(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Wolfram 14.3' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 154, 7] NotebookDataLength[ 746814, 13252] NotebookOptionsPosition[ 732821, 13010] NotebookOutlinePosition[ 733221, 13026] CellTagsIndexPosition[ 733178, 13023] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["PHYS 611, Fall 2025", "Title", CellChangeTimes->{{3.9685049490514936`*^9, 3.968504956163288*^9}},ExpressionUUID->"077e4eb2-d171-421b-867b-\ cd2fbf268f1a"], Cell["Theoretical Mechanics", "Subtitle", CellChangeTimes->{{3.968517185230608*^9, 3.9685171898882847`*^9}},ExpressionUUID->"718ce460-b110-4ad7-9c30-\ 172c19149919"], Cell[CellGroupData[{ Cell["Homework Set 1. - TA Solution", "Chapter", CellChangeTimes->{{3.968504900718302*^9, 3.9685049115172043`*^9}, { 3.9685147811063128`*^9, 3.96851478675247*^9}},ExpressionUUID->"07e697e4-53cc-402f-a451-\ 5ac477dc5e30"], Cell[TextData[ButtonBox["Thomas (Tom) C. 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CellLabel->"In[8]:=",ExpressionUUID->"d1248f52-ff2f-42fb-b098-6e33f83291ff"], Cell["\<\ Since we\[CloseCurlyQuote]re solving a 2nd order homogeneous ODE, we know we \ can form a general solution as a superposition of the two solutions found in \ (a), so we can specifically take\ \>", "Text", CellChangeTimes->{{3.96850524634711*^9, 3.9685053230491734`*^9}, { 3.9685077648780727`*^9, 3.968507773293*^9}},ExpressionUUID->"c4168bcb-94ad-4d9f-a15e-38081ad72317"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"kp", " ", "=", " ", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], " ", "\[Omega]"}], "/.", " ", RowBox[{ RowBox[{"Solve", "[", RowBox[{"CONDa", ",", " ", "\[Omega]"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], "//", "Expand"}]}], "\[IndentingNewLine]", RowBox[{"km", " ", "=", " ", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], " ", "\[Omega]"}], " ", "/.", RowBox[{ RowBox[{"Solve", "[", RowBox[{"CONDa", ",", " ", "\[Omega]"}], "]"}], "[", RowBox[{"[", "2", 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