(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing 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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", StyleBox["Do this for both resonance and beats, but also use other forcing \ functions too, including one on which undetermined coefficients might not \ work. Report on your results. Part of your report must include \ mathematical, graphical, and physical descriptions of both beats and \ resonance.", FontWeight->"Bold"] }], "Text"], Cell[CellGroupData[{ Cell["Simple Harmonic Motion with No Forcing Function", "Subsubsection"], Cell[BoxData[{\(Clear[x, t, de1, sol1, pp1]\), "\n", RowBox[{ RowBox[{"de1", "=", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "+", \(12\ x[t]\)}], "==", "0"}]}], ";", RowBox[{"sol1", "=", RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{"de1", ",", \(x[0] == 0\), ",", RowBox[{ RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "[", "0", "]"}], "==", "1"}]}], "}"}], ",", \(x[t]\), ",", "t"}], "]"}]}]}], "\n", \(pp1 = Plot[sol1[\([1, 1, 2]\)], {t, 0, 40}, PlotStyle -> RGBColor[1, 0, 0]]\ ;\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Resonance - Forcing Function with Same Frequency as System", \ "Subsubsection"], Cell[BoxData[{\(Clear[x, t, de2, sol2, pp2]\), "\n", RowBox[{ RowBox[{"de2", "=", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "+", \(12\ x[t]\)}], "==", \(Cos[Sqrt[12] t]\)}]}], ";", RowBox[{"sol2", "=", RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{"de2", ",", \(x[0] == 0\), ",", RowBox[{ RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "[", "0", "]"}], "==", "1"}]}], "}"}], ",", \(x[t]\), ",", "t"}], "]"}]}]}], "\n", \(pp2 = Plot[sol2[\([1, 1, 2]\)], {t, 0, 40}, PlotStyle -> RGBColor[0, 1, 0]]\ ;\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Beats - Forcing Function with Frequency Close to that of System\ \>", "Subsubsection"], Cell[BoxData[ RowBox[{ StyleBox[\(Note\ that\ for\ the\ problem\ below\), FontFamily->"Times New Roman"], StyleBox[",", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], RowBox[{ StyleBox["the", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["coefficient", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["of", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[ StyleBox[ RowBox[{ StyleBox[" ", FontSlant->"Plain"], " "}]], FontFamily->"Times New Roman"], StyleBox["is", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox[\(\@12\), FontFamily->"Times New Roman"], StyleBox["which", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["is", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["approximately", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox[\(3.4642 . \ Notice\), FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["what", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["happens", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["when", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["the", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["forcing", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["function", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["has", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["a", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["coefficient", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["close", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["to", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox[\(\(that\)\(.\)\(\ \)\), FontFamily->"Times New Roman"]}]}]], "Text"], Cell[BoxData[{\(Clear[x, t, de3, sol3, pp3]\), "\n", RowBox[{ RowBox[{"de3", "=", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "+", \(12\ x[t]\)}], "==", RowBox[{"Cos", "[", RowBox[{ StyleBox["3", FontColor->RGBColor[1, 0, 0]], "t"}], "]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{"sol3", "=", RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{"de3", ",", \(x[0] == 0\), ",", RowBox[{ RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "[", "0", "]"}], "==", "1"}]}], "}"}], ",", \(x[t]\), ",", "t"}], "]"}]}], "\n", RowBox[{ RowBox[{"pp3", "=", RowBox[{"Plot", "[", RowBox[{\(sol3[\([1, 1, 2]\)]\), ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", StyleBox["40", FontColor->RGBColor[1, 0, 0]]}], "}"}], ",", \(PlotStyle -> RGBColor[0, 0, 1]\)}], "]"}]}], " ", ";"}]}], "Input"], Cell["\<\ You can look at all three solutions together - no forcing function, one \ causing resonance, and one causing beats.\ \>", "Text"], Cell[BoxData[ \(\(Show[pp1, pp2, pp3]\ ;\)\)], "Input"], Cell["\<\ Now let's see what happens as we get closer to the frequency of the system. \ In particular, notice what happens to both the frequency of the beats and the \ size of the amplitude as those coefficients get closer together. 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