Problem

One end of a light elastic string of natural length $a$ and modulus of elasticity $2mg$ is attached to a fixed point $A$ on a rough horizontal surface. The other end of the string is attached to a particle $P$ of mass $m$. The particle and string rest on the surface. The coefficient of friction between $P$ and the surface is $\mu$. The particle $P$ is initially held in equilibrium at a distance $\frac43a$ from $A$. The particle is then released from rest.

(a) Given that the string never becomes slack, find the minimum value of $\mu$.

It is now given that $\mu=\frac12$.

(b) Find the extension of the string when the particle comes to rest.