Why is the shape of the cable a parabola and not a catenary? I've always been taught that the shape a cable makes hanging between two supports is a catenary. Isn't a suspension bridge the same thing?
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.
[0] - https://en.wikipedia.org/wiki/Catenary#Suspension_bridge_cur...
If you hang a single 1lb weight in the middle of a clothes line, it forms very nearly a perfect V. That's because the 1lb totally dominates the 1/2 oz that the rest of the line weighs.
Maybe a linear weight distribution favors the parabola, while a curved weight distribution (i.e. the cable's own weight when hanging free) causes the catenary? That makes sense to me, there's more weight per linear meter at the ends than in the center when considering just the cable's weight.
A catenary forms when self weight is the driving force. In other words, when the weight per unit length of the cable is the same. For a bridge, the weight of the deck is much alrger than the cable. The weight of the deck is the same per unit horizontal length which is not the same as the unit length of the cable due to the curvature.