Why is it that when we fix the two ends of a rope and let gravity act, the rope forms a curve that looks like a parabola?
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The actual curve is called a **caternary**, and is not quite the same as a parabola. I am pretty sure we did the derivation of a caternary in some class at university – either Calculus or Vectorial Mechanics. The shape is defined by the force of gravity on the cable and the tangential tension forces within the cable.
You get a parabola on *suspension bridges,* not on free-hanging cables by themselves. The difference is that when you just have a cable or rope or chain, its mass is distributed evenly *along the length of the rope,* whereas in a suspension bridge, where the mass of the deck dominates the mass of the cable, the mass is distributed evenly *across the horizontal length,* no matter what angle the cable is hanging at in that section.
The rope has a fixed length. Its shape is a balance between gravity that acts at every point, and the tension in the rope. You can see, for example, that if you add a weight at the center of the rope, the shape will change. You can play with a real rope and feel the restoring forces in it whenever you try to give it any different shape.
You can model the rope as basically a string of tiny massive beads that has a fixed length, figure out the general force on one bead in the middle that’s at some angle, and it gives you a differential equation. The solution to that differential equation happens to be the hyperbolic cosine function, which is actually exponential but isn’t that different from a parabola in the middle part.
It’s not actually a parabola, but the reason it forms is that the horizontal stress is the same at all points of the rope, but the vertical component depends on how much weight is being carried at that point. At the lowest point it’s horizontal, because that part is being held up by the parts either side of it. Near the ends it will be steepest, because it’s holding up everything between that and the lowest point, in addition to the sideways force.
The angle at any particular point depends on the ratio of the horizontal force to the vertical force being carried at that part of the cable.
This is not an answer but what you’re talking about is called a catenary. The curve is a hyperbolic cosine not a parabolic.