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We pin \ the ends of a soft and uniform chain of length ", Cell[BoxData[ \(TraditionalForm\`l\)]], " at these points. Assuming the length of the chain is longer than the \ distance between the points, the chain under gravity pull assumes a Catenary \ shape. In conjunction with this setting, we work on two projects. In the \ first one, with the help of ", StyleBox["Mathematica", FontSlant->"Italic"], ", we exercise two mathematical scenarios; leaving one of the pins intact, \ we move the second pin to a set of selected positions on the same plane \ without changing a) the ordinance and, b) the abscissa. Utilizing ", StyleBox["Mathematica,", FontSlant->"Italic"], " we animate both processes. In the second project, we release a bead from \ the mobile pin. Applying ", StyleBox["Mathematica", FontSlant->"Italic"], ", we evaluate the run-time of the bead along the Catenaries of the first \ project. In the latter case, moreover, we consider two practical situations: \ a) a frictionless chain, and, b) a rough chain. We tabulate the run-time as \ a function of the coefficient of kinetic friction. Applying ", StyleBox["Mathematica,", FontSlant->"Italic"], " we animate the movement of the bead along the corresponding Catenaries. \ " }], "Abstract"], Cell[TextData[{ "\[FilledSquare]", StyleBox["Introduction:", FontWeight->"Bold"] }], "Text"], Cell["\<\ It is a classic problem of interest to study the geometrical \ characteristics associated with the assumed shape of a massive uniform line \ such as a laundry rope, a jump rope, a utility power line and or a flexible \ chain pulled by the gravity. The common features of these examples are: the \ lines are uniform, they are pinned at two ends in a vertical plane and that \ gravity is fully in action. The strategies adapted to analyze the problem \ falls in one of the two categories, either Newton mechanics [1] or the \ Calculus of Variations [2]. The end product of these two distinct analytic \ methods, however, is the same -- the lines assume Catenary shapes. Simply \ put, a hyperbolic cosine function describes the shape of the line. \ \>", \ "Text"], Cell[TextData[{ "Building on this classic academic background the author comes to realize \ its practical geometry-related applications. To begin, hold the ends of a \ chain, e.g. a pocket watch chain, in each hand. Keep the hands at the same \ height, and then, without moving the hands up or down bring them together or \ pull them apart. Alternatively, without changing the distance between the \ hands, move one of its ends either up or down. As long as the distance \ between the hands is shorter than the length of the chain, as mentioned, the \ hanging chain assumes a Catenary shape. Although it is easy to toy with the \ chain, it is challenging to simulate the shapes mathematically. The author \ applied ", StyleBox["Mathematica", FontSlant->"Italic"], "'s [3] integrated superb symbolic, numeric and graphic features to \ replicate the problem. Utilizing ", StyleBox["Mathematica", FontSlant->"Italic"], "'s animation, the replicated models resemble real-time cases." }], "Text"], Cell["\<\ Having concurred the proposed Catenary's geometry-related issues of \ interest, we considered a few kinematic and dynamic physics related \ situations. For instance, we select one of the Catenaries, e.g. the one with \ an uneven heights, and release a massive bead from its high point; the bead \ negotiates the Catenary. Applying appropriate physics principles, we \ evaluate the bead's run-time, the bead's horizontal and vertical speeds at \ any given time, and its time dependent \"attack\" angle. We also look into \ dynamic quantities such as the time dependency of the chain's surface \ reaction. Having established this basis, we modify the calculation including \ the kinetic friction.