$8\mathrm{.}17$ A catenary cable is one that is hung between two points not in the same vertical line. As depicted in Fig. P$8\mathrm{.}17$a$,$ it is subject to no loads other than its own weight. Thus, its weight $(\frac{N}{m})$ acts as a uniform load per unit length along the cable. A free$-$body diagram of a section $AB$ is depicted in Fig. P$8\mathrm{.}17$b$,$ where $T_A$ and $T_B$ are the tension forces at the end. Based on horizontal and vertical force balances, the following differential equation model of the cable can be derived:

$\frac{d^{2}y}{d{x}^2}=\frac{w}{T_A}\sqrt[\phantom{2}]{1+(\frac{dy}{dx}{)}^2}$

Calculus can be employed to solve this equation for the height $y$ of the cable as a function of distance $x,$

$y=\frac{T_A}{w}cosh(\frac{w}{T_A}x)+y_0-\frac{T_A}{w}$