The force that takes what it needs

A block sits on a table. A finger pushes it gently and it does not move. The finger pushes harder and it still does not move. Somewhere between one push and the next it goes, and after that it slides easily. Everything interesting about friction is in that sentence, and almost none of it is in the formula usually written down.

The formula $f = \mu N$ describes one point of that curve — the break — and one flat stretch after it. For the whole of the region to the left, which is where a table leg, a knot, a bolted joint and a hillside spend their entire existence, it is not merely inaccurate but the wrong kind of statement. Static friction is not a force with a value. It is a force with a range, and which value inside the range it takes is decided by whatever else is going on.

An unknown, not a datum #

The practical consequence is that a free-body diagram containing static friction has to be solved backwards. Every other force on the block can be written down from its own law — the weight from $mg$, a spring from its extension, a normal force from the geometry. Friction cannot. It is found the way an unknown reaction at a support is found: by demanding that the acceleration comes out right, and then checking afterwards that the answer is one the surface could have supplied.

On a 31° slope the component of weight along the surface is $mg\sin\theta$, fixed entirely by the geometry — and the friction force is whatever equals it. That is the point the essay turns on: $\mu N$ is not the friction, it is the most the friction can be. Below that ceiling the force takes whatever value is required and no equation predicts it in advance, which makes friction an unknown to be solved for rather than a datum to be looked up.

Two conditions are checked and only the second involves μ. The first is equilibrium: the friction needed is $mg\sin\theta$, whatever θ is. The second is whether the surface can supply it: the most it can offer is $\mu_s\,mg\cos\theta$. Setting demand equal to supply gives the angle at which the block lets go, and the mass falls out of both sides.

That $\theta_s = \arctan\mu_s$ is worth pausing on. It is the earliest and still the best way to measure a coefficient of friction, it works for a gram of powder or a tonne of rock, and it needs no force gauge. The whole apparatus is a hinge and a protractor. It is also the reason a scree slope, a heap of sand and a silo of grain all settle at a characteristic angle: the pile grows until its surface reaches the angle at which the top layer can no longer be held, and then it sheds.

The area that is not the area #

The most surprising thing about $f = \mu N$ is what is missing from it. There is no area. Two blocks of the same weight, one with a face ten times the size of the other, need the same force to start them sliding — a result that reads like an error and was published as one, by Amontons in 1699, after Leonardo had found it two centuries earlier and left it in a notebook nobody read.

Once the real area is known the rest follows. The junctions where the asperities meet are cold-welded, and shearing them takes a shear strength $s$ times the real area $A_r = W/H$. So $f = s\,W/H$, and the coefficient of friction is $s/H$: a ratio of two material properties, both of them about the softer surface, and with the geometry gone entirely. It also explains why μ is roughly constant over enormous ranges of load, why it is not constant at all for materials that do not deform plastically, and why a hard oxide film on a soft metal — a thin skin with low $s$ over a substrate with high $H$ — gives the lowest coefficients available.

This is a model, and it is worth naming as one. It assumes plastic junctions, which rubber does not have; for elastomers the real area grows less than proportionally with load and μ falls as the load rises, which is exactly why a wide tyre is used and why the folk explanation for it is wrong about the mechanism while being right about the outcome.

Where the energy goes #

A block dragged a metre across a table at constant speed has had work done on it and has no more kinetic energy at the end than at the start. The work went somewhere, and it went into the one place work goes when it stops being organised.

Sliding friction converts a single coordinated motion — every atom in the block moving together — into an enormous number of uncoordinated ones. The count of ways energy can be spread among the available degrees of freedom is what makes that conversion irreversible: there are overwhelmingly more arrangements with the energy scattered than with it aligned, and nothing forbids the reverse except how few of those arrangements it corresponds to.

