I did a "real world example" physics homework problem last year, where they said a climbing rope's tension is not simply proportional to extension, as a 'physics spring' is. At the end of the problem, we ended up proving that the maximum force on a system is the same for all systems with the same fall factor (i.e. maxForce(4ft drop on 2ft rope) = maxForce(40ft drop on 20ft rope)).

Is this true, or did the problem oversimplify/misstate? If anybody wants the actual F(dx) equation that was in the book, I can try to find it.

...It would be nice if engineering reps from the big gear companies monitored this site to answer questions related to unpublished gear specs.

If ropes actually obeyed Hooke's law, then they would not need to "rest" after long falls, they would also act like bungee cords rather than semi-elastic cords.

In addition, were it not for the facts of the matter, posting a rope's "peak force" with its other descriptive info would be pointless. A linear spring doesn't have a "peak force" per se, nor could a general "peak force" be predicted.

I did a "real world example" physics homework problem last year, where they said a climbing rope's tension is not simply proportional to extension, as a 'physics spring' is. At the end of the problem, we ended up proving that the maximum force on a system is the same for all systems with the same fall factor (i.e. maxForce(4ft drop on 2ft rope) = maxForce(40ft drop on 20ft rope)).

Is this true, or did the problem oversimplify/misstate?

If ropes actually obeyed Hooke's law, then they would not need to "rest" after long falls, they would also act like bungee cords rather than semi-elastic cords.

No, it does not obey Hooke's law. Otherwise you would spring upward after coming to a stop in a lead fall.

In addition, were it not for the facts of the matter, posting a rope's "peak force" with its other descriptive info would be pointless. A linear spring doesn't have a "peak force" per se, nor could a general "peak force" be predicted.

A linear spring certainly has a peak force as a response to a specific load, and this is all that is asserted by the peak force data considered for climbing ropes.

http://www.marlow-ropes.com/...nager.cfm?page_id=21

However, the traditional damping term, which is proportional to velocity, is based on a damping mechanism consisting of a viscous fluid in a cylinder. Whether the mechanisms that damp rope oscillations are analogous to a classical car shock absorber is not addressed in their web presentation and seems to me to require some justification.

At the end of the problem, we ended up proving that the maximum force on a system is the same for all systems with the same fall factor (i.e. maxForce(4ft drop on 2ft rope) = maxForce(40ft drop on 20ft rope)).

A pure Newtonian fluid exhibits viscous behaviour. Mathmatically, this relationship looks very much like Hooke's law, except that it relates shear stress and shear strain, rather than Force and elongation.

In a fluid, shear stress is proprtional to the time rate of strain not shear strain!

For an elastic solid shear stress is proportional to shear strain but in a Newtonian fluid (let's call it a normal fluid) it is proportional to strain rate. This is one of the principal differences betwen a solid and a fluid.

If your end result was the invariance of maximum force for falls of the same fall factor, then your starting point was indeed Hooke's Law, which says that if you plot the tension in the rope against its percentage elongation, you get a straight line through the origin. The result that you quote is a consequence of Hooke's Law, not some "more realistic" replacement for it.


The actual graph is certainly not a straight line for all possible rope deformations, but is reasonably close for many ropes to a straight line for elongations in the working range of the rope, which is up to about 30%. All climbing ropes are engineered to have lower elongations at low loads (presumably by virtue of the internal friction of the core and sheath), and a number of the newer ropes that qualify for multiple UIAA ratings are engineered to have curves that are not straight lines, since the predictions based on the straight-line model do not agree very well with their published impact numbers.

Certain mathematical features of Hooke's Law are clearly not present in ropes. For example, there is no response to a contracting force. In other words, the tension-extension graph ends at the origin. Moreover, the near absence of a perceptible bounce and the relatively long recovery time all bespeak a damping effect. The Italian model uses the traditional differential equation for damped harmonic motion, which also depends on Hooke's Law but adds a damping term. However, the traditional damping term, which is proportional to velocity, is based on a damping mechanism consisting of a viscous fluid in a cylinder. Whether the mechanisms that damp rope oscillations are analogous to a classical car shock absorber is not addressed in their web presentation and seems to me to require some justification.

One can use models other than the linear one; the graph of a cubic equation T=kx^3, rather than a linear equation T=kx is the starting point for the analysis of what is called a "stiff" rope. But the idea that the model should involve an odd power of the extension x is based on the assumption that compression and extension should produce the same force magnitude, a fact that is certainly not true for ropes.

Perhaps the best way to think about Hooke's Law is that it is, in many cases, a very good linear approximation to the true tension-elongation curve. The fall-factor concept, which is a consequence of Hooke's law, has been confirmed in many tests and remains the fundamental concept in the testing and certification of ropes.

The primary real-world feature that is not accounted for in the analysis you did is the effect of rope friction against rock and through multiple carabiners. The Italian computer model is based on a system of differential equations for damped simple harmonic motion and attempts to account for frictional effects at each carabiner and can be adjusted for a certain amount of rock friction. But at the heart of those equations is....Hooke's Law.

english please there are 6th graders here thank you

From this data, if it is representative, we'd have to say kernmantle ropes don't obey Hooke's law very well and the linear spring model for the rope, which is used in a lot of simple analysis of fall forces etc., is not a particularly good model! This is a bit of a surprise to me as it seems to give reasonable results.

[..assumption of a damping force that accounts for the lack of bounce.

....which has the effect of emphasizing the apparent curvature.

No one would expect the curve over this range to be linear---Hooke's law certainly breaks down as the rope reaches its limit.

Moreover, the beginning sections of the curve are also not representative, since under low loads the damping effects of rope construction may well dominate the linear behaviour.

The middle section of the curve is the portion one would expect to exhibit Hooke's Law behavior, and it is the middle section, in the case of climbing ropes, that is involved in catching falls.