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

The involute (a.k.a. the evolvent) of the catenary is the tractrix.
Mathematics Subject Classification
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