The fingers that always meet in the middle

Hold a metre rule level on two index fingers, one at the 10 cm mark and the other at the 60 cm mark, and slowly bring the hands together. Somebody trying this for the first time usually expects the fingers to slide equally, or the rule to slide off whichever side is pushed harder. Neither happens. One finger slides along the underside of the rule while the other stays fixed to it and carries it along. After a while the fingers swap, then swap again, more and more often, and they meet. They always meet at the 50 cm mark, under the rule’s balance point. Start them anywhere and push however unevenly, and the answer is the same.

It is a party trick, and nothing in it is mysterious. But it is also one of the cleanest demonstrations there is that static and sliding friction are two different numbers, and that the difference between them can steer a motion. The force that takes what it needs set out the rule that static friction supplies whatever force is asked of it up to a ceiling of $\mu_s N$, and that once sliding starts the force drops to $\mu_k N$. The rule under the fingers is that ceiling and that drop, applied twice, with the normal forces changing as the fingers move.

Which finger carries what #

The rule is a beam on two supports. Measure positions from the balance point, the centre of mass, and let the left finger be a distance $a$ to one side and the right finger $b$ to the other. Taking moments about each finger in turn gives the share of the weight $W$ each carries:

$$N_L = W\,\frac{b}{a+b}, \qquad N_R = W\,\frac{a}{a+b}.$$

The finger farther from the balance point carries less. That is the fact a see-saw teaches, and it is the whole of the geometry the problem needs. Nothing about the hands enters, and nothing about the rule except where its centre of mass lies.

Now start the hands moving. The rule is pulled two ways, by friction at the left finger dragging it right and friction at the right finger dragging it left. Something has to give, and what gives is the contact whose static ceiling is lower. Both coefficients are the same skin on the same wood, so the lower ceiling belongs to the finger carrying less weight: the farther one. It slips. The near finger holds, and the rule moves with it, with the far finger sliding along underneath.

The left finger in the picture starts 30 cm out and slides first. As it slides inward, its share of the weight grows, because it is getting closer to the centre, and the right finger’s share shrinks. The sliding finger is applying a friction force $\mu_k N_L$ to the rule, and the stationary finger has to balance it, which it can do only while that force stays below its own ceiling $\mu_s N_R$. When

$$\mu_k N_L = \mu_s N_R,$$

the holding finger reaches its limit and breaks loose. Substituting the shares, the condition is $\mu_k b = \mu_s a$, so the left finger stops sliding at

$$a = \frac{\mu_k}{\mu_s}\,b.$$

At that moment the roles reverse. The right finger is now the farther one in the only sense that matters, the one whose ceiling is the lower, and it starts to slide while the left finger takes hold. The same calculation, with the letters swapped, stops it at $b = (\mu_k/\mu_s)\,a$.

Why each turn is shorter #

Write $r = \mu_k/\mu_s$, which is less than one for nearly every pair of dry surfaces, since a contact that has started sliding grips less well than one that has not. Each stage ends with the moving finger at $r$ times the other’s distance. So the larger of the two distances is multiplied by $r$ at every change of roles, and the fingers close geometrically. In the picture, with $r = 0.75$, the first stage takes the left finger from 30 cm to 7.5 cm. The right finger then goes from 10 cm to 5.6 cm, the left from 7.5 cm to 4.2 cm, and so on down, the stair getting finer while keeping its proportions.

This picture is the same motion read as forces rather than positions. The solid line is what the sliding finger demands, the dashed line what the holding finger can supply, and every jump is a change of roles. Immediately after a change the new slider carries a little less than before, because it is the farther finger, and the new holder carries more, so the demand drops and the ceiling rises. Then the slide closes the gap between them again. The fingers are not being steered by anyone. The friction law alone decides which one moves, and it decides in favour of a pattern that cannot end anywhere but at the balance point, because the only state in which neither finger has the lower ceiling is the state in which both carry the same weight and sit at the same distance — zero, since the fingers have met.

That last remark is worth being careful about. The fixed point of the process is the point where the two distances are equal, and the hands are bringing the fingers together, so equal distances and contact can only happen together at the centre. If the coefficients were different at the two fingers, the meeting point would move: a dry finger and a damp one would meet off-centre, at the point where the two products $\mu_s N$ balance. With the same skin on both hands, they meet under the centre of mass.

