Calculus in Climbing

I'm supposed to write a one page paper about how calculus relates to the real world. Figured I'd write on calculus in climbing.

Do y'all have any ideas on how calculus might relate to climbing?

Happy Climbing!

HA! My second attempt at Calculus is coming up! I'll let you know when I pass!

I know it's obvious, but you could talk about falling objects (climbers) and acceleration, or "time release holds" and the change from static to kinetic friction. You could approximate the volume of your climbing helmet. Sounds geeky and fun.

how about this:

Chris Sheridan wrote:how about this:

Oh! Pleeeassseee explain that! I will learn so much better from one post on MP than hours in Calculus class!

Well, you could start with figuring out the weight of a climber with relation to the height of the ground. Figure out the rope weight with relations to your height off the ground. You could then factor in quickdraws, or if you want to get real fancy, trad gear. Rope drag would be a hassle, but its not impossible. Or, you could talk about force in a fall. Peak force, total force, and average force.
Good luck!

Here's a few to get started with (links are old):

Attaway, S. W.
Rope Systems Analysis
International Technical Rescue Symposium
Albuquerque, NM (1996)
lamountaineers.org/xRopes.pdf

---,
The Mechanics of Friction in Rope Rescue
International Technical Rescue Symposium
Fort Collins, CO (1999)
jrre.org/att_frict.pdf

--- and Weber, C.
Predicting rope impact forces using a non-linear force deflection.
International Technical Rescue Symposium
Denver, CO (2002)
web.mit.edu/sp255/www/refer…

--- and Beverly, J. M.
Measurement of dynamic rope system stiffnes in a sequential failure of lead climbing falls.
amga.com/resources/various/…

Bedogni, V.
Computer mathematical models in belaying techniques.
Nylon and Ropes for Mountaineering and Caving
Torino, Italy, (2002)

Bramley, A., Philips, A., and Vogwell, J.,
Forces Generated in a Climbing Rope During a Fall
The Engineering of Sport 6

Custer, D.
An estimation of the load rate imparted to a climbing anchor during fall arrest.
Engineering of Sport, 6th International Conference
Vol I pp 45--50 (2006).

---,
Rope behavior
ocw.mit.edu/NR/rdonlyres/Sp…

Powerpoint version
web.mit.edu/sp255/www/refer…

Leonard, R. M., Wexler, A.,
Belaying the leader
Sierra Club Bulletin 31 (7) (1946)

Manin, L., Richard, M., Brabant, J.-D., and Bissuel, M.
Modeling the climber fall arrest dynamics
ASME International Design Engineering Technical Conferences and Information in Engineering Conference,
IDETC-IEC 2005, pp 1077--1084,
Long Beach, CA (2005)

Manin, L., Richard, M., Brabant, J.-D., and Bissuel, M.
Rock climbing belay device analysis, experiments and modeling,
The Engineering of Sport 6, Vol 1 pp 69--74, Springer (2006)

Pavier, M.
Experimental and theoretical simulations of climbing falls
Sports Engineering 1 (2) pp 79--91 (1998)
www3.interscience.wiley.com…

---,
Derivation of a rope behaviour model for the analysis of forces developed during a rock climbing leader fall, The Engineering of Sport 1. (1996)

Do a search on this site for posts by Brenta.

Don't kid yourself. Calculus doesn't have any applications.

Plenty of options. Most obvious is the effects of a dynamic rope or a screamer in energy absorption. You have a given amount of kinetic energy in a fall and the dynamic parts of that system cause that energy to be absorbed over a longer duration, decreasing peak impact forces. You could also look at the design of cams, which are logarithmic spirals. I've tried (while bored in a meeting/class) to start with a basic sketch of a cam, knowing that the contact point is always a given angle behind the axle, and derive the formula for the cam shape... But i never got it fully worked out. Other options that would be more engineering type problems would be to figure out the stress in components, new I-beam cross section caribiners would be interesting compared to classic round cross section. There was a pretty recent discussion about how the new Totem cams work that might be a bit inspiring for you too.

Holy crap, Rich, at least let him enjoy some school

Obviously I don't know what level calculus you are in, but a few basic applications might me work done on the rope by a given distance of fall. Also just a simple integration of the excess weight the climber will have to climb with depending how high they are off of the ground. You could also prove Chris Sheridan's theory. Unfortunately I believe it is true.

