How much yarn has to hang

A criterion is only useful if somebody can see whether it is satisfied. The condition for a twisted thread to stay straight is a torque against the square root of a bending rigidity times a tension, and every quantity in it is in units nobody carries around.

Converting it into a unit people do carry around turns it from a formula into an observation, and the conversion is one division.

The unit #

A yarn’s weight is its count: twenty tex is twenty grams a kilometre, which is twenty milligrams a metre.

So a force can be expressed as a length. The tension a hanging thread supplies at a point is the weight of everything below that point, and asking “how much tension does this yarn need” is the same as asking “how much of itself has to be hanging under it”.

That is a unit anybody can picture. Two metres of thread is a long arm’s span and a bit; six hundred metres is most of a reel.

The two curves #

The critical tension goes as the square of the twist, and it goes as the square of the torsional rigidity over the bending rigidity, so it reads the bracket every force here carries exactly once. So it carries the stiffness bracket once, not twice — because the torque goes as C and the criterion as the square root of B, and C² over B is one factor of the bracket rather than two.

One factor of the bracket is three hundred and twenty-seven, and the two curves are that far apart at every twist level.

At the free bound — fibres sliding, each resisting on its own — a twenty tex cotton at eight hundred turns a metre needs 0.37 millinewtons, which is 1.9 metres of itself.

At the coherent bound — fibres locked, the yarn twisting as a solid rod — it needs 121 millinewtons, which is 616 metres.

Which one a person sees #

Nobody needs an instrument to tell those apart.

Hold a metre of sewing thread by both ends and bring the hands to half a metre. It snarls. Under the free bound that is expected: half a metre is well under the two metres it needs.

Hang the same thread from a shelf two metres up with nothing on the end. It does not snarl. Under the free bound that is expected too, and it is the more informative half: it is the observation that says the threshold is somewhere near two metres rather than near two centimetres.

Under the coherent bound the second observation is impossible. A yarn needing six hundred metres of itself would snarl on every reel, in every hand, at every length anybody has ever handled — including on the spinning frame, where it would be unspinnable.

So the everyday observation is not merely consistent with the free bound. It refutes the coherent one.

Why the count moves it the way it does #

A coarser yarn needs a larger force and a shorter hanging length, which is worth pausing on because the two move in opposite directions.

The critical tension goes as the fourth power of the yarn diameter, near enough, because both rigidities do. The linear density goes as the square of the diameter. So the hanging length — force over weight per unit length — goes as the square of the diameter, and a coarser yarn should need more of itself hanging.

It does not, and the reason is the twist. Coarse yarns are spun at lower twist per metre than fine ones, because the twist factor rather than the twist is what is held roughly constant, and the twist factor is the twist times the square root of the count. So a sixty tex yarn at the same twist factor as a twenty tex one carries about forty per cent fewer turns a metre, and the threshold goes as the square of that.

The two effects partly cancel and the surviving dependence is mild, which is why the observation is robust: it does not much matter what count of thread anybody picks up. The twist factor doing the work here rather than the twist is the same group that decides a yarn’s surface angle.

Why the fibre moves it a great deal #

The fibre moves it much more than the count does, and it moves it through the ratio rather than through the rigidity.

The threshold goes as C² over B, which is B times the square of the stiffness ratio. The ratio runs from 0.044 for an aramid to 0.8 for a wool — a factor of eighteen, squared, is a factor of three hundred and thirty.

So a wool yarn at a given twist is enormously more lively than an aramid one, and a great deal of that difference is the ratio rather than the stiffness. That is a prediction with a familiar answer: wool is famously lively, wool yarns are steamed as a matter of course, and hand spinners talk about wool’s tendency to snarl in a way they do not about flax.

The square law, checked as an exponent #

The criterion says the threshold rises as the square of the twist. That is worth checking as a relation rather than at a point, because a relation that holds at one pair of twist levels holds for any monotone function.

