(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 137338, 5452]*) (*NotebookOutlinePosition[ 138368, 5485]*) (* CellTagsIndexPosition[ 138324, 5481]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ VELOCITY PROFILE OF A PARTICLE MOVING ON AN IRREGULAR PATHWAY WITH FRICTION \ USING A COMPUTER ALGEBRA SYSTEM\ \>", "Text", CellMargins->{{32, 34}, {Inherited, Inherited}}, PageBreakBelow->Automatic, TextAlignment->Center, FontFamily->"Courier New", FontSize->16, FontWeight->"Bold"], Cell["\<\ Charles B. Wakefield Department of Science, Mathematics, and Computer Science University of Texas of the Permian Basin Odessa, Texas 79762 e-mail: Wakefield_C@UTPB.edu\ \>", "Text", CellMargins->{{32, 34}, {Inherited, Inherited}}, PageBreakBelow->Automatic, TextAlignment->Center, FontFamily->"Courier New", FontSlant->"Italic"], Cell["ABSTRACT", "Text", CellMargins->{{32, 34}, {Inherited, Inherited}}, PageBreakBelow->Automatic, TextAlignment->Center, FontFamily->"Courier New", FontSize->14, FontWeight->"Bold"], Cell[TextData[{ "\tIn this study, using the computer algebra system ", StyleBox["Mathematica", FontSlant->"Italic"], ", we consider the evaluation of the velocity at any point along an \ irregular pathway. We assume there is friction which is a function of the \ horizontal distance along the pathway. Since there is friction, we can't \ assume conservative forces. Also, because the pathway is not simple we must \ use numerical integration to evaluate the distance traveled. This is \ necessary because for nonconservative forces, the force is dependent on the \ path.\n\n\tThe pathway data is fit with a fifth degree polynomial. The \ distance along the path is calculated with the numerical integrator for \ several different values of the independent variable. These distances are \ plotted and then fitted with a cubic fitting function. This gives us a \ function which will allow us to calculate the distance along the path at any \ point on the path. Using the work energy relation for nonconservative \ forces", Cell[BoxData[ \(TraditionalForm\`\^1\)]], "\n\t\n\t ", Cell[BoxData[ \(TraditionalForm \`w\_\(nc\ \)\ = \ \(\(-\ f\_k\)\ \ \(d(x)\)\ = \ \[CapitalDelta]K\ + \ \[CapitalDelta]U\)\)]], ",\n\nwe derive an expression for the final velocity as a function of x. \ This ", Cell[BoxData[ \(TraditionalForm\`v\_f\)]], " is then plotted for the region of interest. We then show a graphic which \ illustrates how the velocity will change with a range of different values of \ the coefficient of friction. \n\n\tThe problem is understandable to students \ but not doable without a Computer Algebra System. ", StyleBox[ "This problem illustrates vividly the Calculus in Calculus Based \ Engineering Physics", FontSlant->"Italic"], ". ", StyleBox[ "It can be used to illustrate the use of a computer algebra system and the \ basic Calculus involved in the Engineering Physics course and the Calculus II \ course. ", FontSlant->"Italic"] }], "Text", CellMargins->{{32, 34}, {Inherited, Inherited}}, PageBreakBelow->Automatic, FontFamily->"Times New Roman"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Load needed packages."]], "Subsection"], Cell[BoxData[ \(\(Needs["\"]; \)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Load the irregualr pathway data."]], "Subsection"], Cell[BoxData[ \(\(xydata\ = \ {{1, 1.6}, {1.2, 0.97521}, {1.4, 0.38768}, {1.6, 0.213284}, {1.8, 0.398258}, {2, 0.670916}, {2.2, 0.798382}, {2.4, 0.730886}, {2.6, 0.578476}, {2.8, 0.483784}, {3.0, 0.50677}}; \)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Find the fitting function for the pathway data and show the plot of the \ function and the data so we can see how well the fitting function fits the \ data."]], "Subsection"], Cell[BoxData[ \(\(t1\ = \ ListPlot[xydata, PlotStyle -> {PointSize[ .025], Red}, DisplayFunction -> Identity]; \)\)], "Input"], Cell[CellGroupData[{ 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Background->RGBColor[1, 1, 0]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Calculate the derivative of the fitting function and the square of the \ derivative for use in