19.3: The Catenary

Consider a chain elevated above ground, attached only at its two ends, both ends at the same height, and hanging under its own weight. The chain will sag, forming a hyperbolic cosine curve called a catenary. With a coordinate system defined as shown in Figure $\PageIndex{1}$, the equation of the catenary is found to be

Figure $\PageIndex{1}$: A chain hanging under its own weight, forming a catenary curve.

$$y=a \cosh \left(\frac{x}{a}\right)-a$$

where $a=H / w, H$ is the horizontal tension in the chain at the pole (in newtons), and $w$ is the linear weight density of the chain (in newtons per meter).

The arc length $s$ of the catenary from $x=0$ to $x$ is given by

$$s(x)=a \sinh \left(\frac{x}{a}\right)$$

so that if the poles are separated by a distance $d$, the total arc length $s_{t}$ is

$$s_{t}=2 a \sinh \left(\frac{d}{2 a}\right)$$

Note that if the horizontal tension $H$ is very large (the chain is pulled very taut), then $a=H / w$ is very large, $d / 2 a$ is very small, and so $\sinh (d / 2 a) \approx d / 2 a$, so that $s_{t} \approx d$, as expected.