Two consequences follow from that framing rather than from the formula. First, the heat is generated at the interface, in a layer microns thick, which is why brake discs glow at the surface while their centres stay cool and why the flash temperature at a single asperity can exceed the bulk melting point of the metal for a microsecond. Second, static friction does no work at all — the two surfaces are not moving relative to one another, so there is no distance for the force to act through.

The same loss in a single event is a collision with a coefficient of restitution below one: momentum conserved exactly, kinetic energy not. Friction is that loss spread continuously over a slide rather than concentrated in an impact, and the bookkeeping is identical — which is why a restitution coefficient and a friction coefficient are both empirical numbers standing in for the same unmodelled microphysics.

A car accelerating on dry tarmac is driven forward entirely by static friction, and none of the engine’s energy goes into it. The same is true of the force that holds a car on a banked curve: the tyre is not sliding, so the friction there is an unknown to be solved for, and the maximum speed of the turn is the speed at which the demand reaches the ceiling.

The gap that makes the noise #

Everything so far has treated the drop from $\mu_s$ to $\mu_k$ as an untidy detail. It is not a detail; it is a mechanism, and it is responsible for most of the noises the world makes.

Set $\mu_s = \mu_k$ in that calculation and the sawtooth vanishes; the block slides smoothly at the drive speed and nothing is heard. The gap is the whole mechanism. It is a violin string driven by a bow, a door hinge, a chalk squeak, a brake shudder, a wet finger round a wineglass — and, on a geological scale, an earthquake, which is a stick-slip event on a fault whose spring is the elasticity of half a continent.

The frequency of the sawtooth is set by the drive rather than by any resonance: a stiffer spring or a faster drive gives more slips per second. That is a testable difference from a resonance and it is how the two are told apart in the field. A brake that squeals at the same pitch however slowly it is applied is resonating; one whose pitch tracks the pedal is sticking and slipping.

Stick-slip read as a landscape is a stuck joint sitting in a local minimum with a barrier in front of it. The spring tilts the landscape until the barrier disappears, the joint slips, and it settles into the next minimum along — so the motion is a sequence of jumps rather than a slide, and the gap between the static and kinetic coefficients is what makes the barrier exist at all.

Rolling, which is a different question #

The friction at a rolling contact deserves its own paragraph because it is constantly confused with the sliding case, and because the confusion produces a sign error.

What makes a body roll rather than slide is static friction, and it does no work at all: the contact point is instantaneously at rest, so the force acts through no displacement. That is why three bodies of the same mass and radius arrive in an order fixed by their mass distributions and by nothing about the surface — the friction is essential to the motion and takes nothing from it.

Rolling resistance, which is a real and separate effect, has nothing to do with the coefficient of friction at all. It comes from the fact that a real wheel and a real road deform, and the deformation is not perfectly elastic: energy goes in at the front of the contact patch and less comes back at the rear. Its coefficient is one or two hundredths of the sliding one, it depends strongly on inflation pressure and on temperature, and it is proportional to the load in a way that has a quite different explanation. It also depends on speed, which sliding friction to a first approximation does not, and the dependence comes from how a material dissipates rather than from anything happening at the surfaces. A cyclist fighting rolling resistance and a skidding cyclist are up against different physics that happen to be described by similarly shaped formulae.

The rope round the post #

There is one arrangement in which a small friction becomes an enormous one, and it is worth deriving because the result is exponential and almost nothing in elementary mechanics is.

Wrap a rope round a fixed post and pull on one end. Take a short arc of the contact subtending an angle $\mathrm{d}\theta$: the tension on either side of it pulls inward with a resultant $T\,\mathrm{d}\theta$, which is the normal force pressing that piece of rope against the post. Friction there can supply up to $\mu T\,\mathrm{d}\theta$, so the tension can change by that much across the arc:

$$\frac{\mathrm{d}T}{\mathrm{d}\theta} = \mu T \quad\Longrightarrow\quad \frac{T_{\text{load}}}{T_{\text{hold}}} = e^{\mu\theta}.$$

The tension the rope can hold grows exponentially with the angle wrapped, and the numbers are severe. For a coefficient of 0.3, one full turn multiplies the holdable load by 6.6, three turns by 285, and five turns by twelve thousand. A person pulling with a hundred newtons holds more than a meganewton after five turns — which is why a single sailor at a bollard can hold a ship, and why a climbing belay works with one hand.

