The part of the wrap that is actually gripping

The force that takes what it needs derives the capstan equation and reads off the consequence everybody quotes: the tension ratio a wrap can hold is $e^{\mu\theta}$ , the drum’s radius does not appear, and three turns multiply a hand’s pull by a hundred.

That is a statement about a limit, and almost nothing spends its life at a limit. A belt drive carrying a quarter of the torque it could carry is not slipping; a mooring line holding a boat that is not straining is not on the verge of anything. The question this rung asks is what the wrap is doing in between — where along it the friction is being used, and what that costs.

1 turn of rope, holding 5 times what is pulled. A rope wrapped 1 turns round a fixed drum, with the slack end at the bottom and the loaded end where the arc ends. The rope is drawn at a radius that grows with the tension it is carrying, so the widening spiral is the tension itself. Each element of rope is pressed onto the drum by the two tensions at its ends, which are not parallel, and friction can hold at most µ times that press — so on the verge of slipping the tension multiplies by exp(µ dφ) for every element, and by exp(µθ) over the whole wrap. At µ = 0.25 and 1 turns that is 4.8, so a hand pulling with eighty newtons holds 385. Neither the drum's radius nor the rope's thickness appears anywhere in the result, which is why the same three turns work on a bollard a metre across and on a winch drum the size of a fist. — Fig. 1 One turn of rope round a fixed drum, drawn at a radius that grows with the tension it is carrying — so the widening spiral is the tension itself. The slack end is at the bottom and the loaded end where the arc closes. At a coefficient of 0.25 a single turn multiplies by 4.8, so eighty newtons of hand holds three hundred and eighty.

An advantage with no length in it

One feature of the limit is worth taking further than the earlier rung does, because it is what makes the rest of this essay’s arithmetic portable.

The exponent is $\mu\theta$ , and both factors are dimensionless: an angle and a ratio of forces. So the machine has no length in it anywhere, and that is unusual. Almost every other mechanical multiplication has a length in it — a lever’s advantage is a ratio of arms, a hydraulic press’s is a ratio of areas, a screw’s is a ratio of pitch to circumference. A machine whose advantage contains no length is a machine that can be scaled to any size without recomputing anything, and that is why the same rule of thumb — three turns and hold the tail — is taught for a ship’s bollard and for a hand winch.

What the numbers are

How much a wrap holds, against how many turns it is. The ratio of the two tensions a rope can hold across, against the number of turns it is wrapped, for coefficients of 0.15, 0.25, 0.4. The axis is logarithmic because the law is exponential, so each line is straight and its slope is the coefficient. At µ = 0.15 one turn multiplies by 2.6, two turns by 7 and three by 17 — so a person pulling with the strength of one arm holds a load that a small crane would be needed to lift. The practical consequence is the one a sailor states as a rule: turns are cheap and each is worth as much as the one before it, which is a statement about a constant factor rather than a constant force. — Fig. 2 The ratio of the two tensions a wrap can hold across, against turns, for three coefficients. The vertical axis is logarithmic because the law is exponential, so each line is straight and its slope is the coefficient. At 0.25 one turn gives 4.8, two give 23 and three give 111 — which is a hand’s eighty newtons holding nearly nine kilonewtons.

The reading that matters when deciding how many turns to take is not “more is better” but “each turn is worth the same factor as the one before it”. Going from two turns to three multiplies the holding power by five; going from five to six multiplies it by five again. A rope arrangement in which the load is ten times too large for the hand is fixed by two extra turns rather than by a stronger sailor — and by two, not by ten.

It also means the coefficient matters more than anything else, because it sits in the exponent. Dropping from 0.25 to 0.15 — which is what wet rope on a polished steel bollard does — takes three turns from 111 down to 17. That is a factor of six from a change nobody can see, and it is the reason mooring practice specifies turns generously rather than exactly. The same sensitivity in the other direction is why a rough drum or a rope with a grippy jacket transforms a marginal hold into a comfortable one.

