4. One end \(A\) of a light elastic string \(A B\), of modulus of elasticity \(m g\) and natural length \(a\), is fixed to a point on a rough plane inclined at an angle \(\theta\) to the horizontal. The other end \(B\) of the string is attached to a particle of mass \(m\) which is held at rest on the plane. The string \(A B\) lies along a line of greatest slope of the plane, with \(B\) lower than \(A\) and \(A B = a\). The coefficient of friction between the particle and the plane is \(\mu\), where \(\mu < \tan \theta\). The particle is released from rest.

1. Show that when the particle comes to rest it has moved a distance \(2 a ( \sin \theta - \mu \cos \theta )\) down the plane.
2. Given that there is no further motion, show that \(\mu \geqslant \frac { 1 } { 3 } \tan \theta\).
   [0pt] [Question 4 Continued]