(*********************************************************************** 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[ 5796, 156]*) (*NotebookOutlinePosition[ 6637, 183]*) (* CellTagsIndexPosition[ 6593, 179]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["The Pendulum", FontColor->RGBColor[0, 0, 1]]], "Section"], Cell[TextData[{ "In this project you will investigate the built-in ", StyleBox["Mathematica", FontSlant->"Italic"], " commands to solve a differential equation by looking at equations which \ model the motion of a pendulum. You will look at some increasingly \ complicated versions of the problem and investigate the behavior of the \ model." }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ " ", StyleBox["Model 1 - a linear model", FontColor->RGBColor[1, 0, 0]] }], "Subsection"], Cell[TextData[{ "A pendulum is made of a \"bob\" which hangs from a rigid stick of length, \ say L. The other end of the stick is free to rotate on a frictionless pin. \ \n\nWe assume that initially, i.e., at T=0, the bob is given an angular \ displacement of ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\_0\)]], " and then released. The problem is to determine the position of the bob \ over time.\n\nThe equation for the angle \[Theta] can be written as:\n\n \ \[Theta] '' = - (G / L) * sin( \[Theta] )\n \n\ where G is the gravitational acceleration constant. If \[Theta] is small, \ we can approximate sin( \[Theta] ) by \[Theta] , obtaining the linearized \ pendulum equation\n\n ", StyleBox[" \[Theta] '' = - (G / L) * \[Theta] ", FontColor->RGBColor[1, 0, 1]], "\n\nwith initial connditions\n \n ", StyleBox[" \[Theta] (0) = ", FontColor->RGBColor[1, 0, 1]], Cell[BoxData[ \(TraditionalForm\`\[Theta]\_0\)], FontColor->RGBColor[1, 0, 1]], StyleBox["\n \[Theta] '(0) = 0 ", FontColor->RGBColor[1, 0, 1]], StyleBox[" \n \n", FontColor->RGBColor[0, 1, 0]], "In order to use most ODE packages, we need to write this second order ODE \ as a pair of first order equations. Let U be another name for \[Theta] and \ V denote \[Theta] '. We then have\n\n ", StyleBox[ " U ' = V\n V ' = - ( G / L) * U\n U(0) \ =", FontColor->RGBColor[1, 0, 1]], Cell[BoxData[ \(TraditionalForm\`\[Theta]\_0\)], FontColor->RGBColor[1, 0, 1]], StyleBox["\n V(0) = 0", FontColor->RGBColor[1, 0, 1]] }], "Text"], Cell[TextData[ "1. For this simple system of equations we can obtain both an exact and an \ approximate solution.\nIn these calculations use G=386.09 inches/second and \ L=10 inches and \[Theta]=\[Pi] /6. Choose the final time so that the \ pendulum swings back and forth at least 3 times. In each case plot the \ displacement and estimate the period of the pendulum graphically, i.e., the \ length of time it takes before its motion repeats.\n\na. Use the command \ DSolve to find the exact solution of the first order system."], "Text"], Cell["b. Use the command NDSolve to find an approximate solution.", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ " ", StyleBox["Model 3 - pendulum with a driving force", FontColor->RGBColor[1, 0, 0]] }], "Subsection"], Cell[TextData[{ "It is interesting to consider the case where the pendulum has a driving \ force; that is,\n \[Theta] '' = - ( G - A * ", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(\ \[Omega]\^2\), "TraditionalForm"], " "}], TraditionalForm]]], "* sin (\[Omega] t ) ) * sin( \[Theta] ) / L" }], "Text"], Cell[TextData[{ "4. Approximate the solution to this problem for the following choices of \ parameters:\n\n A \[Omega] \ ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\_0\)]], "\n_______________________________________ \n 0.5\t \ 5.3 0.05\n 10.0\t 5.3\t 0.05\n \ 10.0 5.3 0.5 \n \nDiscuss \ your results. " }], "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 1152}, {0, 850}}, WindowSize->{520, 740}, WindowMargins->{{62, Automatic}, {Automatic, 11}}, MacintoshSystemPageSetup->"\<\ 00<0001804P000000]P2:?oQon82n@960dL5:0?l0080001804P000000]P2:001 0000I00000400`<300000BL?00400@0000000000000006P801T1T00000000000 00000000000000000000000000000000\>" ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1731, 51, 83, 1, 50, "Section"], Cell[1817, 54, 366, 8, 63, "Text"], Cell[CellGroupData[{ Cell[2208, 66, 115, 4, 44, "Subsection"], Cell[2326, 72, 1772, 37, 446, "Text"], Cell[4101, 111, 545, 7, 126, "Text"], Cell[4649, 120, 76, 0, 30, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[4762, 125, 129, 4, 44, "Subsection"], Cell[4894, 131, 368, 9, 47, "Text"], Cell[5265, 142, 503, 10, 158, "Text"] }, Open ]] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)