We shall now analyze motion on an inclined plane with friction. Our purpose is to determine how we can use the inclined plane to measure the coefficients of friction.
First we consider the brick at rest. You can easily imagine that
if I increase the inclination angle there will be a critical
angle beyond which the block will begin to slide.
I will write the equations for the net force at this critical
inclination angle 
Again we choose the coordinate system with 
parallel to the incline
and resolving forces along the directions of the coordinate
system I get:
If there is no motion in the 
direction we have
We insert this in Eq. 1 to get
The condition that the brick just manages to stay static is
Similarly if we consider the brick in motion we get a condition for it
to be on the verge of stopping by requiring that 
under the
influence of kinetic friction vanishes. The critical angle, 
is again related to the corresponding coefficient of friction by
We perform the experiments to confirm that this way of measuring gives the same coefficients of friction as in the direct experiment using the spring scale.