The most surprising feature is what is absent. The radius of the post does not appear. A thin capstan and a thick one hold the same load for the same wrap angle, because a thinner post gives a larger normal force per unit length over a proportionally shorter length, and the two cancel exactly. That is the same indifference to geometry that removes the area from the sliding law, arriving here for a completely different reason.

The result is the most used piece of friction physics there is. It sets the capacity of every belt drive — which is why a V-belt is wedge-shaped, since squeezing the belt into a groove multiplies the normal force and therefore acts as an increase in the effective coefficient. It is why a capstan winch works with a loose tail rather than a clamp. It is why a knot holds at all: a knot is a rope wrapped round itself, and the exponential is what makes the friction of a few centimetres of contact exceed the strength of the rope. And it is why a rope belay pays out smoothly when the tail is slackened and locks when it is pulled — the control is on the small end of an exponential.

Coulomb’s prize essay of 1785, mentioned above, was on exactly this: the friction and stiffness of ropes in machines, commissioned because the navy needed to know what its rigging would hold.

The angle that decides whether a machine stays put #

The slipping angle $\arctan\mu$ has a second use, and it decides whether a mechanism stays where it is put when the power is switched off.

A screw thread is an inclined plane wrapped round a cylinder. Its lead angle — how steeply the helix climbs — plays the part of the slope’s angle, and the same criterion applies: the load holds itself if the lead angle is below the friction angle, and runs back down if it is above.

The numbers make an ordinary bolt self-locking with room to spare. A coarse metric thread of eight millimetres’ mean diameter and 1.25 mm lead has a lead angle of about 2.9°, and dry steel on steel with a coefficient of 0.15 has a friction angle of 8.5°. The bolt does not undo itself under load, and it would need a lead three times as steep before it did.

The same criterion runs through a great deal of machinery. A screw jack holds a car up with no brake. A worm gear drives a wheel and cannot be driven backwards by it, which is why a worm-geared winch needs no ratchet. A wedge splitting a log stays in the log. A Morse taper holds a drill bit by friction alone, and is released by a wedge rather than by pulling. All of them are the block on the slope, and all of them work because the slope is shallower than $\arctan\mu$.

The price is efficiency, and the trade is exact rather than approximate. Working out the efficiency of a screw gives $\tan\lambda/\tan(\lambda+\phi)$, with $\lambda$ the lead angle and $\phi$ the friction angle, and putting $\lambda < \phi$ — the self-locking condition — into it gives an efficiency below one half, always. A mechanism that will not run backwards necessarily wastes more than half of what is put into it.

That is a genuine constraint and it is visible in what gets built. A screw jack is self-locking and about thirty per cent efficient. A ball screw, which replaces the sliding contact with rolling balls and reaches ninety per cent, is not self-locking, back-drives freely, and must be fitted with a brake — which is a straightforward statement that the brake is the price of the efficiency, and that the two cannot both be had from the thread.

What the picture cannot show #

The response curve at the top of this page has three properties that no real surface has.

The break is not sharp. Drawn as a corner, the transition from stuck to sliding is actually a gradual one: junctions fail a few at a time, and a joint under a load approaching its limit creeps, moving microns over hours. Bolted connections lose preload this way; so do glaciers. The corner is a limit of a smooth process, taken at a timescale that suits the drawing.

The coefficients are not constants. $\mu_k$ depends on sliding speed, usually falling then rising; $\mu_s$ depends on how long the surfaces have been in contact, because the junctions continue to grow by creep after they form. That second dependence is the reason a brake grabs harder after being parked for a week, and it is the parameter in the rate-and-state laws by which earthquake recurrence is modelled.

