The Curve for a Suspended Line

 A while ago I did some posts on the equilibrium position of a weight hanging from zip line and some similar problems. We can also take a look at the curve for a cord or chain hanging freely. Lagrange states that the center of gravity is a minimum when the cord is in equilibrium and the curve is a catenary. Galileo also mentions the hanging line and notes that it closely approximates a parabola. I used the Calculus of Variations to derive a general equation for the catenary assuming the curve has minimal potential energy for a cord with fixed length L.

One can also try to find the curve for a cord in equilibrium by doing a derivation similar to that found in Lamb's Statics.

A comparison shows that the catenary and equilibrium curve appear identical although the equilibrium curve is dependent on the cord's linear mass density λ.

The difficulty in doing the comparison is fitting the curves exactly so the endpoints match up with the chosen positions for the desired length. In each case one has two condition equations that need to be solved for the unknown parameters. For the catenary one can do a two dimensional search for the values of x_min and A but the surfaces F and G do not intersect in a straight line. The Excel plot below shows the rms deviation of the functions F and G for chosen values of the parameters. It's difficult but one can zoom in on the parameters for the minimum.

The parametric equations are a little easier to work with since the surfaces involved are more nearly planar.

One can see that the curves for Δx=0 and Δy=0 are nearly linear and when plotted as points in the A,B plane appear to be two lines crossing each other. A simultaneous solution for the two condition equations requires that the values of A and B be identical for both lines.

In the plot above the center and widths were chosen and Δx and Δy were computed for a grid of points in the A,B plane. Interpolation was used to estimate the positions of the zeros to get the two lines then the set of points for each line were fit to find a linear equation for each line and their intersection was estimated (shown as a grey circle above). By shifting to the estimated intersection point and changing the widths one can more easily zoom in on the intersection point.

One can do a "notch fit" to show how the distance between the two lines varies with the value of B by estimating the intersection of the two segments confirming the "zeros."

The computed minima for both curves agree to 5 decimal places but the parametric equations are easier to work with.