How do they calculate the coefficent of friction, both static and kinetic?

For static friction slowly tip a plane with a block on it. Record the angle it slid and compute static u.

The force of friction is

f=uN
N- normal to the surface force
u - coefficient of friction (static in this case say us)
So let the force of static friction fs=uN
and
N=mg sin(A)

A- angle of the incline
m - mass of the block
g - acceleration due to gravity

Also th eforce that forces the block down  F is
F=mg cos(A)

As soon as fs <= F the block  will slide
so since fs=usN=usmgcos(A)
and F=mg sin(A)

Then us= F/N= mg sin(A)/ mg cos(A)
us= sin(A)/cos(A)
us=tan(A)

Now dynamic friction. The relationship between forces is the same only now we are dealing with a dynamic friction.

fd=udN

and  fd is alway less then F (the downward force parallel to the sliding plane)

Ft= F-fd  

Ft=ma

S=.5at^2
by measuring the time and distance
a= 2S/t^2
then
ma= mg sin(A) - ud mg cos(A)
ud= m(a - g sin(A))/[mg cos(A)]
ud= (a-gsin(A))/g cos(A)
ud= (a/g)sec(A) - tan(A)  
where a= 2S/t^2