Hold up a chain by both ends and you'll get a curve. What kind of curve is it? You might say it is a parabola  Galileo Galili believed it was a parabola. Yet, Galileo was wrong!!!! That curve is NOT a parabola. It is a catenary.
It makes sense that you would think that the curved chain is a parabola. Both the catenary and the parabola have similar properties. Both curves have a single low point. They both have a vertical line of symmetry, they at least appear to be continuous and differentiable throughout, and the slope is steeper as we move away from the low point, but it never becomes vertical.
So, how is the curve of the cable in a suspension bridge a parabola? When the structure is being built and the main cables are attached to the towers, the curve is a catenary. But when the cables are attached to the deck with hangers, it is no longer a catenary. The curve of the cables become the curve of a parabola. Unlike the catenary, which is curving under its own weight, the parabola is curving not just under its own weight, but also curving from holding up the weight of the deck. The cable of a suspension bridge is under tension from holding up the bridge.
Therefore, the cables of a suspension bridge is a parabola, because the weight of the deck is equally distributed on the curve.
Proving that the Curve of a Suspension Bridge's Cable is a Parabola
If the deductive reasoning is not enough for you, there is another way to prove that the curve of the cable in a suspension bridge is a parabola. By finding the equation of the curve of the cable in the suspension bridge, you can prove its a parabola. To get more indepth and more into calculus (of which I do not yet have an understanding), go to Hanging with Galileo, a comprehensive webpage that compares the equations of the catenary (a hyperbolic cosine) and the parabola in relation to a suspension bridge.
1. This example of ropes that are spanning two cliffs shows what basically is a catenary.
2. This is a picture of the master mechanic E.F. Farrington traveling the length of the newly installed cable of the Brooklyn Bridge. The cable is an example of a catenary, curving under the weight of itself (the weight of Farrington is insignificant).
3. This picture shows the deck being added to the cables of the Brooklyn Bridge. The catenary is slowly becoming a parabola.
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