(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 10.4' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 48664, 1014] NotebookOptionsPosition[ 47934, 985] NotebookOutlinePosition[ 48290, 1001] CellTagsIndexPosition[ 48247, 998] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"Student", " ", "Name"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ RowBox[{"June", " ", "28"}], ",", " ", "2017"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{"MITES", " ", "Physics", " ", RowBox[{"III", ":", " ", RowBox[{ "Intro", " ", "to", " ", "Oscillations", " ", "and", " ", "Waves"}]}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ "Various", " ", "Plots", " ", "of", " ", "Damped", " ", "Oscillations"}], " ", "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ "Write", " ", "code", " ", "which", " ", "documents", " ", "the", " ", "three", " ", "different", " ", "types", " ", "of", " ", "damped", " ", "motion", " ", "possible"}], " ", "*)"}]}]], "Input", CellChangeTimes->{{3.706438832926828*^9, 3.706438866641247*^9}, { 3.706438907051383*^9, 3.706438919944001*^9}, {3.706439072568179*^9, 3.706439089533779*^9}, {3.706439258924308*^9, 3.706439289865704*^9}, { 3.7064393456960297`*^9, 3.706439351887888*^9}, {3.706439447169464*^9, 3.706439450270192*^9}, {3.706439639024536*^9, 3.7064397032785587`*^9}, { 3.706439776532929*^9, 3.7064399922180643`*^9}, 3.7064401468014183`*^9, { 3.70644020350401*^9, 3.706440231525341*^9}, {3.706440284268818*^9, 3.706440284303707*^9}, {3.706440354737877*^9, 3.706440355031473*^9}, { 3.706440460576838*^9, 3.706440473598736*^9}, {3.7064409668876553`*^9, 3.7064410153161507`*^9}, {3.707005411225128*^9, 3.7070054165262423`*^9}, { 3.707005471415758*^9, 3.7070054740367413`*^9}, {3.707005517221489*^9, 3.707005537363543*^9}, 3.7070056116301517`*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{"(*", "Underdamped", "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"x0", " ", "=", " ", "1.0"}], ";"}], " ", RowBox[{"(*", " ", RowBox[{ "Defines", " ", "the", " ", "initial", " ", "position", " ", "of", " ", "the", " ", "mass"}], " ", "*)"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"xdot0", " ", "=", " ", "3.0"}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Defines", " ", "the", " ", "initial", " ", "velocity", " ", "of", " ", "the", " ", "mass"}], " ", "*)"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"e", " ", "=", " ", "0.01"}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Defines", " ", "how", " ", "much", " ", "we", " ", "increment", " ", "by", " ", "time"}], " ", "*)"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"\[Omega]", " ", "=", " ", RowBox[{"Sqrt", "[", "5", "]"}]}], ";", " ", RowBox[{"\[Gamma]", " ", "=", " ", "0.5"}]}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"xvalues", "=", " ", RowBox[{"ConstantArray", "[", RowBox[{"0", ",", "600"}], "]"}]}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Defines", " ", "an", " ", "empty", " ", "array", " ", "of", " ", "values", " ", "for", " ", "our", " ", "position"}], " ", "*)"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"xdotvalues", "=", " ", RowBox[{"ConstantArray", "[", RowBox[{"0", ",", "600"}], "]"}]}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Defines", " ", "an", " ", "empty", " ", "array", " ", "of", " ", "values", " ", "for", " ", "our", " ", "velocity"}], " ", "*)"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"xvalues", "[", RowBox[{"[", "1", "]"}], "]"}], " ", "=", " ", "x0"}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Sets", " ", "first", " ", "value", " ", "of", " ", "position", " ", "array", " ", "to", " ", "the", " ", "defined", " ", "initial", " ", "position"}], " ", "*)"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"xdotvalues", "[", RowBox[{"[", "1", "]"}], "]"}], " ", "=", " ", "xdot0"}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Sets", " ", "first", " ", "value", " ", "of", " ", "velocity", " ", "array", " ", "to", " ", "the", " ", "defined", " ", "initial", " ", "velocity"}], " ", "*)"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"For", "[", RowBox[{ RowBox[{"i", "=", "2"}], ",", " ", RowBox[{"(*", " ", RowBox[{ "Sets", " ", "initial", " ", "value", " ", "of", " ", "for", " ", "loop"}], " ", "*)"}], " ", "\[IndentingNewLine]", RowBox[{"i", "<", "601"}], ",", " ", RowBox[{"(*", " ", RowBox[{ "Condition", " ", "for", " ", "for", " ", "increment", " ", "of", " ", "for", " ", "loop"}], " ", "*)"}], " ", "\[IndentingNewLine]", RowBox[{"i", "++"}], ",", " ", RowBox[{"(*", " ", RowBox[{ "Increment", " ", "for", " ", "loop", " ", "by", " ", "a", " ", "single", " ", "index"}], " ", "*)"}], " ", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"xvalues", "[", RowBox[{"[", "i", "]"}], "]"}], " ", "=", " ", RowBox[{ RowBox[{"xvalues", "[", RowBox[{"[", RowBox[{"i", "-", "1"}], "]"}], "]"}], "+", RowBox[{"e", "*", RowBox[{"xdotvalues", "[", RowBox[{"[", RowBox[{"i", "-", "1"}], "]"}], "]"}]}]}]}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Defines", " ", "next", " ", "x", " ", "value", " ", "as", " ", "a", " ", "first", " ", "order", " ", "Taylor", " ", "series", " ", "approximation", " ", "of", " ", "the", " ", "position"}], " ", "*)"}], " ", "\[IndentingNewLine]", RowBox[{ RowBox[{"xdotvalues", "[", RowBox[{"[", "i", "]"}], "]"}], " ", "=", " ", RowBox[{ RowBox[{"xdotvalues", "[", RowBox[{"[", RowBox[{"i", "-", "1"}], "]"}], "]"}], "-", " ", RowBox[{"e", "*", RowBox[{"(", RowBox[{ RowBox[{"\[Omega]", " ", RowBox[{"xvalues", "[", RowBox[{"[", RowBox[{"i", "-", "1"}], "]"}], "]"}]}], "+", RowBox[{"2", " ", "\[Gamma]", " ", RowBox[{"xdotvalues", "[", RowBox[{"[", RowBox[{"i", "-", "1"}], "]"}], "]"}]}]}], " ", ")"}]}]}]}]}]}], " ", "]"}]}], " ", RowBox[{"(*", " ", RowBox[{ "Defines", " ", "next", " ", "velocity", " ", "value", " ", "as", " ", "a", " ", "first", " ", "order", " ", "Taylor", " ", "series", " ", "approximation", " ", "of", " ", "the", " ", "velocity"}], " ", "*)"}], " ", "\[IndentingNewLine]", RowBox[{"ListPlot", "[", "xvalues", "]"}], " ", RowBox[{"(*", " ", RowBox[{ "Plots", " ", "the", " ", "list", " ", "of", " ", "xvalues", " ", "to", " ", "yield", " ", "a", " ", "position", " ", "as", " ", "a", " ", "function", " ", "of", " ", "time", " ", "plot"}], " ", "*)"}], " "}]}]], "Input", CellChangeTimes->{{3.707005131093*^9, 3.7070052851683598`*^9}, { 3.707005368630376*^9, 3.7070053736676817`*^9}, {3.70700554699152*^9, 3.707005554524253*^9}, {3.7070055936933393`*^9, 3.707005593787342*^9}}], Cell[BoxData["0.5`"], "Output", CellChangeTimes->{ 3.707005107556694*^9, 3.70700521802346*^9, {3.707005257294613*^9, 3.7070052862282543`*^9}, 3.707005374528186*^9, 3.707005594105255*^9}], Cell[BoxData[ GraphicsBox[{{}, {{}, {RGBColor[0.368417, 0.506779, 0.709798], PointSize[ 0.0055000000000000005`], AbsoluteThickness[1.6], PointBox[CompressedData[" 1:eJxV2Hk4lHvYB3Cn3WlTslSq0YJUGi2Sim+p0GYqRFnGvjMMY+eZOdJe6GhR Mqm0OUWckMoobRyytB3EpJKSKNURlbf3cj/vdb39ketz3X6+bs88v9/9PJou gZvcBygoKNj++u9/v/b/6zT5/18VEDXuktmLSNbDUKNQFJ/WxVoJftt6Zs33 +Ugeh8Pck4+ljazVETxqwLLXvE9kDZgVmKFPxpqDGzv15O16XeSpmLf7ukl2 Kuvp2L+u+9SqQZ/JWvgSd0L4ly9rHaRve2Lwpoq1LgyfvTr0af4X8myo3Wl0 ePgnaz1Ii+1EMV2sudAZp9H23fIrWR8ZB8pb1l9gPQ8eFsdyBQP+I8/HNr/u a+62rBfgnyEG2+dksV4ITuuiwvs/WBvg9kENDcP13eRF8OC1JEansjbEv1un qR9+zXoxzpimPNyp941shIOxTz9uCmW9BEtnDX7ytZD1Utyf5NgR+p31MrzW a/unemkP2Rhj6jTHj4hibYIF+juOauWzBnbN3qQ37SOZAR47jKwZOLOX6stR PdLwaJkjmVmOy2UtORHJbH0Fjga4uiqXsvUV+HH6n5GpXWzdFJ5f8mYqaX6n uinytpaYha8jK6zEOn8zpcehbH0lhmu3rZiZxtZXYbvZ0qtht9j6Kmg7bd5y +zVbX42HOwJmjh76g+qrsaF63ypHLbKCGdZ6ja7LNWXrZmiLWaur7MTWzXHw 69z9ceFs3RwTWuqDvh9g6xZIazKy3neGrVtg8RRp+cJCtr4GNw46R30rY+tr EFkZPKmhjq2vhUFTX/SLVra+FkZTh4WN/MLW1yF4yLdvrn1sfR2Cqs0Hvx36 k+rrYeQ55rx0FJlZD2bixcEHlNn6Bmgkm60tUGXrG2CVt7eQo87WLZEkm5v+ jxpbt0RfRKKdTIWt85ASY2WlMJYMHl6VDFBMGsF+Pw+DzdJ/igaTZTx09q5+ cPU7+/tvhG9juaL1JzI2omDrRn+bFrafjVDrbM0qfkaWbQRnytx9KQ/Y9Zvw NrP1eUMBu34THqlII1Iz2fWbIL9iuqo2mV2/CQtLb8/dG8Ou34zji3L0yzzY 9ZshyWlYl7yBXb8Zo4Z+3fN2Abt+Myq0Zw94PJ5dbwVeVEWZ3Q/6PMEKnPzW YTFN7OfNCiv0P7SYysgyK+zN3XgoN539/FmjsXa2ZXUMu94ato9fWGZsZddb Y49qccMsA3a9NfYqnTIRKrHrbXD3nfRezFu6X2CDjWfP3OeVsPeTDf5om3Ts 42GyzAZ8r+PH3P3Z+2sLIlsY3SvL2fVbUNidfrFhHLt+C/6wa9v+roXub9kW HPwyYdqL/7v/bVF+1qbjzg4ybLHovNz2hA27H9jCrUD2MHA6u94W6r2Bd0w+ svuNHQLTtmcp3yDDDtB36+7cQWbskFP1RuP5RrLMDrlDVPY/n8Cu34ppzP3i /5pp/8NWzN/pOUP/ApnZinObkvQPCMiyrfBVrbZWM2D3y22Q9/AGPOyh/Rbb sLlu35mim2RmG8rrh2c1i8mybQhqGc5YrGT3Z3tc71y+68tgdr09XObsnf3+ Lu3/jD2mqF8sn7eDLLPHAree0srV7PnggD8qdYV3BpPhgOGPv8/WKKXzhXFA vPX3lc/EZJkDbo1bpthnzJ4/jvDjqt9O6aXzC45Ii3539WQ+mXFERWXtNB0h WeaIyEVhpnp67HnnhJpZ1y0KW+m8hBNOSxPdKjPIjBPK1LKKgu3JMifMOLlZ cFGFPV/5WFI1OS2mks5jDh+iszesPySQwcf7iOLMASZkPh8HSremFH+l853h I7lv4RTdS2QpH4Ve+kZm7mQZH6vu6T+aqEGW86Gi+74mq4adH5yxU3pFuW8n meOMYx+W+6qZkOEM5wjO2i9vOyjfGUUJi4dvsyYzzlB5w+z2ln2gfGesMeib qTuLLHOG9sjCRZkp7ZTvjDHGNh9a+t5TvguShpeHfPEmc1xQYTuwsa62jfJd UFN7QEe6jMx3ATPe2njD2XeU74LimqWf25XIUhekl7Uq7ox8S/kuKPvuOXPG q1bKd8GlY+0tFevJCq5IH7lbtjv/DeW7Qln+epKbJhmuUMz+aemyp4XyXSEZ 0pK478tryndFp7Zk4AcnstQV6x/zcveUvaJ8V2z/MevPgIVkuSuKHJQkJ6Qv Kd8NdaNaMG0EmeOG6p9rogeFN1O+G65udrpn8eoF5bvhx9WDx3t4ZMYNTxIi HCbflFO+Gy63/bnp+iyyzA2idGeb/PlNlO+GM37ePUPGNFK+O2QvuneFdjRQ vjuGOLqWKT2sp3x3NFduWN9yuY7y3TFVL0AwMPlfyneH71kTh9DQZ5Tvjt1O lcMstj6lfHeoPgubvwdPKN8dJW/zvMx1HlO+B4ItW37uHfOI8j2w7Pp0Nffv NZTvgVGicWkNrdWU74GKgOur+55UUb4HVC+W6OTse0j5HvCad0v6RlBB+R7Q z5o3uWBLOeV74MeG/Q6GKx5QvidOZL28FMO9R/meyCi6+OX41DuU74mE2LPn LqjfpnxPjDUqrb42toTyPaFmO0LjluZNyveErUfXBGeta5TviQVXrLXC51+l