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 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.
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