Various centroids and some characterizations of catenaries

Abstract

For every interval [a, b], we denote by (xA, ?A) and (xL, ?L) the geometric centroid of the area under a catenary curve y = k cosh((x - c) k) defined on this interval and the centroid of the curve itself, respectively. Then it is well-known that xL = xA and ?L = 2?A. In this paper, we fix an end point, say 0, and we show that ?L xL = 2?A/xA for every interval with an end point 0 characterizes the family of catenaries among nonconstant positive C2 functions. ? 2017 Pushpa Publishing House, Allahabad, India.