Two points \(A\) and \(B\) lie on a smooth horizontal table with \(A B = 4 a\). One end of a light elastic spring, of natural length \(a\) and modulus of elasticity \(2 m g\), is attached to \(A\). The other end of the spring is attached to a particle \(P\) of mass \(m\). Another light elastic spring, of natural length \(a\) and modulus of elasticity \(m g\), has one end attached to \(B\) and the other end attached to \(P\). The particle \(P\) is on the table at rest and in equilibrium.
Show that \(A P = \frac { 5 a } { 3 }\).
The particle \(P\) is now moved along the table from its equilibrium position through a distance \(0.5 a\) towards \(B\) and released from rest at time \(t = 0\). At time \(t , P\) is moving with speed \(v\) and has displacement \(x\) from its equilibrium position. There is a resistance to motion of magnitude \(4 m \omega v\) where \(\omega = \sqrt { } \left( \frac { g } { a } \right)\).
Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \omega \frac { \mathrm {~d} x } { \mathrm {~d} t } + 3 \omega ^ { 2 } x = 0\).
Find the velocity, \(\frac { \mathrm { d } x } { \mathrm {~d} t }\), of \(P\) in terms of \(a , \omega\) and \(t\).