I’m a physicist with an interest in rock climbing, and recently I’ve been getting interested in the physics of the conditions under which friction is strong enough to make a knot hold. There is a huge mathematical literature on knot theory, but much less on actual physical knots and how friction makes them hold together. There is a little bit of published work on this topic – I’ve been able to find two papers, which I’ve referenced below for anyone who wants to look them up. (Neither is available online, unfortunately, except behind a paywall.) Both of these papers are almost 100% theoretical, and it seems to me that it would be worthwhile and fun to test whether their predictions are even correct.
Two predictions in the Maddocks paper are that a square knot fails if the coefficient of friction is less than 0.24, whereas the magic number for a sheepshank is 0.09. They also have some predictions about hitches. I’m posting because I’m interested in finding out if folks here can give me any pointers in trying to test this stuff, e.g., in how to locate cheap ropes with a variety of coefficients of friction.
To test the predictions, the materials I was able to come up with around the house were some nylon rope with a coefficient of friction of about 0.22 and some teflon plumber’s tape, which I twirled to form it into a thin cord. Teflon on teflon is supposed to have a coefficient of friction of about 0.04.
Square knots held with the nylon but failed with the teflon. Although this appears to be contrary to the prediction of the theory in the case of the nylon, I’m not sure what the margin of error is on my own determination of the coefficient of friction, and the paper clearly presents the prediction only as a rough estimate.
The sheepshank held with both the nylon and the teflon. This seems to clearly contradict the prediction, since the teflon’s coefficient of friction is less than half the claimed minimum.
It would be interesting if anyone could try similar tests with other materials. For example, dyneema (a.k.a. spectra) has more friction than teflon but less than nylon, so it would be interesting to see whether a dyneema rope (without a mantle of some other material) can hold a square knot. I see that there’s already a thread on whether a sheepshank should be expected to hold in dyneema: http://igkt.net/sm/index.php?topic=4072.0 Very few materials are slippery enough to make a square knot fail, so it might be interesting to find out whether other materials fail when made into weaker knots (e.g., a granny). There may also be knots or hitches such as a Munter that are not normally expected to hold with ropes made of ordinary materials, but that might hold, for example, when tied in a high-friction material such as a rubber o-ring. Mountaineers often have to work with icy ropes, and I wonder if it’s possible to get any useful insight into what knots would hold under such conditions. I’m not really sure that my tests with the teflon tape are fair, since the stuff doesn’t really behave like normal cordage. There does seem to be teflon-coated cordage commercially available, such as some thin line sold for ice fishing. Does anyone have any teflon-coated cord that they could try knotting?
In general, the calculations for hitches seem to be a lot more solid than the ones for knots, because in a hitch, the cylindrical post forms the rope into a predictable circular shape, whereas a knot controls its own complicated shape in three dimensions. Testing the calculations for hitches gets a little more complicated, though, because there are at least three variables: the coefficient of friction between the rope and the post, the coefficient for rope on rope, and the ratio of the rope’s diameter to that of the post. (For example, it’s easy to verify that a clove hitch always fails when the rope is too thin compared to the post, and the theoretical calculations get this fact right.)
Bayman, Am J Phys, 45 (1977) 185
Maddocks, J.H. and Keller, J. B., “Ropes in Equilibrium,” SIAM J Appl. Math., 47 (1987), pp. 1185-1200
