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Re: [armsrforclimbing] Spring constant of a rope:
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rgold
Oct 8, 2008, 4:36 AM
Views: 13594
Registered: Dec 3, 2002
Posts: 1804
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Arms, the report you and others are referring to is an account I wrote up after explaining and re-explaining the various issues over and over. However, other than being re-read by me, it has not been refereed and, as far as I know, has never had its substance, as opposed to its conclusions, discussed online. I am honored to have it a called a "paper," but considering the absence of expert review, I think we should find a less lofty term, and as you can see I've settled on "report." I have tried to free said report of typos and mathematical errors, and certainly any that remain are my responsibility, but the conclusions have been obtained over and over again by many different people and are, in my estimation, reliable.
Relative stretch is, of course, defined by the equation y/L, there's nothing more to understand. The values taken on by y/L are, again by definition, what I called units of relative stretch. So when y=L, the equation becomes
T (in Kn) = k (in kN per unit relative stretch) X 1 (unit of relative stretch)
and there is no problem with any mismatch of units, and no problem with the meaning of the equation. But your point is well-taken, it would have been clearer for me to say that T=k kilonewtons, rather than saying that T=k.
Petsfed, there isn't a mathematical model in the world that incorporates every detail of the reality being modeled. If one did, it would in some sense be the reality and not a model of it. Perhaps the essential ingredient in every model is the fact that it ignores details, so your criticism is in fact a foregone conclusion. I'd love to see your model incorporating the mass of the rope, and I would be interested to see, after the mass is properly taken into account, how much effect it actually has on the results. Please do post up.
By the way, the fact that the real dynamical system is much more complicated than the model does not automatically disqualify the model from providing useful data. One has to demonstrate that the additional complexities, if they are incorporated, will significantly change the predictions. Assertions to this effect, without supporting arguments, are not evidence.
Not to be outdone in knocking what I wrote (which never pretends to be any more than it is, by the way), I should mention that the source of all deductions, Hooke's Law, is most likely just the linear part of a more accurate higher-order formula for rope tension, so defects are built into the model from the very beginning. And believe me, I could go on about other defects. Nonetheless, it is a deep misunderstanding of the pervasive role of mathematics in science and engineering to equate the fact that the model has ignored details of reality with an automatic claim that the model is invalid.
All this windy justification notwithstanding, I do think Petsfed is right to cite the complexities of reality in regard to the use of a simple model to calculate how far above your gear you can climb. Mtnrock, you can calculate the spring constant of the rope as I indicated in my report, but it will be of no use in your practical quest to know how far you can climb above your gear. Indeed, even within the realm of the theoretical model, that question has a conditional answer.
Edit: In the meantime, Arms has posted another comment.
It is true that the usual statement of Hooke's Law assumes a spring of fixed length, but that is not, and could not be the version I used in the report, where relative stretch (you could think percentage elongation if that makes it any clearer) has to replace the absolute change in length of a fixed-length spring. Look, there can be varying amounts of rope out in the system, so knowing that there has been two feet of stretch tells you nothing.
The version of Hooke's Law that I (and everyone else) used makes k the same constant regardless of how long the rope is. The x-axis in Arms' comment should be marked in percentages, not in linear units. Otherwise, it is Arms' formulation that subverts the intrinsic nature of k.
(This post was edited by rgold on Oct 8, 2008, 4:52 AM)
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Post edited by rgold
() on Oct 8, 2008, 4:52 AM
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