(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 72935, 1769] NotebookOptionsPosition[ 69237, 1642] NotebookOutlinePosition[ 69682, 1661] CellTagsIndexPosition[ 69639, 1658] WindowFrame->Normal ContainsDynamic->True *) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Morin 4.22 (Projectile on Spring)", "Subsection", CellChangeTimes->{{3.4507107110545154`*^9, 3.4507107202422504`*^9}}], Cell["\<\ A projectile of mass m is fired from the origin with speed v0 and angle \ \[Theta]0 in the presence of gravity g. It is also attached to a zero-length massless spring with constant k=m \ \[Omega]^2 (with the other end at the origin). (a) Find x(t) and y(t) (b) Show that the motion is the expected parabola when the spring is weak, \ and the expected harmonic motion when the spring is strong. (And note that \ \"weak\"and \"strong\" refer to the strength relative to gravity.) (c) Find the \[Omega] such that the projectile hits the ground going straight \ down.\ \>", "Text", CellChangeTimes->{{3.450710724742366*^9, 3.4507109081845617`*^9}, { 3.4507109880616064`*^9, 3.4507110402191916`*^9}, {3.4507113399768653`*^9, 3.450711370462021*^9}, {3.450711924429327*^9, 3.4507119287575626`*^9}, { 3.4507122230932226`*^9, 3.450712249234517*^9}}], Cell["\<\ Begin as always with the forces: there is a -mg in the y direction, and a \ spring force of magnitude k*r with x-component -kr*Cos[\[Theta]] = -kx and y component -ky:\ \>", "Text", CellChangeTimes->{{3.4507118163640604`*^9, 3.4507119412891335`*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"eqnx", " ", "=", " ", RowBox[{ RowBox[{"m", " ", RowBox[{ RowBox[{"x", "''"}], "[", "t", "]"}]}], " ", "\[Equal]", " ", RowBox[{ RowBox[{"-", "m"}], " ", RowBox[{"\[Omega]", "^", "2"}], " ", RowBox[{"x", "[", "t", "]"}]}]}]}], "\[IndentingNewLine]", RowBox[{"eqny", " ", "=", " ", RowBox[{ RowBox[{"m", " ", RowBox[{ RowBox[{"y", 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CellChangeTimes->{3.4510416933744597`*^9}] }, Open ]], Cell["We launch with given conditions:", "Text", CellChangeTimes->{{3.4507119860402794`*^9, 3.4507120004000216`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"s", " ", "=", " ", RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{"eqnx", ",", "eqny", ",", RowBox[{ RowBox[{"x", "[", "0", "]"}], "\[Equal]", "0"}], ",", RowBox[{ RowBox[{"y", "[", "0", "]"}], "\[Equal]", "0"}], ",", RowBox[{ RowBox[{ RowBox[{"x", "'"}], "[", "0", "]"}], "\[Equal]", " ", RowBox[{"v0", " ", RowBox[{"Cos", "[", "\[Theta]0", "]"}]}]}], ",", " ", RowBox[{ RowBox[{ RowBox[{"y", "'"}], "[", "0", "]"}], "\[Equal]", " ", RowBox[{"v0", " ", RowBox[{"Sin", "[", "\[Theta]0", "]"}]}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"x", "[", "t", "]"}], ",", RowBox[{"y", "[", "t", "]"}]}], "}"}], ",", "t"}], "]"}]}]], "Input", CellChangeTimes->{{3.450712003118841*^9, 3.4507120708237*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"x", "[", "t", "]"}], "\[Rule]", FractionBox[ RowBox[{"v0", " ", RowBox[{"Cos", "[", "\[Theta]0", "]"}], " ", RowBox[{"Sin", "[", RowBox[{"t", " ", "\[Omega]"}], "]"}]}], "\[Omega]"]}], ",", RowBox[{ RowBox[{"y", "[", "t", "]"}], "\[Rule]", FractionBox[ RowBox[{ RowBox[{"-", "g"}], "+", RowBox[{"g", " ", RowBox[{"Cos", "[", RowBox[{"t", " ", "\[Omega]"}], "]"}]}], "+", RowBox[{"v0", " ", "\[Omega]", " ", RowBox[{"Sin", "[", "\[Theta]0", "]"}], " ", RowBox[{"Sin", "[", RowBox[{"t", " ", "\[Omega]"}], "]"}]}]}], SuperscriptBox["\[Omega]", "2"]]}]}], "}"}], "}"}]], "Output", CellChangeTimes->{3.4510416941401043`*^9}] }, Open ]], Cell["\<\ Let's package up the solution in the form of a vector-valued function of \ time:\ \>", "Text", CellChangeTimes->{{3.4507121508101225`*^9, 3.4507121667949066`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"pt", "[", "t_", "]"}], " ", "=", " ", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"x", "[", "t", "]"}], ",", RowBox[{"y", "[", "t", "]"}]}], "}"}], "/.", RowBox[{"s", "[", RowBox[{"[", "1", "]"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.4507121692012177`*^9, 