PlanetPhysics/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 [[../Bijective/|graph]] of the hyperbolic cosine [[../Bijective/|function]] in a suitable coordinate [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|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 $\phi$ with the positive direction of $x$-axis.\, Then, $\tan \phi =\frac{dy}{dx}=y^{\prime }.$ In the point, a certain tension $T$ of the wire acts in the direction of the tangent; it has the horizontal component\, $T\cos \phi$\, which has apparently a constant value $a$.\, Hence we may write $T=\frac{a}{\cos \phi },$ whence the vertical component of $T$ is $T\sin \phi =a\tan \phi$ and its differential $d(T\sin \phi )=ad\tan \phi =ad{y}^{\prime }.$ But this differential is the amount of the supporting [[../Thrust/|force]] acting on an infinitesimal portion of the wire having the projection $dx$ on the $x$-axis.\, Because of the [[../InertialSystemOfCoordinates/|equilibrium]], this force must be equal the weight\,

$\sigma \sqrt{1+(y^{\prime }(x){)}^2}dx$

(see the arc length).\, Thus we obtain the [[../DifferentialEquations/|differential equation]]

$\begin{array}{c}\sigma \sqrt{1+y{\text{'}}^2}dx=ad{y}^{\prime },\end{array}$

which allows the [[../SeparationOfVariables/|separation of variables]]: $\int dx=\frac{a}{\sigma }\int \frac{d{y}^{\prime }}{\sqrt{1+y{\text{'}}^2}}$ This may be solved by using the substitution $y^{\prime }:=\sinh t,d{y}^{\prime }=\cosh tdt,\sqrt{1+y{\text{'}}^2}=\cosh t$ 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/|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

$\begin{array}{c}y=a\cosh \frac{x}{a}\end{array}$

of the catenary.

Some properties of catenary

• $\tan \phi =\sinh \frac{x}{a},\sin \phi =\tanh \frac{x}{a}$
• 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 $x$-axis.
• The catenary is the [[../Catacaustic/|catacaustic]] of the exponential curve reflecting the vertical rays.
• If a parabola rolls on a straight line, the focus draws a catenary.
• The involute (or evolvent) of the catenary is the tractrix.