Climb on!

-Mac

"Climbing doesn't kill people, gumbies kill people"
"If the answer is 13 RURPs and a bat hook, what was the question?"

Perhaps explain the difference between a static and dynamic rope. You could create a very simplified model of how a dynamic rope makes falls less harsh on the climmber and the gear. With some simplifying assumptions that whole system turns into a physics 1 problem. You've got a spring-mass system, the mass initially having some kinetic energy.

To get you started on the assumption you could set up your model something like this:
1) Model both ropes as overly dampened springs, with the static rope having a much higher k than the dynamic rope. Maybe 10x higher to make the numbers easier.
2) Model the belayer and anchor as fixed. Hell, ignore those altogether and model it like the spring is attached to a fixed object if you like.
3) Model the climber as a rigid object having some mass with velocity along the spring's axis when the spring starts to apply force.
4) There's a couple more you'll need to finish simplifying it. You just have to make sure you're assumptions don't change the concept of the problem. You just use them so people can concentrate on the concept rather than getting bogged down in the math.

You could take this model and graph force(time) for different k values. This could lead to a number of calculus concepts:

1) You can use derivatives to find the max force applied and show that for lower k there is lower peak force. You could mention what a screamer is and how it uses the same concept to make gear or ice less likely to fail when catching a fall.
2) You can integrate and show that impulse is constant with regard to k-value of the spring.
3) You can apply Newton's laws to your force(time) curves and come up with velocity(time) for each k value. You'll need to remember your initial velocity when you do that. Then you can show how the trade-off for a softer catch is falling further.
4) You can come up with max force as a function of k-value. That would be kinda cool because then you could just plug in a k-value and get max force; you wouldn't be limited to a discrete number of k-values anymore.

Remember, what's always key is coming up with assumptions that simplify things without changing the concept of the problem. Without that you end up bogged down in heavy calculations instead of figuring out the concepts at hand. Also important is being able to explain to your audience why it's acceptable to make the assumptions that you made.

I got a bit off topic there, but that should give you a couple ideas. Hope my rant was helpful.

Mark Nelson wrote:Holy crap, Rich, at least let him enjoy some school

Oopsie. My bad.

Hey, you could look at rockclimbing.com/cgi-bin/fo…;postatt_id=746. It's just a summary I wrote several years ago of the well-known elementary facts though; more or less the same thing is in Attaway, for example. The first method mentioned has nothing more than a very simple integral in it; the second method uses the basic second-order differential equation for simple harmonic motion.

As for cams, there's no calculus involved if you stipulate that the so-called logarithmic spiral has the constant angle of contact property---this is what most of the derivations on the web do; see for example vainokodas.com/climbing/cam…. There is, of course, calculus involved if you derive the logarithmic spiral formula from the constant contact angle condition.

JJ Brunner wrote: Oh! Pleeeassseee explain that! I will learn so much better from one post on MP than hours in Calculus class!

Brunner,

The integral sign means "add it all up". So, add up the time you spend on mountain project, is it equal to the negative "time rate of change" of your climbing ability? Units always have to match. Good luck with math.

Thanks for the links rgold. Love the Euro-vars in the Yowie factor, I would have used something much more boring than ÿ as a variable name.

the funny thing about calculus is that you do a lot of work, and then end up just realizing that algebra will basically get the job done. the only thing worth remembering from calculus is that the integral of e to the x is a function of u to the n. but then, any poly sci major could probably tell you that....

it's all ball bearings....

XOG wrote:Thanks for the links rgold. Love the Euro-vars in the Yowie factor, I would have used something much more boring than ÿ as a variable name.

XOG, the variable there is y. The expression ÿ denotes the second derivative of y with respect to time. One dot would be dy/dt. It's Newton's original notation, from before Liebnitz won the notation wars.

Dusty wrote:Don't kid yourself. Calculus doesn't have any applications.

Ha! Dusty, you are thinking of Algebraic Geometry. THAT doesn't have any applications.

Bobby Hanson wrote: Ha! Dusty, you are thinking of Algebraic Geometry. THAT doesn't have any applications.

How else would we know that there are 503840510416985243645106250 rational degree 9 curves on a general quintic threefold?

Determine the cam angle that gives the Maximum holding power. Should be fairly easy to get data from BD, come up with function and find your Critical Points.