A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(l\) and modulus of elasticity \(3 m g\). The other end of the string is attached to a fixed point \(O\) on a rough horizontal table. The particle lies at rest at the point \(A\) on the table, where \(O A = \frac { 7 } { 6 } l\). The coefficient of friction between \(P\) and the table is \(\mu\).
Show that \(\mu \geqslant \frac { 1 } { 2 }\).
The particle is now moved along the table to the point \(B\), where \(O B = \frac { 3 } { 2 } l\), and released from rest. Given that \(\mu = \frac { 1 } { 2 }\), find
the speed of \(P\) at the instant when the string becomes slack,
the total distance moved by \(P\) before it comes to rest again.