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It has long been known that when a cable of specified length is hung between two poles, it takes the shape of a catenary—a hyperbolic cosine function. In this paper, we study a variation of this problem. First, we consider a cable hanging between two poles in which one end of the cable is fixed to one pole; the other end of the cable runs over a pulley, attached to the other pole, and then down to a table. Here, the length of the cable can vary as the pulley rotates. For a specified horizontal distance between the two poles, we vary the height of the fixed cable end. We then determine both experimentally and analytically the stability of the resulting catenary-cable shapes. Interestingly, at certain heights there are two catenaries of different lengths—we use Newtonian mechanics to show that only one of these is stable. Below a certain critical height, no catenary exists and the cable is pulled down to the table. Finally, we explore a related problem in which one end of the cable runs over a pulley, but the other end can now freely move vertically along a pole. These experiments nicely lend themselves as teaching tools in a classroom setting.

Mareno, A., and L.Q. English. The Stability of the Catenary Shapes for a Hanging Cable of Unspecified Length. European Journal of Physics 30, no.1 (2009): 97-108. http://iopscience.iop.org/article/10.1088/0143-0807/30/1/010

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