The rod that friction cannot decide about

The force that takes what it needs found that static friction has no value of its own: it supplies whatever equilibrium demands, up to a limit, and only at the limit does a coefficient of friction mean anything. Sliding friction is simpler. Once a contact slides, Coulomb’s law says the friction force is the normal force times a coefficient, opposite to the sliding, and that is the whole of it. Put that law together with Newton’s laws for a rigid body, and the motion of anything sliding on a floor ought to follow.

In 1895 Paul Painlevé, a mathematician who later became prime minister of France, pointed out that it does not always follow. For some bodies, at some angles, with enough friction, the equations have no solution — no motion at all is consistent with them. In other states they have two. The argument is a few lines long and has nothing exotic in it, which is why it was resisted for decades: either Coulomb’s law was wrong, or rigid bodies were impossible, or Newton’s laws were incomplete. The answer turned out to be the second, in a precise sense, and it predicts something that can be heard.

A contact that digs in when pressed #

Take a uniform rod of mass $m$ and half-length $l$, leaning at an angle $\theta$ above the floor, its lower end touching the floor and sliding along it. Two forces act at the contact: the normal force $N$ pushing up, and friction $\mu N$ along the floor, opposing the sliding. The question at each instant is how large $N$ must be.

It is decided by what the contact point would otherwise do. Without any contact force, gravity and the rod’s rotation would accelerate the lower end vertically by some amount $A$ — downward, into the floor, if the rod is falling. The normal force exists to stop that. Each newton of $N$ changes the contact point’s vertical acceleration by an amount $B$, and the floor supplies exactly the $N$ that makes the total zero:

$$\ddot y_c = A + B N = 0, \qquad N = -\frac{A}{B}.$$

For a frictionless floor $B$ is positive: pushing up on the rod’s end accelerates it upward, both directly and by turning the rod. Friction changes that. The friction force acts at the end, along the floor, and it twists the rod about its centre. If the end is sliding out from under the rod, friction pulls the end back towards the rod’s centre, and that twist drives the end downward. For a uniform rod

$$mB = 1 + 3\cos\theta\,(\cos\theta - \mu\sin\theta),$$

and the friction term can win. Then a larger normal force makes the contact accelerate into the floor. Pressing harder digs the end in further.

No answer, or two #

When $B$ is negative, the floor’s usual job becomes impossible. If the rod is falling onto its end — $A$ negative, the contact accelerating downward without help — the floor needs a positive $N$ to stop it, but every positive $N$ makes the downward acceleration worse. No $N \ge 0$ makes $\ddot y_c$ zero, and lifting off is impossible because the end is accelerating into the floor. The equations have no solution.

If instead the rod is lifting — $A$ positive — two answers are consistent. The end can leave the floor with $N = 0$, accelerating upward as $A$ says; or it can stay on the floor with $N = -A/B$, which is positive because both $A$ and $B$ have the wrong sign, the large normal force and the friction it brings holding the end down. Newton and Coulomb together do not say which happens.

The region in which $B$ is negative needs a high coefficient. Its boundary, $\mu = (\tan^2\theta + 4)/3\tan\theta$, has a minimum of exactly $4/3$ at $\tan\theta = 2$, a rod about 63° from the floor. Painlevé’s critics pointed out that a coefficient of friction above one is unusual, and concluded that the paradox was academic. The conclusion does not follow, for two reasons. Rubber on rough dry surfaces does exceed one, as do many soft materials on hard ones. And the threshold of $4/3$ belongs to a uniform rod. It depends on how the body’s mass is arranged.

A worked case makes the failure concrete. Take a uniform rod at 63° above the floor, with $\mu = 1.6$, its lower end sliding outward and the rod not yet turning. Then $\cos\theta = 0.454$ and $\sin\theta = 0.891$, the bracket $\cos\theta - \mu\sin\theta$ is $-0.972$, and $mB = 1 - 1.32 = -0.32$. Since the rod is not turning, the end’s free acceleration is simply $A = -g$, downward. The floor would need $N = -A/B = -3.1\,mg$ — a pull, not a push. A floor cannot pull, and a rigid floor cannot let the end through. Nothing in Newton’s laws, the rod’s rigidity or Coulomb’s law offers a third possibility. With $\mu = 1.0$ instead, $mB = +0.40$, the floor pushes with $2.5\,mg$, and the motion is perfectly ordinary: the jump from an ordinary motion to no motion at all happens between those two coefficients, at $\mu = 1.33$ for this angle, almost exactly the least value any angle allows.

