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If I hang a rope from two points that are at the same height above the ground, what is the mathematical function that describes the shape of the rope between the two points? Assuming the mass of the rope is evenly distributed between the two points.

Upon visual inspection, it appears to be parabolic, but looks can be deceiving. I want to be able to calculate how much rope I'll need to span the distance, assuming that I want it to sag by a given amount. If I know the function of the rope's shape, I think I can use an integral to calculate the length of the curve.

Qmechanic
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brianmearns
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2 Answers2

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This is a classic problem in the calculus of variations, and the shape is not, in fact, a parabola. The curve is called a catenary and its basic equation is $$ y=a \cosh(x/a). $$ For more details, see the catenary page at Wikipedia.

Emilio Pisanty
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It can be shown using the calculus of variations that this is indeed a catenary. Written as

$$ y = a \, \cosh \left ({x \over a} \right ) = {a \over 2} \, \left (e^{x/a} + e^{-x/a} \right )\, $$

A slightly more interesting problem arises in stretching soap film between two concentric circular wires. Due to the axial symmetry of this problem the solution is the catenoid. This is the surface of revolution obtained by rotating the catenary.

I would highly recommend Gelfand and Fomin "Calculus of variations" (Dover Publications) for further reading on such problems.

Shane O'Mahony
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