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Newton's Laws question : On springs and pulley system

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Poll closed Jul 21, 2011.
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  1. Jul 11, 2011 #1
    A Particle is attached by a string to a point in a rough inclined plane of elevation [itex]\theta[/itex]. Co-efficient of friction is [itex]\mu[/itex]. Originally, the string is unstretched and lay along the line of greatest slope.Prove that the condition for particle to oscillate is [itex]\mu[/itex] < 1/3tan[itex]\theta[/itex].
    Note : Tension in the string = [itex]\lambda[/itex].[itex]\Delta[/itex]l/l
    [itex]\Delta[/itex]l = Change in length
    l = Original length
     
  2. jcsd
  3. Jul 12, 2011 #2
    If friction steals away all the energy of the particle before it reaches at most an equilibrium position of [itex] x_e=\frac{mg(sin \theta +\mu cos\theta)}{k} [/itex], where xe 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:
    [tex]mgsin \theta +\mu (mg cos\theta)-kx_e=0 [/tex]
    Then solve:
    [tex]\mu (mg cos\theta) x_e<\frac{1}{2}k (x_e)^2+mg (x_e sin \theta) [/tex]
    by substituting in the expression for xe to get the condition for oscillation.
     
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