\ \>", "Text"], Cell["\<\ This article is composed of two sections. In section 1, we study \ the proposed geometry-related issues discussed in the first part of the \ introduction. In section 2 we analyze the physics of the problem \ symbolically, conducive to a set of comprehensive formulas. Section 2 \ includes also numeric tabular values of quantities of interest along with \ corresponding graphs. We conclude the article by animating the movement of \ the bead. We close the article with a few conclusive remarks.\ \>", "Text"], Cell[TextData[{ "\[FilledSquare]", StyleBox["Section 1: Geometrical Analysis", FontWeight->"Bold"] }], "Text"], Cell[TextData[{ "Denoting the uniform linear weight density of the line by ", Cell[BoxData[ \(TraditionalForm\`w\_0\)]], ", the tension at the bottom of the line by ", Cell[BoxData[ \(TraditionalForm\`T\_0\)]], ", according to [1] in a vertical upright Cartesian coordinate system with \ the y-axis through the minimum of the Catenary, the equation of the Catenary \ is given by ", Cell[BoxData[ \(TraditionalForm\`y(x) = T\_0\/w\_0\ \(cosh(\(w\_0\/T\_0\) x)\)\)]], ". Since we are interested in the geometry of the problem, we suppress the \ physical quantities, and define a characteristic length ", Cell[BoxData[ \(TraditionalForm\`c = T\_0\/w\_0\)]], ". Furthermore, motivated by the practical settings, we choose a \ coordinate system such that the vertical axis passes through the left-end of \ the chain; to do so, we introduce an adjustable parameter \[Lambda]. We also \ slide the x-axis by \[Alpha]. In this coordinate system the Catenary \ becomes, ", Cell[BoxData[ \(TraditionalForm\`\[Lambda] + y(x) = c\ \(cosh(\(x - \[Alpha]\)\/c)\)\)]], ". We determine the values of ", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\)]], " and \[Lambda] according to our needs. " }], "Text"], Cell[TextData[{ "In practice, we would be dealing with a piece of a chain with a given \ length ", Cell[BoxData[ \(TraditionalForm\`l\)]], ". Applying Catenary equation, symbolically, we evaluate its length in \ terms of its relevant parameters. To do so, we integrate a short arc length, \ ", Cell[BoxData[ \(TraditionalForm\`d\ l\)]], ", along the curve. Applying Pythagorean theorem yields, ", Cell[BoxData[ FormBox[ RowBox[{"l", "=", RowBox[{ FormBox[\(\[Integral]d\ l \[Congruent] \[Integral]\@\(dx\^2 + \ dy\^2\)\), "TraditionalForm"], "\[Congruent]", \(\[Integral]\_0\%x\(\@\( 1 + y'\^2\)\) dx\)}]}], TraditionalForm]]], " = ", Cell[BoxData[ \(TraditionalForm\`c\ [ sinh(\(x - \[Alpha]\)\/c) + sinh(\[Alpha]\/c)]\)]], ". " }], "Text"], Cell[TextData[{ "The length and the coordinate of the left-end of the chain are at our \ disposal; we select ", Cell[BoxData[ \(TraditionalForm\`l = 1.4\ units\)]], " and set ", Cell[BoxData[ \(TraditionalForm\`\(\(p\_1\)\(=\)\)\)]], Cell[BoxData[ \(TraditionalForm\`\(\({x\_01, y\_01} = {0, 0.8 units}\)\(,\)\)\)]], " respectively. The mobile ", "right-end", " of the chain according to the first scenario of interest described in the \ introduction is to move the hands back and forth without changing its height. \ For instance, confining its horizontal movement to ", Cell[BoxData[ \(TraditionalForm\`0.2 \[LessEqual] x\_2 \[LessEqual] 0.8\ units\)]], ", and considering the fact that the distance between the ends of the chain \ is to be