fE9UOKx+N8viCuV7wT4nPwgPsyjfCwHDdo3rnHSG8r0QLTrRZf0jlfK9UOiZ nFV5dwfle+Fzd6sib49XcX++F35OzHuQLk/qt8wLLnbx+5tWSvst90Li26Al Ogln+63gjTNJ2//tyL3Ub443qpRfD+hrudJveGNS07XBo3yu9pvvjdJtvkG/ vyzsN+ONh1EqR+o23qB8bxQ5Hamat01G+d7Q+Suo1GLYLcr3RtC7zANa2bcp 3wdS7ZaBNzfdoXwf3CtVtR/ceZfyfXCnrWPJpx33Kd8H3LG7L0dOKKN8H+gd atNNOVtO+T4Ytytxl+HcCsr3wY1r7SPtcyop3wcb94Zl96pWUb4v3hYWmItr yBxfxMrrFZfsqaZ8X7z+5M+fsbyG8n0hPDipbelnMuOLQTMCI2NP11K+L24X Byi92fiI8n0RqMyXC7+T5b5ohd+GqacfU74f7uWt7+gwf0L5fng2omdH3Tsy /LB27JqR8l1PKd8PvjXpp37OeEb5fihq2n1iQTFZ6gf7Oz4MY/0v5fth99XR PfJWstwPmvF276wi6ijfH781P3KoH1JP+f7QH7JlvzCJDH+cvXi4WWN8A+X7 Y4/kQfuTNDLjj3o1/cCTU55Tvj+WG1/bGH2CLPNH3297L3pOaKR8f1zTP3bD 9SBZIQC925d0CRSbKD8A84MuvzgQTUYAFDOumpW8J/MD8FS6b8jAOXLKD8Bv 3yefnOtClgbgiFPV6DcpZFkAlpkXDNC9T5YHoP1bwrJv3WSFQAw3XnPYWucF 5Qci5Di3x8SGjEDYj1vnWigh8wNRbGfVfOsvMhMIzeNeRx2ekqWB2FxfdX13 H1kWiM+VN5PXajVTfiAc3EcJT64lKwiQ1jj43r5AspIAs3aqKI5NJnMEaJl0 uHRuLpkrQO+jvSmtNWQIsEqFr2P8kcwTIPN1+uJ5o17S7yvAq9HFKXdnkgUC xCnUqP8wJTMCfPZYd+mpPTlRgOepN3m2IWSpABZlyYri3eRsAa5+edC5OZ0s E+DTjYLZtVfIVQIYDSv9qHCHLBfgcfQkSdNjcqcAPq/i5gS9JisEAV5tvIIu slIQkgt/nVcKr6j/IEyanpEnGEHmBuHzgdcfWlTJCMJVUeqfGhwyLwiaVZzl 43TI/CAo9Gw/VqVHFgTBPR02VgvITBAmnnqvl2FITgyC9pihH64vIUuDMGfi BY+zy8jZQdg1zULkZkyWBaGmzm/Qf2y9KggSYeAAl6VkeRD8Ogx9zy0mdwbh 2O7eFRULyQrBMDDrPFTLJSsFoy1NOeamLtt/MLxtmpWTprH9B8Nqj6pkw0S2 /2BMe+729dsYtv9gGD9dfPbQULb/YOTvDqvR/s5e/2A0OM+/kd3BXv9gvJkb krugmb3+wfgU1zk0v5a9/sGw06pQXlLKXv9gnFGzNyjNZa9/MAburH9kncFe /2BY2yzc2nWAvf7BuH/npoE0mr3+wWjc2d3k5MVefyFKOZZVCzez11+Ihwfr E2YsI3OEqIuZvmOeFpkrxOqYJls++/mGEKv6ruy48oX9/AuhGtCdMbuBzBfi 1u3YZTUlZIEQL/5Y356ZSWaEaJ/stfbSbnKiEDtKkrra/MlSISYkj4l35ZGz hdB3CktTn0eWCfGPjN88SplcJcTHS8b/WXTR/S8X4uKBiZKKGnKnEKdvOuum 5ZAVQuCbmHHk+gGyUgjCah1Ec/3Z/SgEbVUP/L9ZkLkhGPnw2MzJWuz+FIIt atVWJ38j80IwUBKemtBA+xs/5NffO/pM9VWyIARaL/05CYns/hkCbtaGv896 kxNDIFR1HGm0gt1PQ5CV9aoKE8jZIVi4asCGuJe0H8tCUO7f5hOSRK4Kgbnl e3gZk+UheB36fpzLO9rvO0PQW/hR2zuF3f9DkXG++Gq8CVkpFNFTw78VtdL5 wQnF5MnLpqkmkbmh+CO31jnVkIxQVFuaP7dsovOIF4qLOn/mL9vOnleh0H1n OsJXlywIRULGlM7GSjrfmFA4t9cKpMHkxFDE84//eVWFLA1F/ZIOO50COi+z Q7HRUTf/ux1ZFgot8fJUo146X6tCYVjk39p6jD1vQ5G1vjFm1FJyZyiMinzm XKmn81pBBJ9Zq2V1kWQlEWom5H3ZP57MESHY7qbkcT6d/1wRJtq+H/O3NRki VN/pXbGgi+YFnggtMakX7BLJfBGS72nWzJpDFohw6s2mRTkPaP5gRHg+SGVd pzs5UQSh0wJRx29kqQhKDVNH/51G80u2CFeaC13NFrPzjQjHk37kZj2i+adK hNiVL+a1BZLlInQFjhyiOJzcKYJZdeaqEZk0PymEQfxJ9XsPyEphiBrwZEZ9 Pc1fnDA8VzDIuSIic8OQuWxu7I4x7HwWhuZZeo5OWTS/8cIwfUezMszI/DCE HJqS6Vj6kPoPw/DeG4x7D82DTBhO7ndtTueSE8PwSNb+RN2T5kdpGIKT+9bU p/1D/Yfh38zLwz89onlTFoZrnYZ3PUeQq8KwtPuvoWYraT6Vh8Gn4NLS1OgH 1H8Yzo3RzPH9m+ZZhXDUVrYduPPhHvUfjoNGec05OmROOBatOJFk5ErzMDcc /tNGu3qfoHkZ4fh0MrjXrK6U+g/HilsDnjepkvnhWDIk2dPIiuZtQThCdGyf 8ZNpHmfCMWhziYV9dQn1H46VlSa5C8eQpeEYOZ3z9stGmuezw3HMs9hwrXYx 9R8OT8/hf6dm0fxf9SvP4MeEtvnXqf9wFLzq+rruxjXqPxyioSn7qszp+UEh As61SfXbn+RT/xFoaxOkRHvQ8wYnAr519tql3XnUfwTqZvyX57svl/qPQPNv 3PGS6fS8wovAxL1TW1RuZlP/EdDTaHvgk0fPN4IIeISKZz3/eJH6j8Coa+8d Tyw4T/1HYFZjhsmD6EzqPwJRGomq4WWnqP8I7E8YlFo66ST1H4HeDVb1B9qP U/8R+Nt86KqFFw9T/xHg7W7xPx+STP1HIN905jaDP3dT/5EoOS7xF2xhqP9I DBlpXuRVQc9znEgkXs/3z7rhaNLffyTsbz+PeJQc1W/8+v4zj1ODTtLzIC8S w8zLH924nNhvfiR4txfapolT+i2IxJoGbpr+anqeZCKxdTLzu82PE/1OjET2 kryDwtqT/ZZGYt734PXOmqf7nR0Jpb3eKa5emf2WRULlEePgc+5cv6si8fTJ VC2PFxf6LY/EIe3CH2bKf/W7MxLu8dOZ34wv91shCrwnqr63srP7rRQFz7th W5tv5vSbE4Ujfml6Wx/Q8zI3CtwjERXTqnKp/yjc6Zjnsrg2j/qPwltuhX1K zd/UfxS2ba0MmldJz9+CKByqkB8dfjef+o9CcUjwGI1rBdR/FLq+8WydLhRS /1H4T8NUv/oQPc9nR0G4IP2kf1wR9R+Fu28aO7lu16n/KGTqcc+prbpB/Uch 59TA5MlT6f1AZxRmLzpSuKKXrBCN6ZbxPuKqYuo/Gur6k9Xf82TUfzRG7Cv3 ycgnc6NRu137iosGvX9ANIKHLR87O47Mi8a+mWqVP5rI/Gi82mlj9tT4FvUf jW9HVX7mHyMz0Yh5diAx/Ss5MRrcjqZr+yzpfYc0GuVPC2wkmeTsaKT3TNoW 00uWRUNiNaEhdkMp9R+Nc+dFv+9IJ8ujMeCfgPmHPpA7o1HU6y7LWkLvVxRi 4OKValCeQFaKge2/Tw0+PSRzYqAYPzNmqtpd6j8GYap75Pb2ZMQg7fuRTRnp ZF4MZrwxaeqSk/kx+HDUN2sTh97vCGJQMezvb8UOZCYG3++P5iw7Sk6MQfpz UVJ5DVkag4LXR575/X6f+o/B25w/jkwFWRYDa3PLvR+E5Kpfdfv7CTVnyPIY WK6cEFz1mNwZgwee4ZbvBrLvn2IxHfcMpnLJSrHoeL7SJnIrmROL0YP1P32W kLmx+PdFiNuf58mIxXmHNRrbKsm8WCiPf5m29iOZH4uUMaMlXmPLqP9Y3O1u Ds3TJzOxaF3haDTPkpwYi5xPSqdf+ZClsdDSKUmtjCdnx2J1uNL4ruNkWSzM lTVnW+eSq2Ixstio6797ZHksoi4GpjfWkTtjsVZruXDEe7JCHOrnr7wV30tW ikP82LudaxTpfR0nDs1/CYQuKmRu3K/zo6OycgoZcfh5NOnwQR0yLw5l1z8J 8uaS+XFYP+LYyIULyYI4TH1+Rm/8YjITh2M33U+5LSEnxqFvjKLFxKVkaRyE Rca/L2fr2XE48LGu8V9DsiwOO7jR5W8XkKvicNea+yKEzZfHobx1/LI49vfr jMPgQyrDRnLICgxc5U6iqarkYQyMRi25d/t3shIDJZuUlV9+0N9HncFTy8sr ijrIHAaKl29NmSAn6/zymvOmqg/JXAbawvgpedfJhgwSzrrN6T5HBoPJRz52 tBwkmzMYlDT2U3wMmffr55kevvvEnWzL4HaXfOiLdWQ+g3a1vVvOzSN7/fp+ +wmO+mrs542B5pS0MqaHPo/hDLxc+ZuONZAZBuGGh7K33yDvZFCeMSBzZRo5 kYHKoHPdDVHkIwy+Dg0RbrYjSxn8od7+Omsh+RyD3x3l0z4okbMZeK5XGqzR RvdfAQPd+y/dlpSy9y+D+9UqUzceJ99n0NPgPtnl/+5nBjZ55sZCC/IzBnN/ S/XaNZm9vxlkCjcknP9E+0Urg9r9WpJnd8idDConuphPPELuZuC+OuSuyJus