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# catenary

A chain or a homogeneous^{} flexible thin wire takes a form resembling an arc of a parabola when suspended at its ends. The arc is not from a parabola but from the graph of the hyperbolic cosine function in a suitable coordinate system.

Let’s derive the equation $y=y(x)$ of this curve, called the catenary, in its plane with $x$-axis horizontal and $y$-axis vertical. We denote the line density of the weight of the wire by $\sigma$.

In any point $(x,\,y)$ of the wire, the tangent line of the curve forms an angle $\varphi$ with the positive direction of $x$-axis. Then,

$\tan\varphi\;=\;\frac{dy}{dx}\;=\;y^{{\prime}}.$ |

In the point, a certain tension $T$ of the wire acts in the direction of the tangent; it has the horizontal component^{} $T\cos\varphi$ which has apparently a constant value $a$. Hence we may write

$T\;=\;\frac{a}{\cos\varphi},$ |

whence the vertical component of $T$ is

$T\sin{\varphi}\;=\;a\tan{\varphi}$ |

and its differential

$d(T\sin{\varphi})\;=\;a\,d\tan{\varphi}\;=\;a\,dy^{{\prime}}.$ |

But this differential^{} is the amount of the supporting force acting on an infinitesimal portion of the wire having the projection $dx$ on the $x$-axis. Because of the equilibrium, this force must be equal the weight $\sigma\sqrt{1+(y^{{\prime}}(x))^{2}}\,dx$ (see the arc length). Thus we obtain the differential equation

$\displaystyle\sigma\sqrt{1\!+\!y^{{\prime 2}}}\,dx\;=\;a\,dy^{{\prime}},$ | (1) |

which allows the separation of variables:

$\int dx\;=\;\frac{a}{\sigma}\int\frac{dy^{{\prime}}}{\sqrt{1\!+\!y^{{\prime 2}% }}}$ |

This may be solved by using the substitution

$y^{{\prime}}\;:=\;\sinh{t},\qquad dy^{{\prime}}\;=\;\cosh{t}\,dt,\qquad\sqrt{1% \!+\!y^{{\prime 2}}}\;=\;\cosh{t}$ |

giving

$x\;=\;\frac{a}{\sigma}t+x_{0},$ |

i.e.

$y^{{\prime}}\;=\;\frac{dy}{dx}\;=\;\sinh\frac{\sigma(x\!-\!x_{0})}{a}.$ |

This leads to the final solution

$y\;=\;\frac{a}{\sigma}\cosh\frac{\sigma(x\!-\!x_{0})}{a}+y_{0}$ |

of the equation (1). We have denoted the constants of integration by $x_{0}$ and $y_{0}$. They determine the position of the catenary in regard to the coordinate axes. By a suitable choice of the axes and the measure units one gets the simple equation

$\displaystyle y\;=\;a\cosh\frac{x}{a}$ | (2) |

of the catenary.

Some properties of catenary

- •
$\tan\varphi=\sinh\frac{x}{a},\quad\sin\varphi=\tanh\frac{x}{a}$ (cf. the Gudermannian)

- •
The arc length of the catenary (2) from the apex $(0,\,a)$ to the point $(x,\,y)$ is $a\sinh\frac{x}{a}=\sqrt{y^{2}\!-\!a^{2}}$.

- •
The radius of curvature of the catenary (2) is $a\cosh^{2}\frac{x}{a}$, which is the same as length of the normal line of the catenary between the curve and the $x$-axis.

- •
The catenary is the catacaustic of the exponential curve reflecting the vertical rays.

- •
If a parabola rolls on a straight line, the focus draws a catenary.

- •
The involute (a.k.a. the evolvent) of the catenary is the tractrix.

## Mathematics Subject Classification

53B25*no label found*51N05

*no label found*

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