The exponential form also explains why the capstan is one of the few machines that is safer than its rating suggests when overloaded a little and useless when overloaded a lot. There is no gradual yield: below the ratio the rope does not move at all, above it the rope runs, and the two states are separated by a factor a few per cent wide in load. Compare the same abruptness in a static friction force that supplies whatever is needed until it cannot, of which this is the wrapped version — the capstan equation is the saturated limit of that indeterminacy, and only the limit.

Half a turn is already useful

0.5 turns of rope, holding 3 times what is pulled. A rope wrapped 0.5 turns round a fixed drum, with the slack end at the bottom and the loaded end where the arc ends. The rope is drawn at a radius that grows with the tension it is carrying, so the widening spiral is the tension itself. Each element of rope is pressed onto the drum by the two tensions at its ends, which are not parallel, and friction can hold at most µ times that press — so on the verge of slipping the tension multiplies by exp(µ dφ) for every element, and by exp(µθ) over the whole wrap. At µ = 0.35 and 0.5 turns that is 3.0, so a hand pulling with eighty newtons holds 240. Neither the drum's radius nor the rope's thickness appears anywhere in the result, which is why the same three turns work on a bollard a metre across and on a winch drum the size of a fist. — Fig. 3 A rope over a branch: half a turn at a coefficient of 0.35, which is about what hemp on wood gives. The factor is only three, but three is the difference between lowering a load and dropping it. The tension spiral is much less dramatic than a full turn’s, and it is the same exponential, sampled over half as much angle.

The half-turn case matters more than the dramatic one because it is the case that occurs by accident. Any rope passing over anything is a capstan, and every such pass multiplies the tension a hand has to supply by a factor depending on which side the load is. A rope thrown over a branch to lower a weight is a factor of three in the operator’s favour on the way down and a factor of three against on the way up, which is the whole reason a pulley is worth carrying: a pulley turns the same wrap angle into a rolling contact and removes the exponent.

That is a useful way to see what a pulley is for. It is not there to change the direction of a force — a smooth post does that. It is there to make the friction irrelevant, so that the tension is the same on both sides and the mechanical advantage of a block and tackle is the count of rope falls rather than the count of falls modified by an unknown exponential. A seized pulley reverts to being a post, and a tackle with a seized sheave loses far more than the one sheave’s worth of friction, because the loss is multiplicative.

The belt that is not slipping

The capstan equation is a statement about the verge of slipping, and almost nothing spends its life on the verge of slipping. A belt drive carrying a quarter of the torque it could carry does not slip; nor does it grip everywhere with a tension that somehow adjusts itself. Something has to change the tension between the tight side and the slack side, and friction is the only thing available.

How much of a belt's wrap is actually slipping. The fraction of the wrap over which a belt creeps, against the fraction of its capacity it is being asked to transmit, at µ = 0.35 over 0.5 of a turn. A belt working below capacity does not grip everywhere and slip nowhere: the tension has to change somewhere along the arc, and friction is the only thing that can change it. So the wrap divides into an idle arc, where the belt rides at constant tension and does nothing at all, and an active arc where it creeps backwards over the pulley and the tension climbs at the full exponential rate. At a quarter of capacity the active arc is 37% of the wrap; at 75% it is 83%. It reaches the whole wrap only at capacity, which is the moment the belt lets go — so a belt that squeals is one whose active arc has just eaten the last of its idle one. — Fig. 4 How a belt divides its wrap when it is working below capacity. Along the idle arc the belt rides at constant tension and does nothing at all; along the active arc it creeps and the tension climbs at the full exponential rate. At a quarter of capacity the active arc is a fraction of the wrap; it reaches the whole wrap exactly at capacity, which is the moment the belt lets go.

The resolution is that the wrap divides into two arcs. Over the idle arc the belt sits at whatever tension it arrived with and no friction is called on. Over the active arc it creeps — moves slowly relative to the pulley — and there the friction is fully mobilised and the tension climbs at the exponential rate. The length of the active arc is set by how much tension change is demanded, and it grows as the torque does until it consumes the whole wrap. That last moment is capacity, and past it the belt runs.

Two practical facts follow, and both are the sort that get attributed to imperfection. A belt drive always loses a little speed, typically under a per cent, even in perfect condition and even far below capacity: the creep over the active arc is real relative motion, and the belt genuinely travels slightly slower leaving the driver than arriving. And a belt squeals just before it lets go rather than throughout its life, because the squeal is the active arc eating the last of the idle one and the transition is abrupt for the same reason the rope’s is. Neither is wear.