The turns are a measurement #

The geometric shrinking means the number of turns is a logarithm. To close from 40 cm either side to 1 cm, the larger distance has to fall by a factor of 40, which takes

$$n = \frac{\ln 40}{\ln(1/r)}$$

stages, rounded up.

Read backwards, the curve turns the trick into an instrument. Mark where each stage stops, measure the two distances from the balance point, and their ratio is $\mu_k/\mu_s$ for those fingers on that rule. It is a cheap measurement of a quantity that usually needs a force gauge and a carefully controlled slide. It also explains what different surfaces feel like. A clean, dry, slightly rough contact, with a large gap between the two coefficients, gives a few long, distinct slides with a definite clunk at each change. A polished rule on oily skin, where the coefficients are close, gives so many tiny alternations that the fingers seem to glide in together, each one stuttering too finely to feel. The motion is the same in both cases. The difference is only how many stairs it takes.

The stutter is also a familiar sound. The chatter a stiffer holder removes followed stick-slip in a tool and a drawn bow, where a spring stores the force that a static contact holds until it breaks. Here there is no spring. The rule is the coupling, and the stored quantity is weight: each time a finger sticks, the normal force is shifting towards it, and each time it slips it is handing that weight back. The alternation is a stick-slip oscillation with a geometric amplitude envelope, and it ends not because the energy runs out but because the space does.

Infinitely many turns in a finite slide #

The logarithm has a curious consequence at the bottom of the stair. To close the last gap completely, from any distance to zero, takes infinitely many turns, since a geometric sequence never reaches zero in a finite number of steps. Yet the hands travel only a finite distance, the initial separation of the fingers, and they arrive. The infinitely many turns are packed into the final few millimetres, and they take the hands a finite time to cover at any steady speed.

The same structure appeared in the bounces that add up to a stop, where a ball losing a fixed fraction of its speed at each bounce makes infinitely many bounces in a finite time, the clicks speeding up into a buzz before it comes to rest. The fingers make the same buzz. The last centimetre of a real trial is not a sequence of distinct slides at all but a fine, rapid shuffle, felt as a vibration rather than as separate events. A fixed fractional loss per cycle is all it takes, whether the fraction is a coefficient of restitution or a ratio of friction coefficients, and the sum of the cycles is a geometric series in both.

What stops the series in practice is the finite size of things. A bouncing ball stops when its bounce height is smaller than the roughness of the floor, and the fingers stop when the remaining gap is smaller than the width of a fingertip. Neither process reaches its mathematical limit, and neither needs to, because by then the answer is already known.

Going the other way #

Run the hands apart and the experiment is not reversed at all.

The finger that breaks loose is still the one with the lower ceiling, the farther one. But now it slides outward, and as it does, its share of the weight falls further and the other finger’s rises. The sliding finger’s demand, $\mu_k N_L$, gets smaller while the holding finger’s ceiling, $\mu_s N_R$, gets larger, so the condition for a change of roles is moving away rather than closer. It is never met. The rule rides on the near finger, with the far finger sliding out along it, until the far finger reaches the end and the rule tips off.

There is a one-line way of saying the difference. Moving together, sliding makes the slider heavier, which is self-limiting. Moving apart, sliding makes it lighter, which is self-reinforcing. In the first case the system returns to sharing the work, and in the second it settles into one finger doing all of it. The same sign of feedback separates stable from unstable behaviour in a great many systems, and here it is visible in one’s own hands, with the difference between the two directions of a single motion.

A balance that finds its own point #

The meeting point is the centre of mass, and nothing in the argument needed the centre of mass to be at the middle of the rule.

Hang 100 g near one end of a 100 g rule and the centre of mass moves to 70 cm. Slide the fingers together from anywhere, and they meet at 70 cm. This is a practical tool. It is how a carpenter finds the balance point of a long plank or a fishing rod without putting it on a knife-edge, and how an archer can locate the balance of an arrow, where a centimetre’s shift changes how it flies. Nothing needs to be balanced to do it. The fingers find the balance point by a process that never passes through a balanced state on the way, and the friction law does the searching.

It also gives a quick way to weigh one thing against another. With a known mass hung at a known point, the meeting position fixes the rule’s own mass, and the other way around, using no more equipment than a ruler and two fingers. Where a hand applies a force through a contact it can hold by friction alone, the arithmetic of ceilings decides everything, as it did for a rope wrapped round a post in the turn that multiplies what a hand can hold. It is the same principle as the steelyard, the old market balance with a sliding weight on an unequal arm, except that the sliding is done by friction and the reading is wherever the fingers stop.