IIamR8G7jsXkYWKYirRP71EkK4mxwmRU84ZntJ+pi7FmQelB40wyRwztoa7n nIRkHTF+fo4dlW/C7pdijK9syrUYTjYUY1CR0XaNJ+z7bTH0St77L5CSzcWY pHh1yyFvMk+Mla2humbzyLZixGa/KV/bQ/s5X4xhZvwp50vIXmK8iN872G0n WSDGFheuvZg9H8LFCOpo+TF4HJkRo3PCy5Kvz+g82SmGwCInyjaNnCgGbn7r metMPiIG/6npcMl09nz61d/ka8LNb+j8OieGZE7H13PnydliGAlObdrhSy4Q I+PqHpOu2WSZGBcz1YM+ttN5eV8MnyEmKQmXyFViCJthlR9AfiZGd2IYb6ce WS7GqeTJWj/b6XxuFWPgpim+k/4id4rhdOv5pXe+5G4xchsHR/rokhUk+F05 7/iKajr/h0kQeXzQeu317DwggaZiYdue+zQvqEug9lcqL8KUzJHAcfP0ld03 aN7QkUDrqXbcGEMyV4Ky+NALFTk0nxhKMNt0u+PCWWRI8Ga75rSNp2meMZeg cti4wJmTyDwJPls9Ons3heYfWwkiRhtq6o8i8yWYO3bGpKAEmpe8JBixYVX5 /p80Twkk2BJ23TxZRA6XwDKqKf2PD+z8JcG3m+33fDzIOyW4ovXPactGmtcS f9VflwwxsiEfkeCJ6qIL+pU030kl0FUPXWS8mnxOApUrXis8b9I8mC1BSp/7 mjwDcoEENhofCmdfpvlRJsHpFFHLU+1ck/8BS66J7g== "]]}, {}}, {}}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->{True, True}, AxesLabel->{None, None}, AxesOrigin->{0., 0}, DisplayFunction->Identity, Frame->{{False, False}, {False, False}}, FrameLabel->{{None, None}, {None, None}}, FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}}, GridLines->{None, None}, GridLinesStyle->Directive[ GrayLevel[0.5, 0.4]], ImagePadding->All, Method->{"CoordinatesToolOptions" -> {"DisplayFunction" -> ({ (Part[{{Identity, Identity}, {Identity, Identity}}, 1, 2][#]& )[ Part[#, 1]], (Part[{{Identity, Identity}, {Identity, Identity}}, 2, 2][#]& )[ Part[#, 2]]}& ), "CopiedValueFunction" -> ({ (Part[{{Identity, Identity}, {Identity, Identity}}, 1, 2][#]& )[ Part[#, 1]], (Part[{{Identity, Identity}, {Identity, Identity}}, 2, 2][#]& )[ Part[#, 2]]}& )}}, PlotRange->{{0., 600.}, {-0.6326162634151028, 1.8808905838043488`}}, PlotRangeClipping->True, PlotRangePadding->{{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.05], Scaled[0.05]}}, Ticks->{Automatic, Automatic}]], "Output", CellChangeTimes->{ 3.707005107556694*^9, 3.70700521802346*^9, {3.707005257294613*^9, 3.7070052862282543`*^9}, 3.707005374528186*^9, 3.707005594164318*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(*", RowBox[{"Critically", " ", "Damped"}], "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"x0", " ", "=", " ", "1.0"}], ";"}], " ", RowBox[{"(*", " ", RowBox[{ "Defines", " ", "the", " ", "initial", " ", "position", " ", "of", " ", "the", " ", "mass"}], " ", "*)"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"xdot0", " ", "=", " ", "3.0"}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Defines", " ", "the", " ", "initial", " ", "velocity", " ", "of", " ", "the", " ", "mass"}], " ", "*)"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"e", " ", "=", " ", "0.01"}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Defines", " ", "how", " ", "much", " ", "we", " ", "increment", " ", "by", " ", "time"}], " ", "*)"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"\[Omega]", " ", "=", " ", RowBox[{"Sqrt", "[", "5", "]"}]}], ";", " ", RowBox[{"\[Gamma]", " ", "=", " ", "5.0"}]}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"xvalues", "=", " ", RowBox[{"ConstantArray", "[", RowBox[{"0", ",", "600"}], "]"}]}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Defines", " ", "an", " ", "empty", " ", "array", " ", "of", " ", "values", " ", "for", " ", "our", " ", "position"}], " ", "*)"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"xdotvalues", "=", " ", RowBox[{"ConstantArray", "[", RowBox[{"0", ",", "600"}], "]"}]}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Defines", " ", "an", " ", "empty", " ", "array", " ", "of", " ", "values", " ", "for", " ", "our", " ", "velocity"}], " ", "*)"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"xvalues", "[", RowBox[{"[", "1", "]"}], "]"}], " ", "=", " ", "x0"}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Sets", " ", "first", " ", "value", " ", "of", " ", "position", " ", "array", " ", "to", " ", "the", " ", "defined", " ", "initial", " ", "position"}], " ", "*)"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"xdotvalues", "[", RowBox[{"[", "1", "]"}], "]"}], " ", "=", " ", "xdot0"}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Sets", " ", "first", " ", "value", " ", "of", " ", "velocity", " ", "array", " ", "to", " ", "the", " ", "defined", " ", "initial", " ", "velocity"}], " ", "*)"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"For", "[", RowBox[{ RowBox[{"i", "=", "2"}], ",", " ", RowBox[{"(*", " ", RowBox[{ "Sets", " ", "initial", " ", "value", " ", "of", " ", "for", " ", "loop"}], " ", "*)"}], " ", "\[IndentingNewLine]", RowBox[{"i", "<", "601"}], ",", " ", RowBox[{"(*", " ", RowBox[{ "Condition", " ", "for", " ", "for", " ", "increment", " ", "of", " ", "for", " ", "loop"}], " ", "*)"}], " ", "\[IndentingNewLine]", RowBox[{"i", "++"}], ",", " ", RowBox[{"(*", " ", RowBox[{ "Increment", " ", "for", " ", "loop", " ", "by", " ", "a", " ", "single", " ", "index"}], " ", "*)"}], " ", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"xvalues", "[", RowBox[{"[", "i", "]"}], "]"}], " ", "=", " ", RowBox[{ RowBox[{"xvalues", "[", RowBox[{"[", RowBox[{"i", "-", "1"}], "]"}], "]"}], "+", RowBox[{"e", "*", RowBox[{"xdotvalues", "[", RowBox[{"[", RowBox[{"i", "-", "1"}], "]"}], "]"}]}]}]}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Defines", " ", "next", " ", "x", " ", "value", " ", "as", " ", "a", " ", "first", " ", "order", " ", "Taylor", " ", "series", " ", "approximation", " ", "of", " ", "the", " ", "position"}], " ", "*)"}], " ", "\[IndentingNewLine]", RowBox[{ RowBox[{"xdotvalues", "[", RowBox[{"[", "i", "]"}], "]"}], " ", "=", " ", RowBox[{ RowBox[{"xdotvalues", "[", RowBox[{"[", RowBox[{"i", "-", "1"}], "]"}], "]"}], "-", " ", RowBox[{"e", "*", RowBox[{"(", RowBox[{ RowBox[{"\[Omega]", " ", RowBox[{"xvalues", "[", RowBox[{"[", RowBox[{"i", "-", "1"}], "]"}], "]"}]}], "+", RowBox[{"2", " ", "\[Gamma]", " ", RowBox[{"xdotvalues", "[", RowBox[{"[", RowBox[{"i", "-", "1"}], "]"}], "]"}]}]}], " ", ")"}]}]}]}]}]}], " ", "]"}]}], " ", RowBox[{"(*", " ", RowBox[{ "Defines", " ", "next", " ", "velocity", " ", "value", " ", "as", " ", "a", " ", "first", " ", "order", " ", "Taylor", " ", "series", " ", "approximation", " ", "of", " ", "the", " ", "velocity"}], " ", "*)"}], " ", "\[IndentingNewLine]", RowBox[{"ListPlot", "[", "xvalues", "]"}], " ", RowBox[{"(*", " ", RowBox[{ "Plots", " ", "the", " ", "list", " ", "of", " ", "xvalues", " ", "to", " ", "yield", " ", "a", " ", "position", " ", "as", " ", "a", " ", "function", " ", "of", " ", "time", " ", "plot"}], " ", "*)"}], " "}]}]], "Input", CellChangeTimes->{{3.707005229880295*^9, 3.707005229982445*^9}, 3.707005304153468*^9, {3.707005348446888*^9, 3.7070053534521103`*^9}, { 3.707005559373723*^9, 3.707005589739893*^9}}], Cell[BoxData["5.`"], "Output", CellChangeTimes->{ 3.7070052304877234`*^9, 3.707005309118053*^9, 3.707005353942972*^9, { 3.707005585608534*^9, 3.707005590099597*^9}}], Cell[BoxData[ GraphicsBox[{{}, {{}, {RGBColor[0.368417, 0.506779, 0.709798], PointSize[0.007333333333333334], AbsoluteThickness[1.6], PointBox[CompressedData[" 1:eJxV13lUzH/7BvBJkVCyZ6mGREQqIpIusqREaVG0TPuiZdqb1s+MKISILKnG li0JEd9vMlmjZOxLoUIikn2tHr/z3O/nnF9/1Hmdu6trrqYzcxrpHb7UrxuP x3P5++n/vv73o93i/3/lIXFg8YLGBOaeaJZ3uW1sYVbHoYnzZy61/kgeiK3m g9Yt28OsgdLhNuVH25hHQLFM5cvsyZ/IfPBeuGV9DWcehZf1aVzmAebRCNee 5HbuHvMYDO5vvUCzk1kPQ423FC4c9Zk8Hl8s4w5XzmaegK6Nacd+uzEbQNak luodzWwInQEbEpwzmI2wsezIZoOdzMa46v1o6vSDzJPhGP7Q6PxJ5ilYPkrs +etfZhMc3i/3GH+JeSp0NylNFFcxT4PTr5STE2uYTZF3rHH28lrm6fglq+g1 8RbzDKh5DhRX/e9uhq8/j/5jc5N5JlwbXS/ducFsDqcIhTex15hnYcAVfd6S /z0eC0w593ZPeAUzkN503Kf1LJkDBpcXq9z9377Z+JS+tNSkiN1nQ7unpcrI A+w+B7smBLjsyWP3ORj+Y+H5ym3sbolF/WaaZ25gd0skFKxqUlnN7nPRS+Pl vTlJ7D4XNeGy8FlR7D4PugcGOCsGs/s8TC6PSdsuYPf5ePrH3Yu3jN3n40py bfUcW3ZfAKdzWXo+luy+AHo9x4b5TWd3K6iuyJ5kO4ndrXDCT71QU5fdF0KW 9fDc42HsvhDhcb17pamzuzW+eNV/1urB7tZQGPxrR9Fv9vdpg6BI31cGH8mc Ddxzjmw92Mzui3D0RuFUjXp2X4TPXZcmS26zuy2aox7Jm6+yuy0WtxUYLyhn 98XwU3gu2H+C3RdDcHPawq5Cdl8CyaeySpfd7L4E4/LGrD+xmd3tYPhWw6FX Ohl2eDrvzmmfJPb9dsh98tWiIoIss8PeA6rCYQEsb4/Pjq1l8W4sb4/VR5qT H9uzvD1GXfqjab6A5e0x3ubu230zWX4pvD33GqgZs/xSCLKVJySNZfmlmOV6 WattBMsvhXNt0Ezf/izvgJiKz6XPlFneAWFix1PuHfR6wzmgbeJn98ZPZJkD ouI0Hwa1sNcjR9StDDP//pQMRzwao1+09i7LO0IUm2M28jrLO8LKYaZCRQXL O2HtDxN1QSnLO6HDMUCkfITlnXAk4YxNaQHLO6F2n/lW/20s74wMx3gPrfUs 74zwASGldRzLO0M09mZ+fizLO6OoQls3IITllyHm4X1bE2+WX4Zj5sFaKi4s vwwBL+Nzm2xZfhlWPbtXVWnJ8i7YZLT7cOF0lnfB4bMLsXkSy7vg6YIrG8W6 LO+CUrPInLjhLO+K8N0zXaL6sbwrVv/Yfy9ameVdkTK4undSB71fyFxhlBvX kfGZvX8sR1rF6MO5b8hYDqsxPdRKn5O55Uj501f/zn2WXw5Tx47Or9UsvwIr 25wytC+y/AosD4u6svgsy6/Aid5rSlcVs/wKqN7a4VCxn+XdEOm7Tdq5i+Xd oLxxwc65m1neDdEQmGWls7wbXCLi0xqTWd4dQQUDIk2jWd4d44bvVdgWzPLu cB28yvC7gOXdcT21xx+PZSzvgY0lYV41tizvAb0PH90t5rK8Bypq/m0rm8Hy HhghaFOfasTynqj22ld5bizLe8J+wvuOOVos74l5+x5evD2Q5T1xTjFI1b83 ywugL937pEuBzBegbYDiklkfP9DPE2CJY2dj+jOyQABhjk5gfTWZE2DD+OV1 ZufIUgH6/UkzKSwkywQIXLFKOHwruUGAF6eM1uSKyTwvFEwKFo4JJ/O9YGLT fVy5G+v3wtXlTw67WbN+L9QqyN4pm7J+L/QvyGw9r8v6vVB1bLA0ZQDr94Ka pomStQLr98K+/Av9+B/aqN8bp95vvMh7SuZ7I1MtWf3dDTK88Szd72vjWbLA G0Y5uiubCsmcN0Yt373y/Vay1BtNCkfbu60iy7whapvyZVQEucEbE+5rC209 Wb8PQp86eEhsWb8Piv/992ylGev3wYrrc0W9x7N+H9gNe77TU4P1+6BzU/LA ih6s3wfJbwe9GPP1PfX74MpDadfOF+QGH0xRGOY75A6Z54tNQ7heBTIy3xd7 n957Z3icDF8cNdLsfjOPLPDF7kFetlGZZM4X/meOXtBJJEt94RTZ2/lZEOv3 Rf6zzb32ubB+X0yqcXgYsYD1++GeTdAZ66ms3w+XTrzfM1GX9fshf0dX9rCB rN8PP17I1vRTZP1+CL7oHN3/0zvq98Ps42+cNRvJMj/cjTqgZywnN/ghRe/c C7sLZJ4/vE2WSUTFZL4/7i493Xk0jwx/ZK/t49CSSRb4447xTpFBEpnzh+KP XSEpK1m/P5LvOI55tJz1++N3L9U9Ztas3x/fTVUfHprO+gPwYvbei9rjWH8A FriM8JBqsP4AXP1Vu298T9YfgF6f1TdWfG+l/gDkOs4c4vaaLA3AwcTi6YoP