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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This is not an answer but what you’re talking about is called a catenary. The curve is a hyperbolic cosine not a parabolic.
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.
The entire weight of the rope is held at the two ends. If you measure a small amount in, the weight below that point is slightly less so the tension isn’t pulling it down as much but the tension is the same so in order for the system to be balanced the rope has to point the tension at a slightly less steep angle. Measure in slightly more and the cycle continues. Once you get to the center the tension is still the same but their is no weight under it so if the rope pulled at any angle other than horizontal it would be unbalanced so it has to be level.
The reason it is a parabola (some people point out this isn’t accurate but it is very close so let’s stick with it for the 5yo) has to do with how we perceived the earlier part. The balance of the tension and the weight was linear measuring along the rope which is angled, but if we measured horizontally we have to account for a constantly changing amount of weight per unit of horizontal distance. To simplify some complex math, the amount of weight below a specific point reduces by both the horizontal distance and the rope length which makes it look kinda like a square function (parabola).
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