A particle \(P\) of mass \(4 m\) is attached to one end of a light elastic string of natural length \(l\) and modulus of elasticity 3 mg . 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 \(O A = \frac { 4 } { 3 } l\). The coefficient of friction between \(P\) and the table is \(\mu\).
Show that \(\mu \geqslant \frac { 1 } { 4 }\)
The particle is now moved along the table to the point \(B\), where \(O B = 2 l\), and released from rest. Given that \(\mu = \frac { 2 } { 5 }\)
show that \(P\) comes to rest before the string becomes slack.
(5)