Coulomb Friction Force

Leonardo da Vinci wrote it down around 1493, Guillaume Amontons rediscovered and published it in 1699, and Charles-Augustin de Coulomb tested it thoroughly enough in 1785 that his name stuck to it. Friction force is proportional to normal force, independent of apparent contact area, and roughly independent of sliding speed. Three centuries later it is still the first thing anybody calculates about a sliding contact, and it is still an approximation rather than a law of nature.

The independence from apparent area is the part that feels wrong, and understanding why it is true is the most useful thing on this page. Two nominally flat surfaces do not touch across their faces — they touch at the tips of their asperities, and the REAL contact area is a small fraction of the apparent one, often a thousandth. Those junctions carry the whole load, so they are stressed to the material's indentation hardness and deform plastically, which means the real area adjusts itself until $A_r = W/H$. Friction comes from shearing those junctions, so $F = A_r \tau = W\tau/H$, and the ratio $\tau/H$ is what we call $\mu$. Put a block on its side and the apparent area changes; the real area does not, because the load did not. That is the whole explanation, and it was worked out by Bowden and Tabor in the 1940s, a century and a half after the law they were explaining.

Knowing the mechanism tells you exactly where the law fails, which is more valuable than knowing the law. It requires plastic asperity contact, so elastomers break it — a rubber tyre or a rubber seal has a friction that depends on apparent area and on speed, because its contact is elastic and its losses are viscoelastic. Very smooth surfaces in intimate contact break it, because when the real area approaches the apparent one, adhesion takes over and friction rises steeply — which is why gauge blocks wring together and why polishing a sliding pair is not automatically an improvement. And any contact where a film rather than metal is being sheared — a lubricated bearing, a PTFE-lined bushing — is governed by the film, not by this equation.

The most damaging error in practice is treating $\mu$ as a material property. It is not. "The friction coefficient of steel" is a sentence with no referent. There is only the coefficient of THAT steel against THAT counterface, at that finish, in that condition, at that temperature and humidity, at that speed, with whatever film happens to be on it. Steel on steel, steel on bronze and steel on PTFE are three unrelated numbers. The same clean pair, oxidised, oiled, wet, or carrying a fingerprint, is several more. Contamination alone routinely moves $\mu$ by a factor of two, and it moves in the direction that hurts: an oily film where none was designed lowers the coefficient, and the clamp that was holding a load by friction lets go.

Static and kinetic friction are different numbers, and this page will use whichever you give it, so decide first which question you are asking. "Will it move?" is static; "what does it cost while it moves?" is kinetic. Breakaway is generally higher, often by 10 to 40 %, and the gap grows the longer the contact has sat still, because the junctions creep and grow under load. That difference is not a nuisance — it is the mechanism of stick-slip, the juddering that afflicts slow machine slides, brakes and violin strings alike. A contact whose static coefficient exceeds its kinetic one, driven through a compliant drive, cannot slide smoothly at low speed; it grabs, releases, and grabs again. The cures all attack the same gap: a lubricant whose $\mu$ RISES with speed, a stiffer drive, or a PTFE-loaded way material chosen precisely because its two coefficients are nearly equal.

Measure your own where it matters. The tilting-plane method needs no instruments: raise an incline until the specimen slides, and $\mu_s = \tan\theta$. For the kinetic value, pull with a spring scale and read the force while it is moving rather than the peak as it breaks away — reading the peak is how a kinetic coefficient ends up overstated by a third. Record the counterface, the finish, the lubricant, the load, the speed and the temperature beside the number, because a bare coefficient in a notebook will be misapplied within the year.