18.2: The Intrinsic Equation to the Catenary

We consider the equilibrium of the portion AP of the chain, A being the lowest point of the chain (Figure XVIII.1).

It is in equilibrium under the action of three forces: The horizontal tension $T_{0}$ at A; the tension $T$ at P, which makes an angle $\psi$ with the horizontal; and the weight of the portion AP. If the mass per unit length of the chain is $\mu$ and the length of the portion AP is $s$, the weight is $\mu s\text{g}$. It may be noted than these three forces act through a single point.

Clearly,

$$T_{0}=T\cos\psi \label{18.2.1}$$

and

$$\mu s\text{g}=\ T\sin\psi, \label{18.2.2}$$

from which

$$(\mu s\text{g})^{2} + T_{0}^{2}=\ T^{2} \label{18.2.3}$$

and

$$\tan\psi=\frac{\mu\text{g}s}{T_{0}} \label{18.2.4}$$

Introduce a constant $a$ having the dimensions of length defined by

$$a=\frac{T_{0}}{\mu\text{g}}. \label{18.2.5}$$

Then Equations $\ref{18.2.3}$ and $\ref{18.2.4}$ become

$$T\ =\ \mu\text{g}\sqrt{s^{2}\ +\ a^{2}} \label{18.2.6}$$

and

$$s\ =\ a\tan\psi. \label{18.2.7}$$

Equation $\ref{18.2.7}$ is the intrinsic equation of the catenary.