MATHS BITE: Catenary

A Catenary is the idealised shape created when you suspend a chain by its ends and let it hang naturally under its own weight. Any hanging chain will naturally find its equilibrium as the forces of tension from the hooks holding the chain up and the force of gravity pulling it downwards balance out.

Robert Hooke was the first to study the catenary mathematically; in 1675 he announced that he had solved the problem of the optimal shape of an arch by publishing the solution as an encrypted anagram: “As hangs the flexible line, so but inverted will stand the rigid arch.”

Its equation was obtained in 1691 by Leibniz, Huygens and Johann Bernoulli in response to a challenge put out by Jacob Bernoulli to find the equation of the ‘chain-curve’. The curve has a U-like shape and although it is similar in appearance to a parabola, it is not quite a parabola.

Source: IntMath

The equation of the catenary is
{\displaystyle y=a\cosh \left({\frac {x}{a}}\right)={\frac {a\left(e^{\frac {x}{a}}+e^{-{\frac {x}{a}}}\right)}{2}}}


Catenaries for different values of a | Source: Wikipedia

In 1744, Euler proved that the catenary is the curve which gives the surface of minimum surface area for the given bounding circles when rotated about the x-axis.

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