What the energy becomes is molecular agitation, and the temperature rise at a sliding interface is often the quantity that decides an engineering outcome. A brake fades when its friction material gets hot enough to change; a bearing seizes when the film in it boils. The coefficient is the least interesting number in most real problems, and the heat is the one that matters.

Nothing in the curve is reversible. The drawing shows a single-valued function; the real relationship is a history, and running the applied force back down does not retrace the path up.

The domain of validity is therefore narrower than the formula’s confident appearance suggests: dry, unlubricated, plastically deforming solids, at loads well below the point of gross deformation, at ordinary speeds, with surfaces whose state is not changing. Outside it — lubricated bearings, rubber, ice, anything at high speed, anything very clean in a vacuum — the coefficient is not a property of the pair of materials and cannot be looked up. Two atomically clean metal surfaces in high vacuum do not slide at all; they weld.

Three centuries of a law nobody could derive #

The history is unusually clean, because the experimental result arrived first and stayed put while every explanation of it was replaced.

Leonardo, some time before 1500, measured the force to drag blocks across a table and wrote down both of the results that matter: the force is proportional to the weight, and it does not depend on the area. The notebooks were unpublished for three hundred years. Amontons rediscovered both in 1699 and was disbelieved by the Académie, which is a reasonable response to a claim that a quantity obviously about surfaces does not depend on how much surface there is. Coulomb — the same Coulomb, twenty years before the inverse-square law that carries his name — settled the matter in 1785 with a prize essay on the friction of ropes and machinery, established that the kinetic coefficient is roughly independent of speed and smaller than the static one, and proposed the interlocking-roughness explanation that was believed for the next hundred and fifty years.

The interlocking explanation is wrong, and its failure is instructive. If friction were asperities riding up over one another, then polishing a surface would reduce it, without limit; instead, polishing reduces friction to a point and then increases it, and two optically flat surfaces of the same clean metal seize. It also predicts that all the work goes into lifting the block, which would be recovered on the way down: interlocking is conservative, and friction is not. The adhesion model of Bowden and Tabor, in the 1940s, replaced it — and it was they who first measured the real area of contact, by passing a current between the surfaces and reading the resistance of the junctions.

Why it is a hard subject rather than an elementary one #

Friction is taught in the first month of a mechanics course and is not a solved problem. The reason is worth stating plainly: it is the only force in an introductory syllabus that is not fundamental. Gravity, the electrostatic force and the tension in a string are laws or consequences of laws; $f = \mu N$ is a compressed summary of the collective behaviour of an unknown number of contacts between two rough surfaces of unknown history, and every parameter in it is a statistical average over things nobody has measured. In that respect it belongs with pressure and viscosity rather than with weight: a single number standing in for an enormous count.

That is why the coefficient of friction has no theoretical value, why tables of it disagree with one another by tens of per cent, and why the second significant figure of a quoted μ should be regarded with suspicion. It is also why the subject keeps producing surprises: superlubricity between misaligned graphite layers, where the coefficient drops below 0.001 because the two lattices never register; the fact that a coefficient can exceed one, and does for clean rubber on clean glass; and the discovery that the real contact area can be measured directly, by shining light through one of the surfaces and counting the bright spots.

The ladder from here #

Later rungs on this anchor: the adhesion model developed properly, with junction growth under combined normal and shear load; rate-and-state laws and the earthquake cycle; lubrication, from the boundary layer to the hydrodynamic film, where the surfaces stop touching and the whole apparatus above is replaced by the viscosity of the fluid between them; rolling resistance derived from hysteresis in the deforming material; wear as the loss of the welded junctions rather than of the bulk; and friction at the atomic scale, where a single asperity is a tip and the stick-slip has a period of one lattice spacing.

The neighbouring ladders are the free-body diagram, which is where the demand side of every problem here comes from, and entropy, which is what the work done against friction becomes and the reason the process only runs one way.