Project 13: The Catenary

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We want to find the shape and length of electrical wires hanging between telephone poles. Intuitively, we know that we cannot stretch the wire perfectly straight, because there has to be some upward component of the tension force to counteract the weight of the wire and hold it up. The total length is one component of the cost in hanging such wires, but the strength of the wire is a counterbalancing cost. If we want to make the wire very nearly straight, then we need to make it very strong to support the tension required in straightening it. We will develop a model that has an optimal shape (or amount of sag) in terms of the tension (that is, minimal required strength). Our model is based on two simplifying assumptions:

13.1 The Catenary Hypotheses