Two particles \(A\) and \(B\), of equal mass \(m\), are connected by a light elastic string of natural length \(a\) and modulus of elasticity \(4 m g\). Particle \(A\) rests on a rough horizontal table at a distance \(a\) from the edge of the table. The string passes over a small smooth pulley \(P\) fixed at the edge of the table. At time \(t = 0 , B\) is released from rest at \(P\) and falls vertically. At time \(t , B\) has fallen a distance \(x\), without \(A\) slipping (see diagram). Show that $$\ddot { x } = - \frac { g } { a } ( 4 x - a ) .$$ Deduce that, while \(A\) does not slip, \(B\) moves in simple harmonic motion and identify the centre of the motion. Given that the coefficient of friction between \(A\) and the table is \(\frac { 1 } { 3 }\), find the value of \(x\) when \(A\) starts to slip, and the corresponding value of \(t\), expressing this answer in the form \(k \sqrt { } \left( \frac { a } { g } \right)\). Give the value of \(k\) correct to 3 decimal places.