Subject: Re: catenaries vs. parabolas
From: George Dorner (gdorner@harper.cc.il.us)
Date: Wed May 03 2000 - 15:38:57 EDT
I didn't see it mentioned in this discussion that a common catenary is the
curve assumed by a hanging flexible chain with its weight distributed
uniformly along the cable. A not-too-difficult differential equation set up
under this assumption will yield the Cosh as its solution. The same equation
under the assumption that the weight is distributed along the horizontal axis
yields a parabola as its solution.
So when you hold a flexible chain in front of students with your hands close
together and slowly spread them so the curve becomes more "shallow", it
becomes clear that the distribution along the hanging chain "approaches" the
distribution along the axis and the catenary thus approaches a parabola.
The second degree terms of Taylor expansion of the cosh will yield the same parabola.
George Dorner
Harper College
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