Body in Equilibrium on a Horizontal Surface 

(Ok, maybe I should say that the forces acting on the body are in
equilibrium but let's not be too pedantic.) 


The surface needs to exert a force equal but opposite to the force due to gravity. 



This force is called the normal reaction. 

It can be considered to exist as a "reaction" to the force of
gravity pulling the body onto the surface and it acts at 90° to the
surface. 



Because it acts at 90° to the surface, this force has no
component acting parallel to the surface. 







Body in Equilibrium on an Inclined Surface 








If equilibrium
is to be maintained, the surface must continue to exert a
force which is equal but opposite to the force of gravity but
now this force has a component which acts parallel to the
surface. 
















It is often useful to resolve
the reaction force into two components, as shown here, one acting at 90° the
surface and one acting parallel to the surface. 



The component acting at
90° to the surface is still called the normal reaction. 



The
component acting parallel to the surface is called the force of friction
between the body and the surface. 







Similarly, we can resolve the force of gravity into two perpendicular
components. 




To maintain equilibrium: 



1. the magnitude of the force of friction
must be equal to the magnitude of the component of the force of gravity
acting down (parallel to) the slope 



2. the magnitude of the
normal reaction must be equal to the magnitude of the component of the
force of gravity acting at 90° to the surface. 





Therefore,
if the angle between the surface and the horizontal is
θ, the force of friction will have a magnitude given by 



and the
magnitude of the normal reaction is 





The force of friction acting between two (reasonably regular, non "sticky")
surfaces 

1.
opposes their relative motion 

2.
depends on the strength of the force pushing the two surfaces together
(indicated by the magnitude of the normal reaction) 

3.
does not depend* on the relative speed of the two surfaces 

4.
does not depend* on the area of the surfaces in contact. 



* There are, of course, an almost infinite number of things on which the force of
friction does not depend! These two are
mentioned because many people find them surprising... 



The Coefficients of Friction 

Experiments show that the magnitude of the force of friction acting between
two surfaces is directly proportional to the magnitude of the normal
reaction force. 

Therefore 



The constant of proportionality is called the coefficient of friction for
the two surfaces, symbol μ. 





The value of μ depends on the nature of the two surfaces (what
materials they are made of and how "smooth" they are). 

Be careful with this word, smooth. In physics, when we describe
surfaces as smooth, we mean that there is
absolutely no force of friction. 



Therefore, a surface which is described as smooth can only exert a force
at 90° to itself. 



Experiments (or just everyday observations) also show that, for any given
pair of surfaces, we need two different coefficients of friction, μ_{s}
the static (or limiting) coefficient, and μ_{d} the coefficient of
dynamic friction. 

It is usually more difficult to start two surfaces
slipping past each other that to keep them moving once they
have been started (even ignoring the effects of the inertia of the bodies). 



Conclusion: for two given surfaces 





Click here for worked example. 