calculating the distance along the pathway."]], "Subsection"], Cell[CellGroupData[{ Cell[BoxData[{ \(fprime[x_]\ = \ D[fitfun[x], x]\), \(fprimesq\ = \ fprime[x]^2 // Expand\)}], "Input"], Cell[BoxData[ \(\(56.0233081421425715`\[InvisibleSpace]\) - 162.476168458496044`\ x + 153.454822716227745`\ x\^2 - 58.7406606934278219`\ x\^3 + 7.88505108172457624`\ x\^4\)], "Output"], Cell[BoxData[ \(\(3138.61105518945837`\[InvisibleSpace]\) - 18204.9049026099787`\ x + 43592.5989548117609`\ x\^2 - 56447.1955218228416`\ x\^3 + 43519.7908721823445`\ x\^4 - 20590.3411216043696`\ x\^5 + 5870.46345040930628`\ x\^6 - 926.346220283858734`\ x\^7 + 62.1740305614059085`\ x\^8\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "The distance along the pathway must be determined by numerical integration, \ since the expression for the arc length can't be integrated in closed form. \ So, we build a table of values which we will plot and then fit with a \ polynomial. This will provide us with a smooth function for determining the \ distance along the curved pathway."]], "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Ldata\ = \ Table[{c, \ NIntegrate[Sqrt[1\ + \ fprimesq], \ {x, 1, c}]}, \ {c, 1, 3.0, .2}]\)], "Input"], Cell[BoxData[ \({{1, 0}, {1.19999999999999996`, 0.747958935004635705`}, { 1.39999999999999991`, 1.26467452438423055`}, {1.60000000000000008`, 1.5266823158943632`}, {1.8`, 1.75888698256435453`}, {2.`, 2.05808773385719856`}, {2.20000000000000017`, 2.32642921607056107`}, { 2.40000000000000035`, 2.53467395057158517`}, {2.60000000000000008`, 2.80035426339472479`}, {2.79999999999999982`, 3.09123807874857847`}, { 3.`, 3.36830737080602737`}}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(t3\ = \ ListPlot[Ldata, PlotStyle -> {PointSize[ .025], Red}, DisplayFunction -> Identity]; \)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Determine the fitting function for distance along the pathway."]], "Subsection"], 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We will then check one point by \ letting the friction "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[Mu]\)]]], StyleBox[ " be equal to zero and use the fact that without the dissipative force of \ friction the problem should collapse to the simpler conservative force \ problem where "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta]K\ = \ \(-\ \[CapitalDelta]U\)\)]]], StyleBox[". This calculation gets exactly what we get when "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[Mu]\)]]], StyleBox[" is zero. When we increase "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[Mu]\)]]], StyleBox[" the velocity slows down as we would expect."] }], "Subsection"], Cell[TextData[StyleBox["DERIVATION"]], "Text", FontFamily->"Courier New", FontSize->18, FontWeight->"Bold"], Cell[TextData[{ "From the work energy theorem for non conservative forces we know that the \ following is true.\n\n(1)\t\t", Cell[BoxData[ \(TraditionalForm \`\(-\[Mu]\)\ N\ \(dist(x)\) = \ \(1\/2\) m\ \((v\_f\^2\ - \ v\_o\^2)\)\ + \ m\ g\ \((fitfun(x)\ - \ fitfun(x\_0))\)\)]], "\n\t\t\nIn this equation ", Cell[BoxData[ \(TraditionalForm\`N\ = \ m\ g\ \(Cos(\[Theta])\)\)]], ". In this problem ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " changes with x. However, the derivative at any point along the curve \ gives the slope of the curve at that value of x. This slope is equal to the \ tangent of the angle, ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], ", makes with the x axis. So, ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\ = \ ArcTan\ \((fprime(x)\)\)]], ". Therefore,\n\n(2)\t\t", Cell[BoxData[ \(TraditionalForm\`N\ = \ m\ g\ Cos\ \((Arctan(fprime(x)))\)\)]], "\n\t\t\nIf we substitute this result into equation (1) and recall that ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\ = \ \[Sigma]\ \(f(\ x)\)\)]], ", we have:\n ", Cell[BoxData[ \(TraditionalForm \`\(-\[Sigma]\)\ \(f(x)\)\ m\ g\ \(Cos(Arctan(fprime(x)))\)\ \ \(dist(x)\)\ = \ \ 1\/2\ m\ \((v\_f\^2\ - \ v\_0\^2)\)\ + \ m\ g\ \((fitfun(x)\ - \ \ fitfun(x\_0))\)\)]], "\n\nIf this equation is solved for ", Cell[BoxData[ \(TraditionalForm\`v\_f\)]], " we arrive at the final working equation.