Hanging shapes of nonuniform cables

Using a variational approach, I derive the equation governing the equilibrium shape of an inelastic cable with an arbitrary mass distribution hanging freely under uniform gravity except for being supported at both ends. I then develop a parametric solution to this shape equation, giving the coordinates of points on the curve assumed by the cable in terms of the distance along the cable. Using numerical integration to determine values, I plot several examples of the cable’s shape for linearly and exponentially increasing density. I also consider the inverse problem of determining the mass distribution which gives rise to a specified shape. I compare the distributions yielding a semicircle, a parabola, and a catenary.