Why does the coefficient of friction has no units?

The coefficient of friction (COF), often symbolized by the Greek letter µ, is a dimensionless scalar value which describes the ratio of the force of friction between two bodies and the force pressing them together. This is a simple empirical model, which happens to do a reasonably good job in many situations--typically static cases or low velocity cases involving contact between two dry, solid surfaces.

The coefficient of friction depends on both of the materials used; for example, ice on steel has a low coefficient of friction, while rubber on pavement has a high coefficient of friction. Steel on brass will not necessarily have the same COF as steel on polished wood, etc. Coefficients of friction range from near zero to greater than one. A negative coefficient of friction would not make sense as it would generally lead to violations of energy conservation.

##F_f =\mu F_n##

##F_f## is the force of friction exerted by each surface on the other. It is parallel to the surface, in a direction opposite to the net applied force. ##\mu## is the coefficient of friction, which is an empirical property of the contacting materials, ##F_n## is the normal force exerted by each surface on the other, directed perpendicular (normal) to the surface.

If the units of both ##F_f## and ##F_n## are newtons, then µ can not have any units!