Finding the frictional coefficient 
This is going to be very long question.
A 1 kg block situated on a rough incline is connected to a spring constant 100Nm as shown in fig. The block is released from rest with the spring in the unstretched position. The block moves 10cm down the incline before coming to rest. Find the coefficient of friction between the block and the incline. Assume the spring has a negligible mass and the pulley is frictionless. 
The way I tackled the question was by using forces acting at equilibrium. 
The forces acting on the block are:
1.The force due spring up the incline.
2.The force due friction up the incline .
3.Components of gravity.
$R=mg \cos 37$.
The force of friction $f$ will be up the incline
$f=\mu mg\cos 37$.
The component of gravity along the downward slope = $mg \sin 37$.
Putting it all together .
$mg\sin 37 - kx - \mu mg \cos 37 = 0$.
$mg \sin 37-kx = \mu mg \cos 37$.
Here this is get messed up, $\mu$ turns out to negative once you put the values in. If we change the direction of frictional force, the answer comes out to be 0.5.
But the real solution floating on the internet solves the question by using work done .
Work done by friction
 $-\mu xmg \cos 37$.
Work done by gravity $xmg \sin 37$.
Energy stored $\frac{kx^2}{2}$.
Therefore 
$xmg \sin 37 - \mu xmg \cos 37 = \frac{kx^2}{2}$.
Putting values gives the answer around 0.125.
Where did I went wrong? 
 A: The issue is that the problem does not specify if they want the coefficient of static friction or the coefficient of kinetic friction.
You have solved for the coefficient of static friction looking at when the block is in equilibrium. There is are two issues with this "solution", beyond the issue that, based on the solutions, the problem seems to be asking about kinetic friction. First, as you have found, the static friction force is in fact acting down the ramp. This is because at $10\space cm$ down the incline, the spring force is larger than the gravitational force. Therefore, the block would "want" to move up the incline, but if static friction is causing the block to be at rest, then it must be acting down the incline.
The second, larger issue is that static friction is not always equal to $\mu R$, it just cannot be greater than this value. Therefore, you hit a mistake by setting friction equal to this value, when you do not have enough information to say that. The only way you can determine the static friction coefficient is when an object transitions from rest to motion via a smooth change in applied force to the object. This way you know when static friction has reached its limit for you to find the static friction coefficient (i.e. you know that right at that transition the static friction force is actually equal to $\mu R$ rather than being less than that).
But based on the solution it seems like it wants you to find the coefficient of kinetic friction. For this you need to focus on the motion of the block before it comes to rest. This is best done using energy, as the solutions point out. You can use the fact that the elastic potential energy and work done by friction is going to be equal to the initial gravitational potential energy.
Since this is a homework-like problem I will leave the finer details to you. Good luck! (I would check your math on your final answer as well).
