In order to "prove" that rope "works"--i.e., it doesn't fall apart under normal operating conditions--it would be nice to have a simple mathematical model. Here's my first attempt at a "Smith-type" model of rope, in which the tension on the rope itself acts to make sure that the parts of the rope stick together ever more tightly. I've seen cheap clothesline rope that is constructed in the following fashion. Take a yarn made up of a large number N of _straight_ fibers, but the fibers have a particular _mean length_ L, so that within any particular 1-inch length of this yarn, there will be some number of fibers that begin, and an approximately equal number of fibers that end. Surrounding this yarn is a woven outer _sleeve_, which is very compliant in the longitudinal direction. The key point about this sleeve is that this sleeve _carries no load_ itself, but when it is lengthened, it _compresses_ the yarn inside. This lateral compression presses the fibers of the inner yarn together so that the sliding _friction_ of these yarn fibers against one another carries the load of the rope. We now take this model to its simplest logical conclusion: The inner "yarn" now consists of exactly N=2 _straight_ fibers at any point along the rope; each fiber segment is of length exactly L, and the breaks between fiber segments occur at exactly the mid-point of the other fiber segment. The 2 yard fibers are thus arranged to look like this: ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- Obviously, there is nothing whatsoever to hold this 2-fiber yarn arrangement together until the outer sleeve is place around this yarn. Consider now that the sleeve is in place. The sleeve carries no longitudinal force, but any longitudinal force on it is immediately translated into a linearly proportional lateral--e.g., _normal_--force pushing the 2 yarn fibers together. We now have a classic situation in high-school physics (see Wikipedia): "The elementary properties of sliding (kinetic) friction were discovered by experiment in the 15th to 18th centuries and were expressed as ... empirical laws: "Amontons' First Law: The force of friction is directly proportional to the applied load. "Amontons' Second Law: The force of friction is independent of the apparent area of contact." http://en.wikipedia.org/wiki/Friction There is a coefficient of friction which characterizes the friction between our 2 yarn fibers parameterized by the numerical value of the normal force. When the 2 fibers aren't slipping against one another, this friction will tend to be higher--i.e., it is "stiction". http://en.wikipedia.org/wiki/Stiction We can now put together the various pieces of the model. When a longitudinal load is applied to the rope, it lengthens the sleeve while pulling apart the various fibers. Until the lateral force reaches a certain level, the rope will lengthen _without resistance_. However, beyond a certain stretch of the sleeve, enough lateral force is produced on the 2 fibers in the yarn for the friction to start opposing the longitudinal load. Eventually, the sleeve will produce enough lateral force for the friction between the yarn fibers to stop any movement, at which point the fibers themselves will be stretched according to their Young's modulus. The rope then stretches as if it were a single long fiber until it breaks at one of the transition points between one fiber segment and the next fiber segment. The problem with having only 2 fibers/yarn is that the effective strength is halved. However, if we cleverly arrange N fibers so that transitions between fiber segments never occur simultaneously, we can get an effective strength of (N-1)/N of the strength of a single fiber, at a weight of N fibers. Thus, a yarn of 100 fibers will have the strength of 99 fibers at a weight of 100 -- an efficiency of 99%. (This assumes that the sleeve is "free".) At 05:12 PM 10/26/2013, Warren D Smith wrote:
--Actually, the correct formulation of these "theorems" (for now conjectures) would involve the friction coefficients K for their rope surfaces and the "knot does not fail" claim would only be true if K > threshold (and threshold varies depending on the knot type). (The elastic constants of the rope material and the rope radius both seem irrelevant, kind of cancel out of the question.) Albeit the usual knots found in knot manuals evidently have quite safe thresholds versus realistic K values for actual rope materials. I think there are a few knot types which are not considered safe using slippery m(modern synthetic) rope materials but were considered safe using older, less slippery, rope materials like hemp.