ybIAnOxd9eHUVXJDAGa3vZscdobMC0SYfRrPuJDMD0Rs4Y8lXdvICMQMadmI B6vJgkCcNTL2OhPD+gORkFk/ROrH+gPxW8vALNuJ9QdCLXvPlU3zWH8gJteU nthmwvqD0P6q5ed+XdYfBB3jizvOD2L9QbCdVpXxvDvrD8LT4tWXe317S/1B iF2VMd+imSwNgvEtL5XkB2RZEF49PqB8+Sq5IQjLh7wzG1hG5gUjW/eWNPQg mR8Mxep3hvLtZASj+surZrMMsiAYn0s9zpfEs/5g7ENnsUEQ6w+GW1vYmdOu rD8YRlM8audZs/5g/DmQ9PHZDNa/Ers+Jw8T67P+leC/1pg7YQTrX4n+PspB DX1Y/0rkCXqk5XW8of6VqH1/f5NPG1m6EtX609ONn5NlK6Fq+823l5zcsBI5 2Q90WmVkXggMlhRW3DtB5odgd90Uw2t7yQhBY75j1MVssiAEMcp1kqtprD8E knVS17sxrD8EI/dyn1r8WX8IdlS7Wim7sP4Q7Fqt6mywkPWHYvVibrDHDNYf ij17d6bl6LP+UPj9ssx9OIL1h6K0KsRppBrrD4WOSKk4qquF+kMxM6yzoLad LAvFxS67McZN5IZQfI//apZ/l8wLQ9Pglrr+V8j8MOT00FPIOkNGGBJ9ygoH HSILwrDdJPfavp1kLgwPwus8Zqxn/WFwbZMEPE5i/WGoiC6o58JYfxjCi2af MxSw/nC0j0r+0WLP+sMxwSpw22FL1h+OayEj1keasP5whMhO37Mcy/rDoWPn EKo5lPWHI19piF1XL9Yfjs1S3fg3f15Tfzi6XyhsqGsj84Qovn057UEDWV0I C+1jno/vkPlCXK3cFPLyMtlQCN6BgsLvZ8gQojStv8qAw2Q7IdRa+Zum5pIF Qvwy6DHFewNZKERKJO/ztlQyJ8TwPXbXbkeQs4TQC5tRNMiXLBVisoNSnpcz uUQIj43ft562IsuEuMwXbOpnRpYL8agmMz1mItsrxPWBt+IbtcntQrw0TfRw 6s/2R6DDpGXybSW2PwK3LoV/cfzeTPsjEOAQltvwhmwYgbWqlqOj68mIwLDb S9b1vUW2i0CG0c+qk5VkQQSmxhTWuZeShRF4lV15Xv0gmYuAjklZcPVOclYE Sou/Pd2QSZZGoLvip/4uqeSSCGx/rtl9fCRZFgFh7sfjin5keQQ2mLxWe7WM 3BCBpQ6JmrXW5Pa/e2dNfFhhTuZFYlhzyrQyQ7J6JL4//DK9TIftj8TQuYOf VAxm+yNx6kfIgFoVtj8SI7xWN77884r2R8Iqrc1CsZ0siMRsxRnjx70gCyMh HfIn1/kBmYuE8pbqrPXXyVmRKHg0W6mqnCyNxJZXvz72LiGXRGLdvG1Oy/aR ZZEwGHxy/JEcsjwSe5W+hiitIzdEosciJQ3/ZHJ7JH6eyhh7S0jmRUFj08Tt Fr5k9ShIzkm9y5aR+VFw275l9VQbsmEUnAac+V0+i4wojM+TyayN2f4oTNQL lj/XZfujUDsudEzSULY/Clo71lRoq7L9UXBvidl6g8f2R4FT7diX9OUl7Y9C eWxDi0kLuSQKA9Vee3yrI8ui0DD1iULFLbI8Cp41q29mXiI3RGEO/+x57zJy exTMfs6utThK5kVjw+Zvv3QKyOrR2L3tqEXfbDI/Gqmjpu9QSCcbRmPzZRHv dwL5739H3WOto3+Hke2iEZe37kM3H7IgGqv7jAjrt4ws/PvfVGdT6xgbMhcN 38FXfS0tyFnROOFRfM9/MtsfjQvz06dtHsv2R+PnnllrLw5n+6OxrPDylV99 2f5ofHvb+810JbY/Gmf8VT6k/nhB+6NxL6z0bs07Mi8G6ad7ZPMbyeoxyLf6 o5d4n8yPgXZW9ub662TDGLxcdLNqbgUZMTDfXnTj1EmyXQyynk7NHneQLIjB n6YorcJcsjAGv86HBo7LInMx4D00DD2VRs6Kge7rar25IrI0Bte+LdleF0ou iYGRm7xU5E2WxcDiH+9krWVkeQx6dGm9v25DbohBirc2LwlsfwzG+XEnp5iw /bEorxXwvoxj+2NRU9T46pwW2x+LGPsxnmsGsP2xUFnqGOjak+2PxfnV2R3G HU20PxaOSmrDB3wiC2Jxa3R7+c9msjAWI5ICnzfXkblYXKk7mPJETs6KxR67 xpx7V8jSWKSNtB59/x9ySSy+3ByqU3+cLIvFkz7rNr3ZT5bHYkr1A7+OneSG WNwcP2vvkE3k9lh84HizTdPIvDgMcxLM8hSR1eNQ7rk3JzOMzI9Dc4bqfJkP 2TAO3V+/tfrlQkYcdp9fnTtjMdsfhz+J+jPElmx/HHjl+iNrTdn+OFzFo0Uj Ddj+OIR+TTuXoMP2x+F2a7rPEw22Pw5WK6YvgBrbHwfPgs+CIkW2/2//1AHH tH420v44zKj8NnZ7G7khDu+6Om8Oekluj0PenY25ux6TefEwmtZ7g+4tsno8 rKzq8s9cJvPj0W1m4E3bf8iG8bjorTmo9TgZ8fisGxi58QDZLh61Pc42Tssl C+KhnZfi+TqLLIxHn5xJLbvXkLl4PDuJeJckclY8rl0b12dYJFkaj6J/fKWN AeSSeLSLQ42Ou5Nl8TDXufivxIHtj8fik01mbgvZ/niUL7EvNrNg++NRM3ZL 35EmbL8IZhH6bn302X4ReoUcyOrgs/0iLBGsO/xlMNsvQvc8/30f+7D9IjyO rUr40o3tF+HPfEzo+NFA+0X4HiQo7f2BLBRBNOt8X/4rMifChNHvZ8yoI2eJ kJGQYbz8NlkqwuGCcZ9Tr5FLROB/CBEdPU+WiVAta6+sP0WWi1AU4HR9wBFy gwgVhuaZdlJyuwi1XouUs3PIvATYm46bVpdJVk/A7JZEjXGryPwEfN7VdjBJ RDZMwKfQiU33w8lIQF1E86Up/mS7BJgevr5wpxvbnwA35bVCRQe2PwEIfzQl aiHbnwCfk6HZLRZsfwL2b+Kt9ZnK9idA56R1/5cT2P6/LlGeEKzD9iegWeXr na9D2f4EbNa7o5iuzvYnQNVd8K+mMtufgFvnr7yL+v2c9idCmGPuU9ZCVk+E 4oIzd7s/IPMTYbizcprnJbJhInoN/5pxqYSMRDwa8VZmnE+2S8SrYvPHRevJ gkTYFGRVG4nIwkRcjEvfXOlP5hIRmXhjtJsjOSsRlfu7pyrMIUsTMX5fy84T k8gliVB9PCY8RJMsS0SGa9wPo95keSKsm7dO6vbzGe1PROh7c7WnzeT2RCxJ n7Lx4j3y33eGldGzDp+8SFZPwtThuq7HSsj8JPxafXr7yXyyYRLOCmQeFzPJ SEK370MP1SeQ7ZLQYLQrVCGILEjC8W0jDhsuIwuT0AOcY/A8MpeEXa8OBhVP JmclYaRc1PRnJFmaBPP652ed1MklSRDEXWs+2/mU9v99PHkj/ce8J8uTMPRR k2F+HbkhCS3TFWfzb5Dbk6DsmZJVdJbMS8aVIIehlgfJ6sm4yVt9/8U2Mj8Z g2yHXtiYRjZMxp3vQ+WWUWQkI1BvazdFb7JdMrwlO+xq7MiCZPjqmJUWWJCF ydDZkz022YDMJcNd+fx+X01yVjLk9+/rOvchS5Nh2tm13/53Pe1PxrwmwYhl b8myZOTZj1zj/5gsT8b99ND61CpyQzKixscM3VdGbk/G+/krp8kLybwUHLZb Y6icQ1ZPgYI2r2vBajI/BfNaB+VtiSYbpuDS4p7dX/uQkYIqDDea50C2S8HR gTnDj80hC1JwJ+5ShZYxWZiCC+EfB+4aSeZSMNglQkO7HzkrBccatl86xiNL U5DT+7z6/PY62p8CA/2Zv14/J8tSoN8QlZB9iyz/mx9ydb3VBXJDCpao7NBT Pk5uT4GZyVKbW/lkXipmeAvapBvJ6qmoyTXqk5RC5qfC3NC0QBBGNkxFefGb PYs9yEhFxKsKtfmLyXap2Pl8csP8WWRBKgpeXFO3NyALU2Ea0bHLV4vMpSLF YrlIrEbOSoW6WUDhwc4ntD8VBmklIx62kUtSUR9y5UHf52RZKrpiTW/Z3yLL U5G2tICXd4Hc8Pfn123x+3ic3J4KW43jPxZLyTwOl6Zll57OIvfkYLOpMVtX TFbn0Henxeb8CLIGB/4np4N8bzKfwz4XufzoUrIehx+H3XvCkmzIoaXhuNXT yWRTDrcHJGWuGk0Gh3uRCbXGg8hWHJTm+6u0difbcchp7ZhW9O3xf+3CYcCD bk6xr8kCDmvXTV+x8BE5kENFftA83etkIQe/Mne1Xv+Q4zk4rHlR8v0ImeNQ VlU9ti2XnMFh8by3oe8zyVkcpDcGpXxNJu/gMGWinkOPcLKUg3xe52ttAfkQ h5Mf4qbNsSeXcKjuET8zdA75LAee/8uP0slkGQf/VweWPh1NruKQ5nLIddRg spzDlswnCkJl8iMO5fMN5l/98YieXw4+c3aM1H1LbvnrSeqbMuvI7RycS7LW /qkh/+BQ5zKkT0wFmSfG06P7+307Tu4pxuOhM7el7iGrizG+68XmftlkDTHs 9h7oVpRG5osx03hD0+JYsp4Yuy8fnvArgGwohp6jav0xV7KpGB8qalqDbMgQ 4+DlLmcDc7KVGH+UKjV+G5DtxGjsmDhBzie7iPFCJSDzWH+yQAyRx6Yp2Urk QDFW9LqpLf72kJ5/Me5/tloY10KOF0PzoUFRzBMyJ8aRuccsk2rIGWJkFnzt