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In \ this lab we will be considering the behavior of the solutions of some such \ initial-value problems for different values of ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["m", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["c", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["k", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox["0", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSize->10, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox["'", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["0", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSize->10, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[".", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Part 1: Springs without Friction"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Consider the differential equation ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox["'' + 4 ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" = 0, which has the form of the equation above provided ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["c", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" = 0 (so there is no damping force) and ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["k", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" = 4", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["m", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[". The equation has general solution ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox["(", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[") = ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["C", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" cos(2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[") + ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["D", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" sin(2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[ "). The constants can only be determined with additional information. ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" find this solution using the command", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["gsoln = DSolve[y''[t]+4 y[t]==0,y[t],t]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "What happens to a mass attached to a spring modeled by this equation if \ its initial position is one unit above the equilibrium position? It depends \ on the initial velocity. To find a solution when the mass starts at rest (no \ velocity), replace the differential equation in the command above with the \ list of equations: ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["{y''[t]+4 y[t]==0, y[0]==1, y'[0]==0}", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[". Assign your output the name ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["soln[0]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" (rather than ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["gsoln", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox["). ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Here's a little trick to avoid losing the original input and the \ corresponding output: position the cursor just below the output; it should \ flop over on its side and look like a dog bone. Press the left mouse button \ and a horizontal line should appear where the cursor is. Now press down the \ CRTL key and at the same time type L and ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[ " will repeat the last command line. Use the mouse to position the cursor \ in the new input \"cell\" where you would like to edit; click the left button \ again and the blinking \"insertion point\" will appear just where you want to \ add more text.(Note that when we have a list of equations to solve, we \ enclose them in curly brackets, ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["{}", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[".) Enter your edited command. What solution did you get?", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Notice that ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[ " gives the solution in the form of a replacement rule, with an arrow not \ an equal sign. To define a function we use the command", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["a[t_] := y[t] /. soln[0]"], "Input", AspectRatioFixed->True], Cell[TextData[ "To make sure we have defined the function correctly, we can plot it"], "Text",\ Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Plot[a[t],{t,0,20}]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["If we have solved the equation and defined ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["a", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox["(", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[ ") correctly, the solution looks like a cosine curve. That doesn't mean the \ mass is moving along a cosine curve \[LongDash] after all, it's attached to a \ spring and moves up and down. Briefly explain why the curve has two \ dimensions (vertical and horizontal) while the motion only has one dimension \ (vertical).", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["\n\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Solve the inital value problem with inital velocity \[Dash]1, and use it \ to define a function ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["b", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox["(", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox["). Write the definition of ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["b", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox["(", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[") below.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Solve the inital value problem with inital velocity 1, and use it to \ define a function ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["c", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox["(", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox["). Write the definition of ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["c", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox["(", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["t", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[") below.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "We can plot all three solutions at once using the command"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Plot[{a[t],b[t],c[t]},{t,0,20},\n PlotStyle->{Red,Green,Blue}]"], "Input",\ AspectRatioFixed->True], Cell[TextData[ "It turns out that two of the solutions actually describe the same motion, \ the only difference being that the masses are doing the same things at \ different times. (It's just like two people making the same drive on \ different days.) Which two are they? (Hint: In this kind of situation there's \ an old saying: \"What goes up must come down, and with the same speed, \ too.\")"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["How can you tell they're the same, from the picture?"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Part 2: Energy and Differential Equations"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Do this part after the lab period is over!"], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "In Part 1 of this lab we investigated solutions to the differential \ equation ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox["'' + 4 ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[ " = 0, which models the motion of a mass suspended from a spring ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["without damping", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[ ". 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Use ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" to find the general solution that is valid when ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["L", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" = 0.5, ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["R", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[" = 10.0, and ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["C", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[ " = 0.25. 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Write the general solution below.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "We're going to compare the three particular solutions we get with the \ following initial conditions.\n\n(a) ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["q", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox["(0) = 0 and ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["q", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox["'(0) = 0. 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