The wrap as a measuring instrument

Turned round, the equation is a way of measuring a coefficient of friction that is better than the standard one. The standard method is an inclined plane: raise the slope until the block slides, and the tangent of that angle is the coefficient. It works, and it has a poor signal-to-noise ratio, because the angle at which a block lets go has to be read to a fraction of a degree and the block sticks and releases erratically.

A wrap does the same measurement with an exponential amplifier in front of it. Wrap the material over a drum of the second material, hang a known weight on one side, and find the smallest force on the other that starts it moving. The ratio of the two is $e^{\mu\theta}$ , so the coefficient is the logarithm of a force ratio divided by the wrap angle. At three turns a one per cent error in the measured ratio becomes a hundredth of a per cent error in the coefficient — the exponential that makes the machine useful makes the measurement precise, because taking a logarithm divides the error by the exponent.

This is the standard method in the textile industry, where the quantity wanted is the friction of a yarn against a guide and the yarn is far too light for an inclined plane. It is also where the model’s limits bite hardest: a yarn wrapped over a small guide is bent sharply, so the bending stiffness contributes, and the contact pressure is high enough that the coefficient is not constant. The measurement is made at several wrap angles for exactly that reason, and a plot of the log ratio against angle that is not straight is the instrument reporting that the assumption has failed.

Who worked it out

The machine is much older than the equation. Capstans and windlasses are in Roman engineering and the practice of taking turns round a bollard is as old as tying up boats. What was not obvious, and had to be derived, is that the advantage is exponential rather than proportional — that adding a fourth turn to three does something quite different from adding a fourth man to three.

Euler wrote the relation down in 1762 and Eytelwein restated it half a century later, which is why it carries both names in the engineering literature and neither in most textbooks. The interesting thing about the date is what it is later than: Amontons had stated the friction laws in 1699 and Coulomb refined them in 1785, so Euler was working with a friction law that was itself new and not universally believed. The wrap was in part an argument for the law — a rule that predicts a specific and surprising number, testable with a rope and a weight, is a better advertisement than a block on a slope.

The same exponential, elsewhere

What friction returns, against what it is asked for. The friction force on a block under a 120 N normal load, against the force applied to it. Below 66.0 N — the static limit μs·N — friction returns exactly what is asked for and nothing moves, so the curve is the 45° line and the coefficient never appears. At that point the surface gives way and the force drops to μk·N = 50.4 N, where it stays however hard the block is pushed. The gap above the flat line is the surplus that accelerates it: 29.6 N at the right-hand edge of the axis. — Fig. 5 The law underneath all of it: friction supplies exactly what is asked of it, up to a ceiling set by the normal force, and then drops. The capstan is that ceiling integrated round a curve. Everything the wrap does comes from the fact that the ceiling is proportional to the press, and the press is proportional to the tension already there.

The mathematical reason the capstan is exponential is that the thing being limited is proportional to the thing being accumulated: the press at a point is proportional to the tension there, so the increment of tension is proportional to the tension, which is the definition of exponential growth. Any arrangement with that structure produces the same form, and there is a good example a short distance away in this collection.

A silo does not weigh what it holds for exactly this reason. Grain pushes sideways on the walls in proportion to the vertical pressure at that depth; the walls take friction proportional to that sideways push; so the vertical pressure grows more slowly with depth than it would in a liquid, and saturates. Janssen’s analysis of that is the capstan equation with a change of variable — an accumulation limited by something proportional to what has accumulated — and the consequence is the same kind of insensitivity to the obvious variable. A silo’s floor pressure stops depending on how deep the grain is, in the same way that a capstan’s factor stops depending on how big the drum is.

The habit is worth carrying: whenever a resisting force is proportional to the quantity it is resisting, the answer is an exponential and the obvious length scale drops out of it. That is also the shape of a wave losing a fixed fraction of its amplitude per unit distance and of a barrier whose transmission is decided in the exponent, and in all three the practical consequence is the same: small changes in the exponent are large changes in the answer, and the answer is insensitive to almost everything else.