3.4507121789045916`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ FractionBox[ RowBox[{"v0", " ", RowBox[{"Cos", "[", "\[Theta]0", "]"}], " ", RowBox[{"Sin", "[", RowBox[{"t", " ", "\[Omega]"}], "]"}]}], "\[Omega]"], ",", FractionBox[ RowBox[{ RowBox[{"-", "g"}], "+", RowBox[{"g", " ", RowBox[{"Cos", "[", RowBox[{"t", " ", "\[Omega]"}], "]"}]}], "+", RowBox[{"v0", " ", "\[Omega]", " ", RowBox[{"Sin", "[", "\[Theta]0", "]"}], " ", RowBox[{"Sin", "[", RowBox[{"t", " ", "\[Omega]"}], "]"}]}]}], SuperscriptBox["\[Omega]", "2"]]}], "}"}]], "Output", CellChangeTimes->{3.451041694202606*^9}] }, Open ]], Cell["When \[Omega] is small we get:", "Text", CellChangeTimes->{{3.4507121909517746`*^9, 3.45071221399924*^9}, { 3.4507122860167084`*^9, 3.45071230190774*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Limit", "[", RowBox[{ RowBox[{"pt", "[", "t", "]"}], ",", RowBox[{"\[Omega]", "\[Rule]", "0"}]}], "]"}]], "Input", CellChangeTimes->{{3.450712304110922*^9, 3.450712310736091*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"t", " ", "v0", " ", RowBox[{"Cos", "[", "\[Theta]0", "]"}]}], ",", RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"g", " ", SuperscriptBox["t", "2"]}], "2"]}], "+", RowBox[{"t", " ", "v0", " ", RowBox[{"Sin", "[", "\[Theta]0", "]"}]}]}]}], "}"}]], "Output", CellChangeTimes->{3.4510416947963715`*^9}] }, Open ]], Cell["\<\ which sure does look like regular old projectile motion. And when \[Omega] is large, or more precisely when gravity is weak, we get:\ \>", "Text", CellChangeTimes->{{3.4507123170800037`*^9, 3.4507123600342283`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Limit", "[", RowBox[{ RowBox[{"pt", "[", "t", "]"}], ",", RowBox[{"g", "\[Rule]", "0"}]}], "]"}]], "Input", CellChangeTimes->{{3.450712363003054*^9, 3.4507123680031824`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ FractionBox[ RowBox[{"v0", " ", RowBox[{"Cos", "[", "\[Theta]0", "]"}], " ", RowBox[{"Sin", "[", RowBox[{"t", " ", "\[Omega]"}], "]"}]}], "\[Omega]"], ",", FractionBox[ RowBox[{"v0", " ", RowBox[{"Sin", "[", "\[Theta]0", "]"}], " ", RowBox[{"Sin", "[", RowBox[{"t", " ", "\[Omega]"}], "]"}]}], "\[Omega]"]}], "}"}]], "Output",\ CellChangeTimes->{3.4510416949213743`*^9}] }, Open ]], Cell["\<\ which is simple harmonic oscillation along an axis tilted at \[Theta]0.\ \>", "Text", CellChangeTimes->{{3.450712443145731*^9, 3.4507124688807645`*^9}}], Cell["\<\ To be a little more precise about \"large\" and \"small\", this problem has a \ scale defined by the time it takes a projectile to go up and down, i.e. T=2 v0 Sin[\[Theta]0] \ /g. The spring parameter \[Omega] should be compared to 1/T. So let us define the dimensionless parameter \[Alpha]=\[Omega] T and express \ \[Omega] as:\ \>", "Text", CellChangeTimes->{{3.450712521553988*^9, 3.4507125346324477`*^9}, { 3.450712581008635*^9, 3.45071271010569*^9}, {3.45071274234089*^9, 3.450712749684828*^9}, {3.450712822014805*^9, 3.4507128249523797`*^9}, { 3.4507128729848595`*^9, 3.450712891125949*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[Alpha]rule", " ", "=", " ", RowBox[{"{", RowBox[{"\[Omega]", "\[Rule]", " ", RowBox[{"\[Alpha]", " ", RowBox[{"g", "/", RowBox[{"(", RowBox[{"2", " ", "v0", " ", RowBox[{"Sin", "[", "\[Theta]0", "]"}]}], ")"}]}]}]}], "}"}]}]], "Input", CellChangeTimes->{{3.450712711824484*^9, 3.4507127267623663`*^9}, { 3.4507127653571043`*^9, 3.45071277671677*^9}, {3.4507128284837203`*^9, 3.450712866062807*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"\[Omega]", "\[Rule]", FractionBox[ RowBox[{"g", " ", "\[Alpha]", " ", RowBox[{"Csc", "[", "\[Theta]0", "]"}]}], RowBox[{"2", " ", "v0"}]]}], "}"}]], "Output", CellChangeTimes->{3.451041694937*^9}] }, Open ]], Cell["With this