- The fixed point \(A\) is vertically above the fixed point \(B\), with \(A B = 3 l\)
A light elastic string has natural length \(l\) and modulus of elasticity \(4 m g\) One end of the string is attached to \(A\) and the other end is attached to a particle \(P\) of mass \(m\)
A second light elastic string also has natural length \(l\) and modulus of elasticity \(4 m g\) One end of this string is attached to \(P\) and the other end is attached to \(B\).
Initially \(P\) rests in equilibrium at the point \(E\), where \(A E B\) is a vertical straight line.
- Show that \(A E = \frac { 13 } { 8 } l\)
The particle \(P\) is now held at the point that is a distance \(2 l\) vertically below \(A\) and released from rest.
At time \(t\), the vertical displacement of \(P\) from \(E\) is \(x\), where \(x\) is measured vertically downwards.
- Show that \(\ddot { x } = - \frac { 8 g } { l } x\)
- Find, in terms of \(g\) and \(l\), the speed of \(P\) when it is \(\frac { 1 } { 8 } l\) below \(E\).
- Find the length of time, in each complete oscillation, for which \(P\) is more than \(1.5 l\) from \(A\), giving your answer in terms of \(g\) and \(l\)