# Newton's Laws question : On springs and pulley system

## How is this question ?

Poll closed Jul 21, 2011.

100.0%

0 vote(s)
0.0%
3. ### Ok

0 vote(s)
0.0%
1. Jul 11, 2011

### 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

2. Jul 12, 2011

### 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 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:
$$mgsin \theta +\mu (mg cos\theta)-kx_e=0$$
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 xe to get the condition for oscillation.