shorter than its length, the coordinate of the mobile end becomes ", Cell[BoxData[ \(TraditionalForm\`p\_2 = {x\_2, \ y\_01 + \@\(l\^2 - x\_2\^2\)}\)]], ". This data set is entered as:" }], "Text"], Cell[BoxData[ \(x01 = 0. ; y01 = 0.8; length = 1.4;\)], "Input", CellLabel->"In[5]:="], Cell[BoxData[{ \(\(x2 = Range[0.2, 0.8, 0.2];\)\), "\[IndentingNewLine]", \(\(y2 = Range[y01, y01 + \@\(length\^2 - Max[x2]\^2\), 0.2];\)\)}], "Input", CellLabel->"In[6]:="], Cell["\<\ Now the challenge is to search for a set of parameters that are \ subject to a three coupled conditional equations, \ \>", "Text"], Cell[BoxData[ RowBox[{ StyleBox["{", SpanMaxSize->Infinity], GridBox[{ { FormBox[\(\[Lambda] + y\_01 = c\ \(cosh(\[Alpha]\/c)\)\), "TraditionalForm"]}, { FormBox[\(\[Lambda] + y\_2 = c\ \(cosh(\(x\_2 - \[Alpha]\)\/c)\)\), "TraditionalForm"]}, { FormBox[\(\(\ \)\(l = c\ \((sinh(\(x\_2 - \[Alpha]\)\/c) + sinh(\[Alpha]\/c))\)\)\), "TraditionalForm"]} }], "\[IndentingNewLine]"}]], "DisplayFormula"], Cell[TextData[{ "Although, one might be tempted to solve this set of equations for a set of \ selected coordinate ", Cell[BoxData[ \(TraditionalForm\`p\_2\)]], " one at a time, applying ", StyleBox["Mathematica'", FontSlant->"Italic"], "s FindRoot along with Table command, enables us to solve the equations in \ one step. Noting the fact that these equations are non-algebraic and \ transcendental, we apply numeric method. However, the numeric method calls \ for seed search values. By inspection, and by trial and error we \ guesstimate the range of the seed values. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(rootsx = Table[FindRoot[{y01 + \[Lambda] \[Equal] c\ Cosh[\[Alpha]\/c], Min[y2] + \[Lambda] \[Equal] c\ Cosh[\(x2\[LeftDoubleBracket]n\[RightDoubleBracket] - \ \[Alpha]\)\/c], length \[Equal] c\ \((Sinh[\[Alpha]\/c] + Sinh[\(x2\[LeftDoubleBracket]n\[RightDoubleBracket] - \ \[Alpha]\)\/c])\)}, {\[Alpha], x2\[LeftDoubleBracket]n\[RightDoubleBracket]\/2}, {c, 0.1}, {\[Lambda], 0.4}, MaxIterations \[Rule] 200], {n, 1, Length[x2]}]\), "\n", \( (*Table[{c\ Cosh[\[Alpha]\/c] - \[Lambda], c\ Cosh[\(x2\[LeftDoubleBracket]n\[RightDoubleBracket] - \[Alpha]\ \)\/c] - \[Lambda], c\ \((Sinh[\[Alpha]\/c] + Sinh[\(x2\[LeftDoubleBracket]n\[RightDoubleBracket] - \ \[Alpha]\)\/c])\)} /. rootsx\[LeftDoubleBracket]n\[RightDoubleBracket], {n, 1, Length[x2]}]*) \)}], "Input", CellLabel->"In[8]:="], Cell[BoxData[ \({{\[Alpha] \[Rule] 0.1`, c \[Rule] 0.024788236102896346`, \[Lambda] \[Rule] \ \(-0.09956123990095117`\)}, {\[Alpha] \[Rule] 0.2`, c \[Rule] 0.0651545564868717`, \[Lambda] \[Rule] \(-0.096974313249508`\)}, {\ \[Alpha] \[Rule] 0.30000000000000004`, c \[Rule] 0.1229450693905018`, \[Lambda] \[Rule] \(-0.08928522592573396`\)}, \ {\[Alpha] \[Rule] 0.4`, c \[Rule] 0.20693212127687485`, \[Lambda] \[Rule] \(-0.07005417816378515`\)}}\ \)], "Output", CellLabel->"Out[8]="] }, Open ]], Cell["\<\ Supplying the evaluated parameters we animate and display the \ associated Catenaries. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Table[ Plot[Evaluate[\((c\ Cosh[\(x - \[Alpha]\)\/c] - \[Lambda])\) /. rootsx\[LeftDoubleBracket]n\[RightDoubleBracket]], {x, 0, x2\[LeftDoubleBracket]n\[RightDoubleBracket]}, PlotRange \[Rule] {{0, 0.8}, {0, 1.4}}, AxesLabel \[Rule] {"\", "\"}, PlotLabel -> "\< Catenary \n A hanging chain\>", PlotStyle \[Rule] {Thickness[0.008]}], {n, 1, Length[x2]}];\)\)], "Input", CellLabel->"In[9]:="], Cell[CellGroupData[{ Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 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Cell[TextData[{ "A display of a