Talk:PlanetPhysics/Catenary

Original TeX Content from PlanetPhysics Archive

%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: catenary %%% Primary Category Code: 45. %%% Filename: Catenary.tex %%% Version: 1 %%% Owner: pahio %%% Author(s): pahio %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

\setlength{\topmargin}{0.00in} \setlength{\headsep}{0.00in} \setlength{\headheight}{0.00in} \setlength{\evensidemargin}{0.00in} \setlength{\oddsidemargin}{0.00in} \setlength{\textwidth}{6.5in} \setlength{\textheight}{9.00in} \setlength{\voffset}{0.00in} \setlength{\hoffset}{0.00in} \setlength{\marginparwidth}{0.00in} \setlength{\marginparsep}{0.00in} \setlength{\parindent}{0.00in} \setlength{\parskip}{0.15in}

\usepackage{html}

% this is the default PlanetPhysics preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners.

% almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts}

% used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic}

% there are many more packages, add them here as you need them

% define commands here

\begin{document}

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 \htmladdnormallink{graph}{http://planetphysics.us/encyclopedia/Bijective.html} of the hyperbolic cosine \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} in a suitable coordinate \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html}.

Let's derive the equation \,$y = y(x)$\, of this curve, called the {\em 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 \htmladdnormallink{force}{http://planetphysics.us/encyclopedia/Thrust.html} acting on an infinitesimal portion of the wire having the projection $dx$ on the $x$-axis.\, Because of the \htmladdnormallink{equilibrium}{http://planetphysics.us/encyclopedia/InertialSystemOfCoordinates.html}, this force must be equal the weight\, $\sigma\sqrt{1+(y'(x))^2}\,dx$ (see the arc length).\, Thus we obtain the \htmladdnormallink{differential equation}{http://planetphysics.us/encyclopedia/DifferentialEquations.html} \begin{align} \sigma\sqrt{1+y'^2}\,dx = a\,dy', \end{align} which allows the \htmladdnormallink{separation of variables}{http://planetphysics.us/encyclopedia/SeparationOfVariables.html}: $$\int dx = \frac{a}{\sigma}\int\frac{dy'}{\sqrt{1+y'^2}}$$ 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 \htmladdnormallink{position}{http://planetphysics.us/encyclopedia/Position.html} 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{align} y = a\cosh\frac{x}{a} \end{align} of the catenary.

\textbf{Some properties of catenary} \begin{itemize} \item $\tan\varphi = \sinh\frac{x}{a}, \quad \sin\varphi = \tanh\frac{x}{a}$ \item The arc length of the catenary (2) from the apex\, $(0,\,a)$\, to the point\, $(x,\,y)$\, is\,\, $a\sinh\frac{x}{a} = \sqrt{y^2-a^2}$. \item 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. \item The catenary is the \htmladdnormallink{catacaustic}{http://planetphysics.us/encyclopedia/Catacaustic.html} of the exponential curve reflecting the vertical rays. \item If a parabola rolls on a straight line, the focus draws a catenary. \item The involute (or evolvent) of the catenary is the tractrix. \end{itemize}

\end{document}