The argument provoked a long and bad-tempered exchange in the German and French journals of the following fifteen years. Some took it as a refutation of Coulomb’s law, which had never claimed to be more than an approximation; some as evidence that rigid bodies with friction were inconsistent and should be abandoned; Ludwig Prandtl was among those who argued that the elasticity of the contact must settle what happens. Nobody disputed the arithmetic. What was missing was a calculation of what the elasticity actually does, and it took most of a century to make it cleanly.

Bodies that turn easily jam easily #

The twist friction produces matters in proportion to how easily the body turns about its centre, which depends on the ratio $\kappa = ml^2/I$ — the square of the distance from the centre of mass to the contact, divided by the square of the radius of gyration. A uniform rod has $\kappa = 3$. A dumbbell with all its mass at the two ends has $\kappa = 1$. A compact heavy body with a long, light arm reaching to the floor has a large $\kappa$.

The general threshold is $\mu > 2\sqrt{1+\kappa}/\kappa$. A dumbbell needs a coefficient near three; a body whose mass sits close to its centre, touching the floor at the end of a long light arm, needs only about a half. That is the geometry of a robot arm whose motors and gearboxes are near its base and whose fingertip drags across a table, of a pen or a chalk held near its tip and pushed across a surface, of a brake pad mounted on a lever. Each of them is a light lever with a contact at its end and most of its effective inertia elsewhere, and each can reach Painlevé’s region with ordinary friction.

A floor that gives a little #

The resolution is that no floor and no rod is rigid. Every real contact is a little compliant — the surfaces deform by micrometres under load, and the contact that slips before it slides followed the elastic give of two touching bodies that lets part of their contact slip before the whole of it slides — and with any compliance at all, the equations always have a unique solution, because the normal force becomes a function of how far the surfaces have pressed into each other rather than an unknown to be solved for. What the compliance does inside the paradox’s region is the interesting part.

Start the rod in the state where the rigid equations have no answer: falling onto its end, with the end sliding outward, at a friction coefficient above the threshold. Model the floor as a very stiff spring with a little damping, $N = k\delta + c\dot\delta$ for a small penetration $\delta$. The end begins to sink into the floor; the floor pushes back; and because $B$ is negative, the push drives the end further down rather than holding it up. The penetration grows, the force grows with it, and the growth feeds on itself — the end digs in harder the harder it is pushed — until the friction force, rising with $N$, stops the sliding altogether.

The stiffer the floor, the faster and higher the spike. The rod was not dropped; nothing struck it; it was sliding gently and falling under gravity. Yet the floor delivers to it a blow that, as the floor is made more nearly rigid, becomes a larger force for a shorter time.

The impulse that survives #

The limit is the key to the whole paradox. As the floor’s stiffness rises a thousandfold, the peak force and the duration of the jam change by large factors in opposite directions, and their product — the impulse, the total momentum the floor gives the rod — does not change at all.

In the rigid limit, then, the right description is a finite impulse delivered in zero time — an impact. The rigid-body equations failed because they insisted on finite forces, and the physically correct completion allows an impulsive force in a state where there was no collision in the ordinary sense. Léon Lecornu suggested exactly this in 1905, and it was put on a rigorous footing by Yann Génot and Bernard Brogliato in 1999, who called the phenomenon dynamic jamming: the contact jams, the sliding stops instantly, and the rod continues from the post-impact state, usually with its end stuck and the rod pivoting about it, or bouncing off the floor. Collisions are easier than forces argued that a collision is best described by the impulse it delivers rather than by the force during it; Painlevé’s paradox is a case where the impulse is the only thing that survives the rigid limit, and where a rigid theory without impulses is incomplete.

The impulse is not arbitrary. It is exactly the impulse that brings the sliding to a stop: the floor’s blow is whatever it takes to remove the contact’s sliding velocity, with the friction and normal components in the ratio $\mu$, and that fixes its size from the rod’s state alone. Unlike an ordinary collision, there is no coefficient of restitution to choose for the tangential direction — the contact ends up stuck — and in that sense the jam is more determinate than the bounces the bounces that add up to a stop followed, where the restitution is a property of the materials that the rigid theory has to be told.

The indeterminate case, with two consistent motions, is resolved the same way. With a compliant contact the two rigid solutions correspond to two different ways the compliance can respond, and which one the rod takes depends on details — the damping, the exact state — that the rigid model throws away. The paradox is a statement that rigid-body mechanics with Coulomb friction loses information that the motion needs, and that the information lives in how the contact deforms.

