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

$\sigma {\sqrt {1+(y'(x))^{2}}}\,dx$ (see the arc length).\, Thus we obtain the differential equation

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

which allows the separation of variables: $\int dx={\frac {a}{\sigma }}\int {\frac {dy'}{\sqrt {1+y'^{2}}}}$ This may be solved by using the substitution $y':=\sinh {t},\quad dy'=\cosh {t}\,dt,\quad {\sqrt {1+y'^{2}}}=\cosh {t}$ giving $x={\frac {a}{\sigma }}t+x_{0},$ i.e. $y'={\frac {dy}{dx}}=\sinh {\frac {\sigma (x-x_{0})}{a}}.$ 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

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

of the 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 $x$-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 (or evolvent) of the catenary is the tractrix.