Taking four twist levels — two hundred, four hundred, eight hundred and sixteen hundred turns a metre — and fitting an exponent between successive pairs gives two, to within a part in a thousand million. That is a closed form agreeing with itself, and its value is that it would catch a sign, an offset or a stray factor of the twist anywhere in the chain.

The degenerate case is checked too: an untwisted thread has no torque, so it needs no tension, and the criterion must return exactly nought. It does.

What this does not say #

It does not say a yarn under its critical tension is safe. The criterion is for a uniform rod under uniform tension. A thread hanging under its own weight has a tension that falls to nothing at the bottom, so the bottom of a hanging thread is always below threshold and always wants to snarl. What stops it is that there is nothing below it to coil with: a snarl needs two lengths of thread to wrap round one another, and a writhe is a curve winding about itself rather than about anything else.

That is why the failure in practice happens at a slack loop rather than at a free end, and why bringing two hands together is the reliable way to produce one. The observation to make is not “does a hanging thread snarl” but “how short can a hanging loop be before it snarls”.

And it does not say when a snarl comes out. Forming one is a buckling and clearing one is a contact-and-friction problem at much higher tension. The criterion is one-sided.

What the trade’s own test measures #

There is a standard test for yarn liveliness and it is a measurement of exactly this threshold, though nobody describes it that way.

Take a known length of yarn, hang a small weight on it, bring the two ends together so that a loop hangs free, and count how many turns the loop takes up before it stops. The count is quoted as turns per unit length and is used as a specification.

What that test is doing is finding where the loop’s own tension crosses the threshold. The loop is slack at the bottom and tensioned at the top; it wraps from the bottom upwards until the tension where the wrapping has reached is above critical; and the number of turns it has taken is a readout of where that point is.

So the trade has been measuring this since long before anybody wrote the criterion down for yarn, and the test’s units — turns taken up, per unit length, at a stated weight — are a disguised form of the same quantity. That is worth knowing for two reasons: it means the prediction here can be compared against an existing body of data, and it means the data was collected for a purpose that had nothing to do with settling a stiffness bracket.

Why nobody used it to settle the bracket #

The obvious question is why an existing standard test, run daily in every spinning mill, has not already answered the question this rung answers.

Because the test is used as a comparison rather than as a measurement. A mill runs it to check that today’s yarn is like last week’s, and the specification is a number nobody derives. There is no reason for anybody in that position to convert turns-taken-up into a critical tension, and no reason to convert a critical tension into a bending rigidity.

That is a recurring shape and this collection has run into it before. A cloth’s thickness is measured constantly and almost never compared with what a geometry predicts; an areal weight likewise. The measurements exist and the comparison does not, because the people making the measurements are answering a different question.

So the contribution here is not a new experiment. It is a new use for an old one, and the useful part is knowing which old one.

What was counted, and how #

The criterion is Greenhill’s and is not derived here. Into it go the collection’s own quantities: a bending rigidity at both bounds, a torsional rigidity that is the bending one times the fibre’s own 2G/E, and a twist rate in radians per millimetre from the turns per metre.

The conversion to a hanging length is a linear density from the count and a gravitational acceleration, and nothing else.

The plot sweeps six twist levels from two hundred to sixteen hundred turns a metre, which brackets everything the trade spins for ordinary cloth, and it draws both bounds at every point rather than a band, because a band invites a reader to imagine the answer is somewhere in the middle and the whole point of this rung is that it is not.

What happens between the two bounds #

A reader looking at the two curves will want to know what a yarn part-way between the bounds would read, and the honest answer is that the question is not quite well posed.

The bracket is not a probability distribution over stiffnesses. It is two calculations of the same quantity under two assumptions about whether the fibres slide, and a real yarn does not sit at a fixed point between them: it slides more when it is bent slowly, less when it is bent fast, more when it is wet, and less when it has been set.

So a yarn near the free bound in one deformation may not be near it in another. That is precisely the caveat this collection attached when it first wrote the bracket, and it applies with more force here than usual, because the criterion involves both deformations at once — the torque comes from twisting and the resistance from bending.