Catenary (solid curve) along with the active forces ", Cell[BoxData[ \(TraditionalForm\`m\ g\)]], ", ", Cell[BoxData[ \(TraditionalForm\`N\)]], " and ", Cell[BoxData[ \(TraditionalForm\`f\_k\)]], " on a bead of mass ", Cell[BoxData[ \(TraditionalForm\`m\)]], "." }], "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`\(-N\)\ sin\ \[Theta] + f\_k\ cos\ \[Theta] = m\ \(x\& .. \)\)]]], "NumberedEquation"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`N\ cos\ \[Theta]\ + \ f\_k\ sin\ \[Theta]\ - \ m\ g\ = m\ \(y\& .. \)\)]]], "NumberedEquation"], Cell[TextData[{ "The relevant active forces: weight of the bead ", Cell[BoxData[ \(TraditionalForm\`m\ g\)]], ", the reaction of the chain ", Cell[BoxData[ \(TraditionalForm\`N\)]], ", and kinetic friction ", Cell[BoxData[ \(TraditionalForm\`f\_k\)]], " are shown in Figure 3. Components of the acceleration of the bead, ", Cell[BoxData[ FormBox[ RowBox[{\(a\_x\), Cell[""]}], TraditionalForm]]], "and ", Cell[BoxData[ \(TraditionalForm\`a\_y\)]], " are denoted by ", Cell[BoxData[ \(TraditionalForm\`\(x\& .. \)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(y\& .. \)\)]], " respectively. The auxiliary angle ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " the \"attack angle\" is also shown. We set ", Cell[BoxData[ \(TraditionalForm\`f\_k = \[Mu]\ N\)]], " and manipulate eqs (1,2), we arrive at, " }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`\(1 + \[Mu]\ \(tan(\[Theta])\)\)\/\(\[Mu] + tan(\ \[Theta])\) = \(g + \(y\& .. \)\)\/\(x\& .. \)\)]], " " }], "NumberedEquation"], Cell[TextData[{ "We utilized angle \[Theta] to project the various components of the \ relevant forces along the coordinate axes. Angle \[Theta] is also the same \ angle that at the point of interest a tangent to the Catenary makes with \ respect to x-axis; we set ", Cell[BoxData[ \(TraditionalForm\`tan(\[Theta]) = y'\)]], ". With this linker concept, the geometric characteristics of the Catenary \ comes into the play. Equation (3) becomes, " }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`\(1 + \[Mu]\ y'\)\/\(\[Mu] + y'\) = \(g + \ y\&\(\(..\)\(\ \)\)\)\/\(x\& .. \)\)]], " " }], "NumberedEquation"], Cell[TextData[{ "Applying the linker concept, we generalize our calculation and for time \ being put aside the specificities of the Catenaries; we replace ", Cell[BoxData[ \(TraditionalForm\`y\)]], " with ", Cell[BoxData[ \(TraditionalForm\`f(x)\)]], ". Noting that ", Cell[BoxData[ \(TraditionalForm\`f(x)\)]], " could represent curves other than a Catenary provides opportunities to \ apply our methodology to analyze a vast class of similar problems. In terms \ of ", Cell[BoxData[ \(TraditionalForm\`f(x)\)]], " we evaluate; ", Cell[BoxData[ \(TraditionalForm\`\(y\& . \)( t) \[Congruent] \(\(x\& . \)(t)\)\ f' \((x)\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(\(y(t)\)\& .. \) \[Congruent] \(\(x\& .. \)( t)\)\ f' \((x)\) + \(\(x\& . \^2\)(t)\)\ f'' \((x)\)\)]], " with the primes denoting the spacial derivatives i.e. ", Cell[BoxData[ \(TraditionalForm\`f' \((x)\) = \(\(\(d\)\(\ \)\)\/dx\) \(f(x)\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`f'' \((x)\) \[Congruent] \(d\^2\/dx\^2\) \(f( x)\)\)]], ". Substituting these identities in eq (4) yields," }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`\(x\& .. \)\ [1 + f' \((x)\)\^2] + x\& . \^2\ f'' \((x)\)[f' \((x)\) - \[Mu]] + g\ [f' \((x)\) - \[Mu]] = 0\)]], " " }], "NumberedEquation"], Cell[TextData[{ "One might realize that in the absence of kinetic friction, ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\ = 0\)]], ", eq (5) for a