Catenary Demonstration Experiment

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Catenary Demonstration Experiment; Fun with hyperbolic cosine

This experiment was first performed by the the Mathclub and I in Spring 2006.
Since then it has been a calculus project for students at Mount Rainier Lutheran High, WA where its hung in their commons. A volunteer plans to set it up soon for COSI Columbus.
If you repeat this experiment, I want to know!
Please email me at:

Thanks to all who came by to help and/ or spectate, the experiment was a success. There is a good chance we'll grace our campus with a larger version between the Campus Center and Fenster Hall; date/ time TBA.
Special thanks to Jeanina for taking some photos. Note an identical page is on the Mathclub Website as this was an NJIT Mathclub Event.

Look carefully, the green line shows that all the white strings come down the the same height. This was done without rulers, calculators or numbers (and without cutting the rope afterwards)

The secret revealed, a bit of geometric calculus shows that this trick works, you can ask me at any time as well and I will be glad to explain it. If you were in the Mathclub last fall, you already know why. Note: that's me holding the reins.

Changing the shape of the catenary. Notice the strings are longer here ('a' is larger). This was actually done first, then the strings quickly shortened for part II.

The catenary is less sharp here for the longer strings to line up. The levelness is easily seen next to the blackboard. All 3 have the same shaped curve. Thus demonstrating that Density has no Effect.

Ravdeep Rana explains a few things. (Don't forget to marvel the density having no effect).

Links about y=a*cosh(x/a)
Catenary curves, the shape that a hanging rope makes. Wikipedia. Heres a nice derivation of the hanging cable property.

Printable Directions:
Imagine the dark line is a heavy rope, the light lines are also some kind of rope. We set everything as shown on the left, then hang the rope and magically (mathematically) all the lines will line up straight at the bottom! How cool,

Room for Improvement:

My proof is only geometric; we know let me know if you come up with an elegant complex variables proof.

The length of a catenary is proportional to both its area and its derivative. Try and devise a quick way to measure either- say if the curve is drawn on a blackboard.

Note: Catenaries are also not bad if you want to roll squares along them. Ask me why.

Image courtesy of Mathworld.