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 x2 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 + x2/2, 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