Three turns, and why the number is three

The number in the rule of thumb is worth a paragraph because it is not arbitrary and it is not derived from a safety factor either.

What a sailor needs is a factor large enough that the tail can be held one-handed against any load the line will see, and small enough that the line can still be rendered — paid out under control — by easing the tail rather than by fighting it. Those two requirements pull in opposite directions and the exponential decides the compromise sharply. At an ordinary wet coefficient, two turns gives about seventeen: enough for a small boat, not enough for a ship, and easy to surge. Three gives about seventy: enough for most working loads and still controllable. Four gives three hundred, at which point easing the tail no longer eases the line, because the friction is holding it whatever the hand does, and the line becomes something that can only be let go entirely.

So the practical rule sits at the knee of an exponential, which is why it is a number rather than a range, and why the advice changes to “take another turn” rather than “pull harder” when the load rises. Pulling harder multiplies the answer by the ratio of the pulls; another turn multiplies it by five.

Where the model stops

The coefficient is assumed constant, and for rope it is not. Amontons’ rule — friction proportional to normal force, independent of area — is a good description of metal on metal and a poor one of fibre on metal at the pressures a wrap produces. A rope under three turns of load is being squeezed hard at the inner surface, the fibres flatten, the real contact area grows faster than the load, and the effective coefficient rises with tension. The measured holding of a real wrap is therefore usually better than the constant-coefficient calculation, which is a comfortable direction to be wrong in and not a reason to stop calculating.

Rope stiffness is ignored entirely. A real rope resists being bent, so some of the hand’s force goes into bending it round the drum and out again rather than into tension, and the loss is larger for a stiff rope on a small drum. That is the one place the drum’s radius comes back: not in the friction, but in the bending. It is why wire rope specifications quote a minimum drum diameter and why a wire rope on too small a drum fails at a fraction of its rated load.

The rope is assumed not to move. Everything here is a statics problem on the verge of motion. A rope running over a drum at speed introduces a centrifugal term that lifts it off the surface and reduces the press, so a fast-running belt holds less than a stationary one of the same tension — which is why belt drives have a maximum speed as well as a maximum torque, and the maximum speed has nothing to do with the material’s strength.

And nothing here says how the wrap behaves at the ends. The tension is drawn as though it were defined right up to the point where the rope leaves the drum, and near that point the rope is neither straight nor fully in contact. The correction is small for a wrap of a whole turn and is most of the answer for a wrap of a few degrees, which is why the model is useless for the light contact between a thread and a guide eye and excellent for a bollard.

What the pictures cannot show

The tension spiral draws a scalar at each angle and cannot show the direction of the friction, which is the thing that reverses when the load reverses. A wrap is not a ratchet: it holds a large tension against a small one in whichever direction it is asked, so the same three turns that let a sailor hold a ship also let the ship pull the rope through if the sailor’s end is the tight one. Which end is which is decided outside the figure entirely.

Nor do the figures show the pressure on the drum, which is the quantity an engineer needs. It is $T/R$ per unit length, so it grows along the wrap exactly as the tension does and it is inversely proportional to the radius — the one place the radius matters. A small drum with a big wrap crushes the rope even though the holding factor is unchanged, and that, rather than any friction argument, is what sets the minimum size of a winch.

Where the ladder goes next

The friction ladder began with a force that supplies exactly what is asked of it, went through the chatter a stiffer holder removes and the grip that is not a coefficient at all. This rung asks what a wrap does below its own limit, and finds that most of it is doing nothing. The rungs after it: the knot, where the wrap is a rope on itself and the normal force is generated by the tension being held; lubrication, where a film separates the surfaces and the coefficient stops being a property of either; and rolling resistance, where nothing slides anywhere and the loss is entirely in the material’s hysteresis.

The habit worth carrying away is about where to look for an exponential. When the resistance at a point is proportional to what has accumulated up to that point, the accumulation is exponential and the geometry drops out. The capstan, the silo and the absorbing medium are three subjects sharing one line of arithmetic, and recognising the line is faster than solving any of them again.

The part of the wrap that is actually gripping

An advantage with no length in it

What the numbers are

Half a turn is already useful