Catenary and Parabola ComparisonDate: 04/06/2004 at 13:48:57 From: Curious Student Subject: Catenary and Parabola What is the difference between a catenary and a parabola? I don't know the difference in shape. Why is the St. Louis arch a catenary and not a parabola? Date: 04/06/2004 at 16:48:40 From: Doctor Peterson Subject: Re: Catenary and Parabola Hi, Student. The catenary and the parabola are two different curves that look generally similar, in that they are symmetrical and have a cup shape, going up infinitely on either side of a minimum. The parabola, in its simplest form, is y = x^2 while the catenary is defined by the hyperbolic cosine: y = cosh(x) = (e^x + e^-x)/2 Here is a graph of the parabola (blue) and catenary (red) together, so you can see the difference: I used y = (cosh(x) - 1)/(cosh(1) - 1) in order to move the vertex from (0,1) down to the origin, and to make it agree with the parabola when x = 1. This makes it clear that the catenary is slightly more "flat" at the bottom, and that it rises faster than the parabola for large values of x. In fact, if you graphed the two for a larger domain, the catenary would be far higher than the parabola. As for why the St. Louis arch, or a hanging cable, takes the shape of a catenary, while the cables on a suspension bridge form a parabola, that is just a result of the physics of each situation. Once you have learned calculus, you will be able to see that the catenary is the solution to a differential equation that describes a shape that directs the force of its own weight along its own curve, so that, if hanging, it is pulled into that shape, and, if standing upright, it can support itself. The parabola does not have the same property, but is the solution of other important equations that describe other situations. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 04/06/2004 at 22:08:34 From: Curious Student Subject: Thank you (Catenary and Parabola) Thank you, I now understand and will share this information with my math class. |
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