What is the shape of a suspended rope? Is there some function that describes it?

Answer: it’s the cosh curve!

**Presentation Suggestions:**

This may be seen quite dramatically by putting up a transparency of the catenary

y = cosh x

and suspending a rope in front of the transparency projection so that the rope shadow can be compared!

Now, someone may object and say that the curve of x^{2} will also give a good approximation. If they do this, you can talk about how the Taylor series of cosh begins with a quadratic 1 + x^{2}/2, so it is not surprising!

**The Math Behind the Fact:**

Calculus and modeling are useful here: by breaking the rope into lots of little chunks, and modeling the forces on each chunk, one can obtain a differential equation whose solution is the cosh curve.

**How to Cite this Page:**

Su, Francis E., et al. “Suspended Rope Trick.” *Math Fun Facts*. <https://www.math.hmc.edu/funfacts>.

**Fun Fact suggested by: **

Michael Moody