\n\n(3) ", Cell[BoxData[ \(TraditionalForm \`v\_f\ = \ \ \[Sqrt]\(( \(-2\)\ \[Sigma]\ \(f(\ x)\)\ g\ \(Cos(Arctan(fprime(x)))\)\ \(dist(x)\)\ + \ v\_0\^2\ - \ 2\ g\ \(fitfun(x)\)\ + \ 2\ g\ \(fitfun(x\_0)\))\)\)]] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "For out first calculation, we will use ", Cell[BoxData[ \(TraditionalForm\`f(x)\ = \ x\)]], "." }], "Subsection"], Cell[BoxData[ \(\(vf[x_]\ := \ Sqrt[\(-2.0\)\ sigma\ x\ g\ Cos[ArcTan[fprime[x]]]\ dist[x]\ + \ v0^2\ - \ 2\ g\ fitfun[x]\ + \ 2\ g\ fitfun[x0]]; \)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Using the working equation above turn off the friction and let us see if the \ results gives us the value for a conservative force system which is evaluated \ with the \"Solve\" instruction."]], "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(vf[1.5] /. sigma -> 0.0\)], "Input"], Cell[BoxData[ \(5.16669434330274768`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[ .5\ v0^2\ + \ g\ fitfun[1.0]\ == \ .5\ vfinal^2\ + \ g\ fitfun[1.5], vfinal] // Last\)], "Input"], Cell[BoxData[ \({vfinal \[Rule] 5.16669434330274768`}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Now let us evaluate the velocity when some friction is introduced. 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In this case ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\ = \ \[Sigma]\ x\)]], ", the horizontal distance." }], "Text", Background->RGBColor[1, 1, 0]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "We will next assume that the friction ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\ = \ \[Sigma]\ \(dist(x)\)\)]], ", the actual distance along the irregular path rather than x, the \ horizontal distance. The plot is supressed and will be shown later." }], "Subsection"], Cell[BoxData[ \(Clear[x]\)], "Input"], Cell[BoxData[ \(\(vf2[x_]\ := \ Sqrt[\(-2.0\)\ sigma\ dist[x]\ g\ Cos[ArcTan[fprime[x]]]\ dist[x]\ + \ v0^2\ - \ 2\ g\ fitfun[x]\ + \ 2\ g\ fitfun[x0]]; \)\)], "Input"], Cell[BoxData[ \(\(path2\ = \ Plot[vf2[x], \ {x, 1, 3}, PlotLabel -> "\", PlotStyle -> {Green, Thickness[ .01]}, DisplayFunction -> Identity]; \)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Use the function of ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\ = \ \[Sigma]\ \(dist(x)\)\^2\)]], ", the actual path distance squared. The plot is supressed and will be \ shown later." }], "Subsection"], Cell[BoxData[ \(Clear[x]\)], "Input"], Cell[BoxData[ \(\(vf3[x_]\ := \ Sqrt[\(-2.0\)\ sigma\ dist[x]^2\ g\ Cos[ArcTan[fprime[x]]]\ dist[x]\ + \ v0^2\ - \ 2\ g\ fitfun[x]\ + \ 2\ g\ fitfun[x0]]; \)\)], "Input"], Cell[BoxData[ \(\(path3\ = \ Plot[vf3[x], \ {x, 1, 3}, PlotLabel -> "\", PlotStyle -> {Red, Thickness[ .01]}, DisplayFunction -> Identity]; \)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "In the next plot we see all three functions tried for the friction ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\ = \ \[Sigma]\ x\)]], ", the green curve, ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\ = \ \[Sigma]\ \(dist(x)\)\)]], ", the blue curve, and finally, ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\ = \ \[Sigma]\ \(dist(x)\)^2\)]], ", the red curve." }], "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(\(pathcomposite\ = \ Show[path1, path2, path3, DisplayFunction -> $DisplayFunction]; \)\)], "Input"], 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Comparing the two we see that the velocity rises as the path decends \ and vice versa.\ \>", "Text", Background->RGBColor[1, 1, 0]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Change vf so that we can plot the same graph with several different values \ of friction."]], "Subsection"], Cell[BoxData[ \(Clear[x]\)], "Input"], Cell[BoxData[ \(\(vff[x_]\ := \ Sqrt[\(-2.0\)\ mu\ \ g\ Cos[ArcTan[fprime[x]]]\ dist[x]\ + \ v0^2\ - \ 2\ g\ fitfun[x]\ + \ 2\ g\ fitfun[x0]]; \)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ In the following plot the top curve is the velocity for the friction turned \ off. 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