Slowing down against stopping dead #

The difference between ordinary friction and a jam is plain in how the sliding ends.

Below the threshold, the end slides, the friction decelerates it steadily, and it comes to rest gently — the straight-line decay of speed that the swing that dies in a straight line found for Coulomb friction. Inside the region, the end catches. The rod, with its end suddenly fixed, pivots about it, and a real rod — with its elastic bending and its bounce — can then leave the floor and land again further on, to repeat the cycle. That cycle, catch, pivot, hop and land, is one of the mechanisms proposed for the juddering of a stick of chalk pushed across a blackboard at a steep angle, which leaves a dotted line instead of a continuous one; and for the hopping of robot fingers dragged across a surface, which robotics engineers learned to avoid by keeping the angle and the friction outside the region.

The chatter a stiffer holder removes found a different source of jerky sliding, the stick-slip that arises when friction falls as sliding speeds up and a compliant holder stores and releases energy. The two are distinct: stick-slip needs friction that depends on speed and a soft holder, while Painlevé’s jam occurs with a constant coefficient and a rigid body, and is set by geometry and mass distribution. Both produce chatter, and in real devices both are often present.

What the rigid picture was hiding #

The deepest lesson is about what “rigid” means. Nothing is allowed to be rigid found that relativity forbids a rigid body because signals cannot travel through it instantly. Painlevé’s paradox is a much older and more mundane failure of the same idealisation: a rigid body with a frictional contact is a model in which the contact can transmit unlimited force without deforming, and when the geometry makes the force feed on itself, the model has nothing to say. The fix is not to abandon rigid-body mechanics, which works almost everywhere, but to recognise its boundary: wherever the contact’s acceleration responds to the normal force with the wrong sign, the deformation at the contact, however small, takes over.

The same boundary appears in the analysis of the ladder that lets go of the wall, where a ladder with friction at its floor end can, in some states, meet the same inconsistency; and in any mechanism with sliding joints — clutches, cams, the pivots of linkages — where engineers design so that the operating states stay away from the region. Painlevé’s own statement was a warning about the mathematics. It has become a design rule.

What the model leaves out #

The figures use a uniform rod in two dimensions, a single point of contact, Coulomb friction with one coefficient for both static and sliding contact, and a floor modelled as a spring with a little damping. Real contacts have friction coefficients that depend on speed and on the history of the contact, which change the region’s boundary slightly. Real rods bend, and their flexibility adds its own compliance in series with the floor’s, which lengthens the jam and lowers its peak. The jam’s impulse is the robust prediction: it is independent of the stiffness, as the figures show, and so of the details of the compliance. The peak force is not, and for a real contact it is set by the materials. In three dimensions the sliding direction is free to turn, and the friction force turns with it, so the region becomes a set of directions as well as angles; with two or more contacts, as in a block wedged between a floor and a wall or a drawer pulled by one handle, Painlevé-type inconsistencies combine with the purely static wedging that keeps a drawer stuck even when nothing moves, and separating the two takes care. The rod with one contact is the simplest system that shows the dynamic failure on its own, which is why it has remained the textbook case for a hundred and thirty years. The domain of the drawings is the outward-sliding case with the rod initially not rotating, at angles near the region’s tip.

Still open: how often real mechanisms visit the region #

That the rigid-body equations with Coulomb friction can fail is settled; how the failure is best resolved in numerical simulation is not. Simulators of robots, of granular materials and of machines with many contacts use rigid bodies because compliant contacts are slow to compute, and they must decide what to do when a state enters Painlevé’s region — typically by applying an impulse, and the choice of rule changes the predicted motion. Whether a given chatter in a real machine is Painlevé jamming, stick-slip, or a coupling of vibration modes is decided case by case, and experiments that isolate the jam cleanly — with stiff, well-characterised contacts and high-speed measurement of the force — are rare. The region’s geometry is exact; how much of the jerky sliding in the world lives inside it is not known.

The arithmetic underneath fits in a line. A rod sliding on a rough floor has a contact acceleration A + BN, and once friction twists the rod hard enough that B is negative — for a uniform rod at μ above $(\tan^2\theta + 4)/3\tan\theta$, never below 4/3 — the rigid equations give no motion or two; a floor that gives slightly makes the rod jam instead, with a force 163 times its weight on a stiff floor and an impulse that stays at $0.239\,m\sqrt{gl}$ however stiff. Coulomb’s law and Newton’s are not wrong; the rigid body is, at the one place where it matters.