Remarkable shapes of a catenary under the effect of gravity and surface tension

Abstract

The shape assumed by a hanging chain (the catenary) has been known for 300 years, since Leibnitz first published the correct equation of the catenary around 1690. We have investigated the shape of a hanging string when the area bounded by the string and the supporting rod is covered by a soap film. The competition between the surface tension and the normal component of the gravitational force per unit length determines the shape assumed by the hanging string. When the surface tension is dominant, the string assumes a convex shape similar to the Greek letter gamma. When gravity dominates, the shape is a distorted catenary. In the special case when the surface tension is exactly balanced by the normal component of the gravitational force per unit length, the string assumes a linear shape like the letter V. In this article the governing differential equations are derived for the general case and solved analytically to yield a pair of parametric equations for the x and y coordinates, describing the shape of the string. Numerical results based on these equations account very well for all the experimental features described above.