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The parabola is the equilibrium shape of a cable which is supporting an even load per unit horizontal distance. The catenary is the same for an even load per unit cable length.

As for why, it's been a few years since I had to derive this, but it goes something like this:

* The horizontal component of the force in the cable doesn't change along its length

* This leads to the fact that the extra vertical load picked up in the cable, per unit horizontal length as you approach the support, is proportional to the change in slope of the cable per unit length

* The change in slope is the second derivative of the cable height, and since it is a constant the cable shape is parabolic

Fun extra fact - the parabola is the equilibrium shape because the cable can resist tension but not bending. If you reverse the sign on all the equations then they still work, but your parabola is pointing away from the direction of gravity and you have only compression, no bending, in the material. That's why arch bridges are shaped like a parabola too.




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