A catenary is the curve formed by a flexible cable of uniform density hanging from two points under its own weigh. Cables of suspension bridges and attached to telephone poles hang in this shape. If the lowest point of the catenary is at , then the equation of the catenary is
Approximated by a parabola
Notice that is an even function. The following computation shows that the first term in the Maclaurin series is . For this reason it is often claimed that the shape of a hanging cable is "approximated by a parabola."
The arc length of the curve is found by using the integrand . The length of the catenary over the interval [0,a] is given by the calculation
In order to find a catenary that has width and height all we need to observe that is and even function and that goes through and we also want it to go through the point . On first glance, we see that all we need to do is solve the equation for . However, this is not possible to do analytically with Mathematica, as we can find out by issuing the following command.
Therefore, we must resort to using a numerical approximation for c instead of a "formula." In our exploration, the graphs we used the mysterious constant , which was found by determining the catenary that passes through the origin and the point .
Proof The Catenary The Catenary
Computer Programs The Catenary The Catenary
Example 1. Find the
catenary that goes through the origin and also passes through the two
points with a = 10 and b = 6.
Example 2. Find the
equation of the parabola that goes through the origin and the point
the parabola solution with the catenary solution.
Either a hanging cable is in the shape of a parabola or a catenary, let's look at the history of this controversy. The following paragraph is from the "Concise Encyclopedia of Mathematics" by Eric W. Weisstein.
"The curve a hanging flexible wire or chain assumes when supported at its ends and acted upon by a uniform gravitational force. The word catenary is derived from the Latin word for "chain''. In 1669, Jungius disproved Galileo's claim that the curve of a chain hanging under gravity would be a Parabola (MacTutor Archive). The curve is also called the Alysoid and Chainette. The equation was obtained by Leibniz, Huygens, and Johann Bernoulli in 1691 in response to a challenge by Jakob Bernoulli." Other mathematicians involved with the study of the catenary have been Robert Adrain, James Stirling, and Leonhard Euler.
An article suitable for undergraduates to read is "The Catenary and the Tractrix (in Classroom Notes)", Robert C. Yates, American Mathematical Monthly, Vol. 66, No. 6. (Jun. - Jul., 1959), pp. 500-505.
The St. Louis Arch at the Jefferson National Expansion Memorial was constructed in the shape of a catenary. Visit the National Park Service web site arch history and architecture. Or, go directly to the web page for the precise mathematical formula for the St. Louis arch (catenary curve equation).
The University of British Columbia Mathematics Department has an amusing property of the catenary (Java animation). Which is part of their "Living Mathematics Project" for "Constructing a new medium for the communication of Mathematics''.
Research Experience for Undergraduates
The Catenary The Catenary Internet hyperlinks to web sites and a bibliography of articles.
Download this Mathematica Notebook The Catenary
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(c) John H. Mathews 2004