The Extensible Cable as a Limiting Case of a Very Flexible Rod

For an extensible cable with a free end, the exact nonlinear differential equations that describe the final shape of the cable present difficulties when one tries to integrate them to find the angle of inclination in the vicinity of that free end. To circumvent these difficulties, a method is proposed wherein the cable is replaced by a flexible rod. The moment of inertia of the cross-section of the rod is then allowed to approach zero while the cross-sectional area and the length of the rod remain finite. In that process, the rod approaches the cable as a limiting case, but since the rod has a proper differential equation on the inclination, no singularity occurs. To establish the validity of the method, two cases without a free end are considered first. For these, cable solutions previously obtained by the author are used for comparison. After that, the method is used to solve two cases involving a cable with a free end. In each of these two cases, the cable is suspended in a moving fluid, but they differ in the assumptions made about the drag force. The results obtained appear reasonable and suggest that this method, tantamount to removing the idealization of perfect flexibility from the cable, shows promise as a method of analyzing cables with a free end.