The Catenary

The Catenary

Background

    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."  

    .

    .

    

Arc Length

    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  

    .

Conclusion

    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''.