One end of a light elastic string of natural length \(a\) and modulus of elasticity \(2 m g\) 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 \(\frac{4}{3} a\) 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 \(\mu=\frac{1}{2}\). (b) Find the extension of the string when the particle comes to rest.

A light spring of natural length \(a\) and modulus of elasticity \(20mg\) stands vertically on a horizontal plane. The lower end of the spring is fixed to the plane. A particle of mass \(m\) is attached to the upper end of the spring.

The particle is pushed vertically downwards until the length of the spring is \(\dfrac{3a}{5}\). The system is then released from rest.

Find the maximum extension of the spring in the subsequent motion.

A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring of natural length \(a\) and modulus of elasticity \(5mg\). The other end of the spring is attached to a fixed point \(O\). The spring hangs vertically with \(P\) below \(O\).

The particle is pulled down vertically and released from rest when the length of the spring is \(\dfrac32a\).

Find the distance of \(P\) below \(O\) when \(P\) first comes to instantaneous rest.