Summary Catenaries are the curves formed by a uniform chain or cable suspended from two ends. They are seen in the natural world in structures such as spider webs, and they also occur in engineered structures, including bridges and overhead power lines. Catenaries and catenary arches are also employed in architecture for their beauty and strength. In this article, we look at catenaries where a chain is fixed at one endpoint but free to move over the other. Such catenaries will either be supported by the weight of the tail end of the chain and hence be “stable,” or else unwind off the free end and adopt the configuration of a vertically hanging chain. We develop two models, which are analyzed to determine the conditions under which a particular system will be either stable or unstable and determine the widest span for which a stable catenary exists. The predictions made here may be explored experimentally with relatively inexpensive apparatus. The models presented also invite extensions, such as more general free point locations and the addition of friction, which could dissipate the system energy.
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