The connection to a heavy cupboard #

People who move heavy furniture use the same mechanics without thinking about it. A wardrobe too heavy to slide is “walked”: tilted onto one front corner, so that corner carries most of the weight and the other is light, the light corner swung forward, then tilted onto that one and the other swung. At each step the corner carrying the weight holds by static friction and the light corner moves. The fingers under the rule are doing exactly that, with the weight shifting by geometry rather than by a deliberate tilt. Whether a pushed cupboard slides or tips over in the first place is a separate contest between friction and the height of the push, which slide or topple settled; walking it is the way round both.

A related case is the problem the stack that leans past its own table took up, where the balance condition for each block involves only where the centre of mass of everything above it lies. The rule on two fingers is the same calculation in the other direction: given two supports, where does the load go? The answer depends on the distances to the centre of mass and on nothing else, which is why the result can be read off without knowing the rule’s length, mass or material.

What the picture cannot show #

Everything here is quasi-static. The slides are taken to be slow enough that the rule’s acceleration is negligible, so that at every moment the forces balance, and the change of roles is taken to be instantaneous. Neither is exactly true. When a holding finger breaks loose, the friction on the rule drops abruptly from its static to its sliding value, and the rule jerks; if the hands are moving fast, the jerk can carry the new slider a little further than the static calculation says before it settles. With quick, impatient hands the stair becomes ragged and the fingers can overshoot one another by a few millimetres. The meeting point is still the balance point, because the overshoots are corrected by later turns, but the exact staircase drawn here is a slow-hands result.

The two coefficients are treated as numbers. The grip that is not a coefficient explained why they are not quite constants: static friction grows with the time two surfaces have been in contact, and sliding friction depends on speed. Skin is worse than most surfaces in this respect. It is soft, it sweats, and its friction changes with pressure, so $\mu_k/\mu_s$ for a finger on a rule is not a property of the skin alone. What the figure calls 0.75 is a representative value, and a real pair of hands will drift during a single trial as the contact warms. The ratio read from the stopping distances is therefore the ratio for that trial, not a material constant.

The rule is treated as rigid and the fingers as points. A thin, springy rule sags, which moves the contact points slightly; a broad fingertip spreads its load over a centimetre or so, which smooths the final few stages. Below about a centimetre the idealisation breaks: the fingertips touch each other before the stair has finished, which is why the figures stop at a gap of a few millimetres rather than at zero. Rolling, rocking and the finger’s own compliance — which the contact that slips before it slides found lets a contact creep before it fully lets go — are left out.

And the argument assumes $\mu_k < \mu_s$. If the two were exactly equal, the turns would become infinitely many and infinitely short, and the fingers would slide together continuously. If sliding friction were the larger, which happens on a few lubricated or very soft contacts at low speed, the slider would never give way to the holder, and the first finger to move would slide all the way to the other. The meeting in the middle depends on the familiar inequality, and is a demonstration of it.

Still open: how a finger decides it has started to slip #

The arithmetic above treats the moment of slipping as a sharp threshold. It is not, for skin. Fingertips are full of sensors that respond to the tiny partial slips at the edge of a contact before the whole contact lets go, and the nervous system uses them to adjust grip force within a fraction of a second. That is how a person holds a glass firmly enough not to drop it and gently enough not to crush it, and it is studied closely in robotics, where artificial fingertips with comparable sensitivity are still being developed. A human finger under a sliding rule is doing that adjustment unconsciously the whole time. How much of the smoothness of a real trial comes from the friction law and how much from the hand quietly varying its pressure has not been measured carefully, and the rule-and-fingers trick, simple as it is, would be a good way to measure it.

Under a rule, the finger farther from the balance point carries less weight and slides, stopping when it is $\mu_k/\mu_s$ times the other’s distance, so the fingers take turns in stages that shrink geometrically — fourteen turns to half a centimetre at a ratio of 0.75 — and meet under the centre of mass wherever that is; pulled apart, the far finger only gets lighter, and it alone slides until the rule falls. The habit worth carrying is to ask what a motion does to the forces that permit it. When moving makes the mover heavier, the motion stops itself and a balance is found, and when it makes it lighter, the motion runs away.