 } { 4 } a\). The particle \(P\) is now projected from \(A\) down a line of greatest slope of the plane with speed \(V\). It comes to instantaneous rest after moving a distance \(\frac { 1 } { 2 } a\).
By using the principle of conservation of energy,
- find \(V\) in terms of \(a\) and \(g\),
- find, in terms of \(a\) and \(g\), the speed of \(P\) when the string first becomes slack.