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For the problem of a cable under its own weight we have a differential equation

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For the problem of a cable under its own weight we have a differential equation of:

Using this differential equation and assuming the cable is of uniform weight density, we can determine the function of the curve that will occur when hung by each end at uniform height. By taking a small segment of the curve inside an interval you can use the tension and tangent angels at each end along with the weight of the cable you can find the sag of the cable under it's own weight.

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