PlanetMath: catenary

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catenary

(Derivation)

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 of this curve, called the catenary, in its plane with -axis horizontal and -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 -axis. Then,

$\displaystyle \tan\varphi = \frac{dy}{dx} = y'.$

In the point, a certain tension

of the wire acts in the direction of the tangent; it has the horizontal component $T\cos\varphi$ which has apparently a constant value

. Hence we may write

$\displaystyle T = \frac{a}{\cos\varphi},$

whence the vertical component of

$\displaystyle T\sin{\varphi} = a\tan{\varphi}$

and its differential

$\displaystyle d(T\sin{\varphi}) = a\,d\tan{\varphi} = a\,dy'.$

But this differential is the amount of the supporting force acting on an infinitesimal portion of the wire having the projection

on the

-axis. Because of the equilibrium, this force must be equal the weight $\sigma\sqrt{1+(y'(x))^2}\,dx$ (see the arc length). Thus we obtain the differential equation

$\displaystyle \sigma\sqrt{1+y'^2} dx = a dy',$

(1)

which allows the separation of variables:

$\displaystyle \int dx = \frac{a}{\sigma}\int\frac{dy'}{\sqrt{1+y'^2}}$

This may be solved by using the substitution

$\displaystyle y' := \sinh{t}, \quad dy' = \cosh{t}\,dt, \quad \sqrt{1+y'^2} = \cosh{t}$

giving

$\displaystyle x = \frac{a}{\sigma}t+x_0,$

i.e.

$\displaystyle y' = \frac{dy}{dx} = \sinh\frac{\sigma(x-x_0)}{a}.$

This leads to the final solution

$\displaystyle y = \frac{a}{\sigma}\cosh\frac{\sigma(x-x_0)}{a}+y_0$

of the equation (1). We have denoted the constants of integration by

and

. 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.

$\includegraphics{catenary}$

Some properties of catenary

$\tan\varphi = \sinh\frac{x}{a}, \quad \sin\varphi = \tanh\frac{x}{a}$
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 -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.

"catenary" is owned by pahio. [ full author list (2) ]

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See Also: hyperbolic functions, tractrix

Other names:

chain curve

Also defines:

catenary

This object's parent.

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Cross-references: tractrix, focus, line, straight, rays, catacaustic, normal line, length, radius of curvature, apex, coordinate, constants of integration, solution, separation of variables, differential equation, arc length, positive, angle, tangent line, point, plane, curve, equation, coordinate system, function, graph, parabola, arc
There are 7 references to this entry.

This is version 21 of catenary, born on 2005-06-07, modified 2009-06-03.
Object id is 7146, canonical name is Catenary.
Accessed 15471 times total.

Classification:

AMS MSC:	51N05 (Geometry :: Analytic and descriptive geometry :: Descriptive geometry)
	53B25 (Differential geometry :: Local differential geometry :: Local submanifolds)

Pending Errata and Addenda

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