replacement, our solution is:", "Text", CellChangeTimes->{{3.450712908001381*^9, 3.450712930548833*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"s\[Alpha]", " ", "=", " ", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{ RowBox[{"{", RowBox[{"eqnx", ",", "eqny", ",", RowBox[{ RowBox[{"x", "[", "0", "]"}], "\[Equal]", "0"}], ",", RowBox[{ RowBox[{"y", "[", "0", "]"}], "\[Equal]", "0"}], ",", RowBox[{ RowBox[{ RowBox[{"x", "'"}], "[", "0", "]"}], "\[Equal]", " ", RowBox[{"v0", " ", RowBox[{"Cos", "[", "\[Theta]0", "]"}]}]}], ",", " ", RowBox[{ RowBox[{ RowBox[{"y", "'"}], "[", "0", "]"}], "\[Equal]", " ", RowBox[{"v0", " ", RowBox[{"Sin", "[", "\[Theta]0", "]"}]}]}]}], "}"}], "/.", "\[Alpha]rule"}], ",", RowBox[{"{", RowBox[{ RowBox[{"x", "[", "t", "]"}], ",", RowBox[{"y", "[", "t", "]"}]}], "}"}], ",", "t"}], "]"}], "//", "Simplify"}]}]], "Input", CellChangeTimes->{{3.4507129153296933`*^9, 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}], "Text", CellChangeTimes->{{3.4507132331972055`*^9, 3.450713326949606*^9}}], Cell["\<\ To determine the magical value of \[Omega] which makes the mass hit the \ ground straight down, we first find the time when y=0:\ \>", "Text", CellChangeTimes->{{3.450713344434428*^9, 3.450713394404457*^9}, { 3.4507135331580095`*^9, 3.4507135394862967`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"sT", " ", "=", " ", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"pt", "[", "t", "]"}], "[", RowBox[{"[", "2", "]"}], "]"}], "\[Equal]", "0"}], ",", "t"}], "]"}]}]], "Input", CellChangeTimes->{{3.450713404467215*^9, 3.4507134537028503`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Solve", "::", "\<\"ifun\"\>"}], RowBox[{ ":", " "}], "\<\"Inverse functions are being used by \\!\\(Solve\\), so \ some solutions may not be found; use Reduce for complete solution \ information. \\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", \ ButtonStyle->\\\"Link\\\", ButtonFrame->None, \ ButtonData:>\\\"paclet:ref/message/Solve/ifun\\\", ButtonNote -> \ \\\"Solve::ifun\\\"]\\)\"\>"}]], "Message", "MSG", CellChangeTimes->{3.4510417155625277`*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"t", "\[Rule]", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"t", "\[Rule]", RowBox[{"-", FractionBox[ RowBox[{"ArcCos", "[", RowBox[{"1", "-", FractionBox[ RowBox[{"2", " ", SuperscriptBox["v0", "2"], " ", SuperscriptBox["\[Omega]", "2"], " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]0", "]"}], "2"]}], RowBox[{ SuperscriptBox["g", "2"], "+", RowBox[{ SuperscriptBox["v0", "2"], " ", SuperscriptBox["\[Omega]", "2"], " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]0", "]"}], "2"]}]}]]}], "]"}], "\[Omega]"]}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", "\[Rule]", FractionBox[ RowBox[{"ArcCos", "[", RowBox[{"1", "-", FractionBox[ RowBox[{"2", " ", SuperscriptBox["v0", "2"], " ", SuperscriptBox["\[Omega]", "2"], " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]0", "]"}], 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CellChangeTimes->{3.4510417156250296`*^9}] }, Open ]], Cell["which we demand to be zero:", "Text", CellChangeTimes->{{3.450713897136077*^9, 3.4507139015580654`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{"%", "\[Equal]", "0"}], ",", "\[Omega]"}], "]"}]], "Input", CellChangeTimes->{{3.4507139032456083`*^9, 3.4507139080582314`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"\[Omega]", "\[Rule]", RowBox[{"-", FractionBox[ RowBox[{"g", " ", RowBox[{"Csc", "[", "\[Theta]0", "]"}]}], "v0"]}]}], "}"}], ",", RowBox[{"{", RowBox[{"\[Omega]", "\[Rule]", FractionBox[ RowBox[{"g", " ", RowBox[{"Csc", "[", "\[Theta]0", "]"}]}], "v0"]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.4510417156406546`*^9}] }, Open ]], Cell["\<\ In other words we want our dimensionless \[Alpha] to be 2. 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