Newton's Laws question : On springs and pulley system

[The Stallion]

A Particle is attached by a string to a point in a rough inclined plane of elevation $\theta$. Co-efficient of friction is $\mu$. Originally, the string is unstretched and lay along the line of greatest slope.Prove that the condition for particle to oscillate is $\mu$ < 1/3tan$\theta$.
Note : Tension in the string = $\lambda$.$\Delta$l/l
$\Delta$l = Change in length
l = Original length

 

[RedX]

If friction steals away all the energy of the particle before it reaches at most an equilibrium position of $x_e=\frac{mg(sin \theta +\mu cos\theta)}{k}$, where x_e is the distance from the starting position of the particle and k is the spring constant, then there is no oscillation. The equilibrium position is gotten by solving:
$$mgsin \theta +\mu (mg cos\theta)-kx_e=0$$
Then solve:
$$\mu (mg cos\theta) x_e<\frac{1}{2}k (x_e)^2+mg (x_e sin \theta)$$
by substituting in the expression for x_e to get the condition for oscillation.