slanted incline with an inclination angle \[Theta] with \ respect to the horizontal axis reduces to the expected well known horizontal \ acceleration ", Cell[BoxData[ \(TraditionalForm\`\(x\& .. \) = \(-\((g\ sin\ \[Theta])\)\) cos\[Theta]\)]], ". " }], "Text"], Cell[TextData[{ "Back to our problem, eq (5) for a Catenary ", Cell[BoxData[ \(TraditionalForm\`f(x) \[Congruent] y(x) + \[Lambda] = c\ cosh[\(x(t) - \[Alpha]\)\/c]\)]], " yields, ", Cell[BoxData[ \(TraditionalForm\`f' \((x)\) = sinh\ [\(x(t) - \[Alpha]\)\/c]\)]], ", ", Cell[BoxData[ \(TraditionalForm\`f'' \((x)\) = \(1\/c\) cosh[\(x(t) - \[Alpha]\)\/c]\)]], ". Utilizing the trigonometry identity ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[\(cosh\^2\ \[Beta]\), "TraditionalForm"], "-", \(\(sinh\^2\) \[Beta]\)}], "\[Congruent]", "1"}], TraditionalForm]]], ", eq (5) gives," }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`\(cosh\^2\)[\(x(t) - \[Alpha]\)\/c]\ \(x\& .. \) + \ \(1\/c\) cosh[\(x(t) - \[Alpha]\)\/c]\ {\ sinh[\(x(t) - \[Alpha]\)\/c] - \[Mu]}\ \ x\& . \^2 + g\ {sinh[\(x(t) - \[Alpha]\)\/c] - \[Mu]} = 0\)]], " " }], "NumberedEquation"], Cell[TextData[{ "Equation (6) is a non-trivial, non-linear second order differential \ equation with variable coefficients; ", StyleBox["Mathematica", FontSlant->"Italic"], " fails to solve the equation symbolically. However, it solves it \ numerically with ease! Plotting and tabulating its solutions provides ample \ and insightful information about the kinematics and dynamics of the bead. \ Assuming the bead is released at rest from the high at the right-end of the \ chain, we supply the needed initial condition; ", Cell[BoxData[ \(TraditionalForm\`\(x\& . \)(t = 0) = 0\)]], ". We wrap the Table command around the NDSolve and for a set of \ reasonable practical values of \[Mu], e.g. ", Cell[BoxData[ \(TraditionalForm\`0 \[LessEqual] \[Mu] \[LessEqual] 0.8\)]], " we generate its solutions. For a demonstrative purpose, with no \ prejudice, we select one of the Catenaries, e.g. the one with the highest \ right-end. The characteristic parameters of this Catenary are given by \ rootsy\[LeftDoubleBracket]6\[RightDoubleBracket]; we get, " }], "Text"], Cell[BoxData[ RowBox[{ StyleBox[\(g = 9.80\), FontColor->GrayLevel[0]], StyleBox[";", FontColor->GrayLevel[0]], RowBox[{ StyleBox["\[Alpha]", FontColor->GrayLevel[0]], StyleBox["=", FontColor->GrayLevel[0]], "0.081498"}], StyleBox[";", FontColor->GrayLevel[0]], RowBox[{ StyleBox["c", FontColor->GrayLevel[0]], StyleBox["=", FontColor->GrayLevel[0]], "0.355519"}], ";", RowBox[{ StyleBox["\[Lambda]", FontColor->GrayLevel[0]], StyleBox["=", FontColor->GrayLevel[0]], \(-0.435099\)}], ";"}]], "Input", CellLabel->"In[16]:="], Cell[BoxData[ \(\(sol\[Alpha]c\[Mu] = Table[NDSolve[{\ \(Cosh[\(x[t] - \[Alpha]\)\/c]\^2\) \(x''\)[t] + 1\/c\ \ Cosh[\(x[t] - \[Alpha]\)\/c]\ \((Sinh[\(x[t] - \ \[Alpha]\)\/c] - \[Mu])\) \(x'\)[t]\^2 + g\ \((Sinh[\(x[t] - \[Alpha]\)\/c] - \[Mu])\) \[Equal] 0, x[0] \[Equal] 0.8, \(x'\)[0] == 0}, x[t], {t, 0, 5}], {\[Mu], 0, 0.8, 0.2}];\)\)], "Input", CellLabel->"In[17]:="], Cell[TextData[{ "Solutions of eq (6) are ", Cell[BoxData[ \(TraditionalForm\`x(t)\)]], ". Knowing the time dependency of ", Cell[BoxData[ \(TraditionalForm\`x(t)\)]], " helps to evaluate the vertical position of the bead in its descent as \ well. Applying ", Cell[BoxData[ \(TraditionalForm\`y(x) + \[Lambda] = c\ cosh[\(x(t) - \[Alpha]\)\/c]\)]], " we plot and tabulate ", Cell[BoxData[ \(TraditionalForm\`{x(t), y(t)}\)]], "; the tables are suppressed. 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