The stability of the catenary shapes for a hanging cable of unspecified length

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.