What the observation settles is therefore narrower than it looks. It says a yarn’s torsional and bending behaviour, taken together, in the combination C²/B, sits at the low end. It does not say the fibres are frictionless, and it does not say either rigidity separately is at its lower bound.

That is still a great deal more than the collection had. A factor of three hundred has been closed by somebody letting go of a piece of thread, and no laboratory measurement in the literature closes it.

Where the model stops #

A spun yarn’s torque relaxes, and a yarn that has been set has none at all — which is a separate rung.

A spun yarn’s torque relaxes. The criterion assumes the torque is a constant times the twist rate for ever. A real yarn’s fibres slip, so its residual torque falls over minutes to hours, and a yarn left overnight is measurably less lively. A steamed yarn is less lively still and permanently so, and steaming is exactly the process of letting the torque go without changing the twist.

That is a large omission and it works in one direction: the model over-predicts liveliness for any yarn that has been sitting. So the observation should be made on fresh yarn, straight off the package, which is when the trade complains about it too.

And the fibres are treated as straight and parallel to the axis. They are at a helix angle, and a fibre at a helix angle contributes to both rigidities in a way neither bound captures exactly. That approximation is the collection’s own and biases both bounds in the same direction, which is part of why the discrimination survives it: a factor of three hundred is not closed by a helix correction.

The shear modulus is a working value with a range. Cotton’s runs over a factor of two, so the free-bound curve is uncertain by a factor of four in tension and by the same in hanging length: two metres could be one or four. It could not be six hundred.

The generalisation #

The rung is a worked example of a habit worth having, and the habit is about units rather than about yarn.

A criterion in the wrong units is not an observation. Nought point three seven millinewtons is a number that can be computed, printed, checked and forgotten. Two metres of the yarn’s own weight is a number that can be disagreed with by somebody holding a reel, and a number that can be disagreed with is worth several that cannot.

The conversion cost one division and it changed the status of the result completely: from an arithmetical consequence of two bounds to a discrimination between them.

This collection has other results sitting in units nobody carries. A contact force of 6.8 millinewtons a stitch is one. A through-thickness pressure of 15.1 kilopascals is another — though that one was converted, to 113 millimetres of mercury, and became immediately comparable with a medical compression class, which is what made what a cuff presses with an argument rather than a table.

The rule is: convert every threshold into the unit somebody would notice it in, and see whether it then agrees with what they notice.

The observation, written out as an experiment #

It is worth setting out the test in the form somebody could actually run, because it costs a reel of thread and five minutes and it settles something this collection has carried as an unresolved bracket from its earliest work.

Take a fresh spool of sewing thread of known count — a twenty tex cotton is ideal because it is what the numbers above are computed for. Do not use anything that has been sitting on a shelf for a year, because the torque relaxes.

Hang a length of it from a fixed point with a small weight on the end, enough to keep it straight. Then raise the bottom end slowly, letting the thread go slack from the bottom up, and watch for the point at which a coil starts.

The free bound predicts that a coil starts when about two metres of thread is above the slack point. The coherent bound predicts that no arrangement anybody can build in a room stays straight at all.

The result is not in doubt and that is the point. The experiment is worth describing not because anybody needs to run it but because the prediction was made by a chain of arithmetic that had never been asked to produce an observable, and the observable it produced turns out to be one everybody has already made.

Who found it, and when #

Greenhill’s criterion is from 1883. Yarn liveliness has been measured in the trade by a standard test for decades — hang a loop of known length and count how many turns it takes up — and that test is a measurement of exactly this threshold, though it is not usually described as one.

What is this collection’s own is the conversion and the discrimination: putting its own bracket through the criterion, expressing both ends as a hanging length, and observing that one of the two is refuted by anybody who has ever held a piece of thread.

Where the ladder goes next #

The threshold is one of two things the instability predicts, and it is the less sharp of the two, because it depends on a shear modulus with a factor of two on it.

The other is the size of the coil, and the size of the coil does not depend on either stiffness at all. That makes it the cleanest test in this work and the one that needs only a ruler: a snarl comes in one size.