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.\Deltal/l
\Deltal = 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&lt;\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.