tb6CnCXGpeEzO/aUkHeIkVteNqhyL1kqRneHvUtatpIPifEl0bRwSDq5RIyy /HMDl4jIZ8VwbQ3I2biSLBOj0yVjzAN3cpUYc/d7y3TtyHIx9tjO9kyeQ34k xrXtEZ31U8gNYiitsdw5dyy55e/v84mKfulQcrsY3oOGHdfvQ/4hxomIGv7R zgf0/Etg8Sox0fgjuacEi7XW/FP5gqwugZbE5rHLA7KGBM76k+5+ryLzJQhr ydmX/y9ZT4JhM1rn2RaTDSU4bb/9tOIesqkExX1V2iqzyZDgi8KJt+lryFYS BHh3O+gkIttJMH9QlJZ+CNlFgoKDiYt6epIFEshst094b08OlOBIo+U/j+eS hRKsalT4eHMaOV6C3mGONdfHkzkJ+i6sXHBTk5whQeNduccjdXKWBDdV1VRb Fck7JNi8pcfi7t/v0/Mvwf0d1ppj35IPSXCgPSTW7im5RIJWBUUnsZx8VoI3 N/aXnrtElklQ9uxd1s8z9y3+A3iFMjA= "]]}, {}}, {}}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->{True, True}, AxesLabel->{None, None}, AxesOrigin->{0., 0}, DisplayFunction->Identity, Frame->{{False, False}, {False, False}}, FrameLabel->{{None, None}, {None, None}}, FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}}, GridLines->{None, None}, GridLinesStyle->Directive[ GrayLevel[0.5, 0.4]], ImagePadding->All, Method->{"CoordinatesToolOptions" -> {"DisplayFunction" -> ({ (Part[{{Identity, Identity}, {Identity, Identity}}, 1, 2][#]& )[ Part[#, 1]], (Part[{{Identity, Identity}, {Identity, Identity}}, 2, 2][#]& )[ Part[#, 2]]}& ), "CopiedValueFunction" -> ({ (Part[{{Identity, Identity}, {Identity, Identity}}, 1, 2][#]& )[ Part[#, 1]], (Part[{{Identity, Identity}, {Identity, Identity}}, 2, 2][#]& )[ Part[#, 2]]}& )}}, PlotRange->{{0., 600.}, {0, 1.2380800647479746`}}, PlotRangeClipping->True, PlotRangePadding->{{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.02], Scaled[0.05]}}, Ticks->{Automatic, Automatic}]], "Output", CellChangeTimes->{ 3.7070052304877234`*^9, 3.707005309118053*^9, 3.707005353942972*^9, { 3.707005585608534*^9, 3.707005590149366*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(*", "Overdamped", "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"x0", " ", "=", " ", "1.0"}], ";"}], " ", RowBox[{"(*", " ", RowBox[{ "Defines", " ", "the", " ", "initial", " ", "position", " ", "of", " ", "the", " ", "mass"}], " ", "*)"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"xdot0", " ", "=", " ", "3.0"}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Defines", " ", "the", " ", "initial", " ", "velocity", " ", "of", " ", "the", " ", "mass"}], " ", "*)"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"e", " ", "=", " ", "0.01"}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Defines", " ", "how", " ", "much", " ", "we", " ", "increment", " ", "by", " ", "time"}], " ", "*)"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"\[Omega]", " ", "=", " ", "1.0"}], ";", " ", RowBox[{"\[Gamma]", " ", "=", " ", "5.0"}]}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"xvalues", "=", " ", RowBox[{"ConstantArray", "[", RowBox[{"0", ",", "600"}], "]"}]}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Defines", " ", "an", " ", "empty", " ", "array", " ", "of", " ", "values", " ", "for", " ", "our", " ", "position"}], " ", "*)"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"xdotvalues", "=", " ", RowBox[{"ConstantArray", "[", RowBox[{"0", ",", "600"}], "]"}]}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Defines", " ", "an", " ", "empty", " ", "array", " ", "of", " ", "values", " ", "for", " ", "our", " ", "velocity"}], " ", "*)"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"xvalues", "[", RowBox[{"[", "1", "]"}], "]"}], " ", "=", " ", "x0"}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Sets", " ", "first", " ", "value", " ", "of", " ", "position", " ", "array", " ", "to", " ", "the", " ", "defined", " ", "initial", " ", "position"}], " ", "*)"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"xdotvalues", "[", RowBox[{"[", "1", "]"}], "]"}], " ", "=", " ", "xdot0"}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Sets", " ", "first", " ", "value", " ", "of", " ", "velocity", " ", "array", " ", "to", " ", "the", " ", "defined", " ", "initial", " ", "velocity"}], " ", "*)"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"For", "[", RowBox[{ RowBox[{"i", "=", "2"}], ",", " ", RowBox[{"(*", " ", RowBox[{ "Sets", " ", "initial", " ", "value", " ", "of", " ", "for", " ", "loop"}], " ", "*)"}], " ", "\[IndentingNewLine]", RowBox[{"i", "<", "601"}], ",", " ", RowBox[{"(*", " ", RowBox[{ "Condition", " ", "for", " ", "for", " ", "increment", " ", "of", " ", "for", " ", "loop"}], " ", "*)"}], " ", "\[IndentingNewLine]", RowBox[{"i", "++"}], ",", " ", RowBox[{"(*", " ", RowBox[{ "Increment", " ", "for", " ", "loop", " ", "by", " ", "a", " ", "single", " ", "index"}], " ", "*)"}], " ", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"xvalues", "[", RowBox[{"[", "i", "]"}], "]"}], " ", "=", " ", RowBox[{ RowBox[{"xvalues", "[", RowBox[{"[", RowBox[{"i", "-", "1"}], "]"}], "]"}], "+", RowBox[{"e", "*", RowBox[{"xdotvalues", "[", RowBox[{"[", RowBox[{"i", "-", "1"}], "]"}], "]"}]}]}]}], ";", " ", RowBox[{"(*", " ", RowBox[{ "Defines", " ", "next", " ", "x", " ", "value", " ", "as", " ", "a", " ", "first", " ", "order", " ", "Taylor", " ", "series", " ", "approximation", " ", "of", " ", "the", " ", "position"}], " ", "*)"}], " ", "\[IndentingNewLine]", RowBox[{ RowBox[{"xdotvalues", "[", RowBox[{"[", "i", "]"}], "]"}], " ", "=", " ", RowBox[{ RowBox[{"xdotvalues", "[", RowBox[{"[", RowBox[{"i", "-", "1"}], "]"}], "]"}], "-", " ", RowBox[{"e", "*", RowBox[{"(", RowBox[{ RowBox[{"\[Omega]", " ", RowBox[{"xvalues", "[", RowBox[{"[", RowBox[{"i", "-", "1"}], "]"}], "]"}]}], "+", RowBox[{"2", " ", "\[Gamma]", " ", RowBox[{"xdotvalues", "[", RowBox[{"[", RowBox[{"i", "-", "1"}], "]"}], "]"}]}]}], " ", ")"}]}]}]}]}]}], " ", "]"}]}], " ", RowBox[{"(*", " ", RowBox[{ "Defines", " ", "next", " ", "velocity", " ", "value", " ", "as", " ", "a", " ", "first", " ", "order", " ", "Taylor", " ", "series", " ", "approximation", " ", "of", " ", "the", " ", "velocity"}], " ", "*)"}], " ", "\[IndentingNewLine]", RowBox[{"ListPlot", "[", "xvalues", "]"}], " ", RowBox[{"(*", " ", RowBox[{ "Plots", " ", "the", " ", "list", " ", "of", " ", "xvalues", " ", "to", " ", "yield", " ", "a", " ", "position", " ", "as", " ", "a", " ", "function", " ", "of", " ", "time", " ", "plot"}], " ", "*)"}], " "}]}]], "Input", CellChangeTimes->{{3.70700533077411*^9, 3.707005363043866*^9}, { 3.707005571222979*^9, 3.707005598339395*^9}}], Cell[BoxData["5.`"], "Output", CellChangeTimes->{3.707005332211391*^9, 3.707005363394676*^9, 3.707005598649301*^9}], Cell[BoxData[ GraphicsBox[{{}, {{}, {RGBColor[0.368417, 0.506779, 0.709798], PointSize[ 0.0055000000000000005`], AbsoluteThickness[1.6], PointBox[CompressedData[" 1:eJxVmHlUzG//h7NEIpKkRFKWCCWKRF4qS9GqjYr20jrt21SfGdlFiyUKky2F ZEtlKb6UvqTsWdokIqkUIunpd577/Zzz84fOde55zTXX3B1NJrsGWnkMFBMT s+//6/++/vdP+9L//1UMMbK5KxuiiSWwfGXwr1GfiKUxsKvnb8XqDsay4CXp ui85SSyPi1ENcW6dxBPQlC9Zqr7oG2NlmPoVu/hEEavAdl5OVO1F4im4Umti PP898TSM8uL3DRndyVgN+0ZtDni3kHgm3LJtko0ciWfhdLnT56MxxHMQyo8v tjtIrImra6t1HlwgnosJaWZ2a+4Ra8G0uahtajXxPHS91fBKbyaej9mpJyI/ /SDWxsmD4m9XDOpirIOQroO176WIF2Duznlf2uSIF6Ky7rtkkhKxLvRK8kw+ TSFehMwPljMmzCTWwzfD9a3L5xAvhu0N55/cXOIliFyzeXjTPGJ9XB2VmrpN m3gpklwvv47RIQamK8sYPiXmgIc7zcKP/O98GaoyxyY3055bhkRzf5Pb8+nc AG+fK46bRz7OAFWSe4xX/e/1GCJbe/8cSQ06N4Rl8prpgll0boSaxICSKzPo 3AgRZevLzk6j8+V4tmiok58qnS+HQujooX8m0fkK7F48qXjjBDpfAd+DhWIZ 8nS+EiqZ959fkaXzlbgo87ThvDSdr4JOkbLfzhF0vgpdk956m0rQuTHm310j +EX3yRljdC0Uk/vovk1QrH99o1wPY84ETR0/set/3w+r4eRYHd/RQeerIfXt Xp5JK52vwcCJCkij7yduDVYdlA5/00jnprA2cI8eW0fnplAIjLVb+ZrOzXAm 0ak36Dmdm0HZXs/oQBWdm6O7uObrlQd0bo7Vs66fe1RK5xbY3/FO8f1txrDA KZv8f7/foMdb4HhLmO7gAsYlFvA4nPVm1GXaW0JMa52CQi7tLbFFxtBCOZv2 llgj9ejZ1JO0t8TTdX3fZhyjvRVUNEzbZh2mvRVOWG4YqrGf9lYoyMkP1kyi vRXS/7Vym7uL9msRJD5/iNZW2q9F4cXazVoC2q8F9/GB2Dw+7ddCsbfyyLwI 2lvjQm145Pxg2lsjZ5ndWW1/2lujIt3JbIE37a0RkzsqQteN9jbY6Fq+cPEG 2tvgbUbnyaXraG+DxiztB4bWtLfB46uZhavMaW+Lub1v+WYmtLeF4ltDZZvl tLdFqZdkriNob4vuO+s0PPRob4e+3T7nAnRob4eKSxmzoubS3g7b3+/OT5hF ezusaplukTyd9vaokb3x96gK7e2h9EGi7PxE2tvDb6np+ZvytLfH8/HWVx+N of06GA/l3tWPpP06vOaeL+gaRvt1qDYYXCAhTvt1yD8f6KEkRvv1/d+vOgba Pezfd6yHaf1CC9MfjLn1SJJWTvbsYFyyHjrerkMEX+jngQPU5Y0vZnykvQMW CrpSCt/R3gFecD73sob2Dqh45vn3RzXtHdHUfHfHuGe0d0TmqgnmupW0d0Tg 7h9mjv/S3hGnuLId3D3aO6HHvK3vVAntnTAl4sbFh9dp74RvW/2PdeXT3gkt hVrlEy/RfgOuuWhorDpP+w3YoBnzOOQM7Teg01k1T3SC9htQc0f9waOjtN+I szl7Vf4eov1GyDwwyJ+zn/Yb4ds9d6tzEu03orbaMCV1F+2dMWbzutdlWxkr O2OBlPmGXgE9nzMqpP5MnB/L2NkZ5aPmK/hF0vM7Y2pPqfmpEMYiZ0wMEdyu CyCfM+TVZwco+jCud8a0eck29h7kd0Hr4uCIA87kd8HdVylPnzuQ3wUnL13e JGdHfhckF6Xq2luR3wVDHdsN0k3J74KGzC1b6laR3wV6jiMHTDUivwuylhld 9V1KfleMl6rPvEyfR5Rd8S7tUOkfbfK7ouPHdNWVc8nviltbl11LmUV+V/RI ZAjrppPfFQ4XvyTMViW/K5J0mm/ylcjvivx6g9kVCuR3g9mBvCdKY8nvhq6P 788HSZPfDe/Hp966N5z8bojTChisOJT8bnjUrRYfNJD8bjAaEj6nvJd93ipx g+j+UBmVX4zr3bBgR9IMfhd9HnPHnuA7oS/bGCu7I3i/eee8FsZwx+bWTyeT PzB2doeNh+GO9gbGnDtUXkuJLGoYi9wxaVJP88Vq8rsjTuymm+wz8rtDeYW4 dGQl+T0wJH9L69t/ye8BTq69x6CU/B6YrvljUc5t8nvgRKVxjsxN8nvAoCzD mF9Afg9sahfJf7xMfg80zJIYt/YC+T3Q65VgVJJDfk9kHSwTzTlNfk9Y3tys fjST/J6o+ODzfuQR8ntCWkPjAZdGfk/U3RLWdaaS3xP+FeMme+8lvyeaRaK9 NTvJ74mfeypnWm8lvxdm9Np+eyAgvxc2unxqMoolvxemGOiKF0eS3wt9Or/M F4WS3ws3m9/cyw8kvxcujN7nOd+X/F74FPts7mVP8nsheLjF9Hmu5PdGWE7h iitO5PeGt9PdvTrryO8NLV3VgUXW5PfGao8dR/QtyO+N14o5Lnfp87/IGw+y F5qtXkl+b3wOaHZ9akB+b3TW+B1z1Cf/JpzmRw36qEv+TXjWeHtPsDb5N8Gv +Bf6NMm/CdsfPVNInEX+TQhrHKIwQY38m3Cj0VD/nCr5N2HG/aU7lkwi/yZU ZeT+qhxPfh/ICIz2usmR3weirKvG3aPJ74N423y1PVLk94Hkx9/qU4eR3wff 7i1Ye3Mw+X2wNGTyUVsx8vvASug3oqOH/b5U74PGQxVHdv+k35980Rfyy3JG J2NlX0jWnlQr+8oYvrhekDDZ8zNjZ18odpsuHvKBMecLwYmrkVkNjEW+SHnO e2Fcw7jEF/o35tu0VpPfF0Vp+V3Jz8jvB0Hx2SsLqsjvB9O08n21D8jvB7mj 9/ZtLSO/H0ZutLmi8Q/5/ZDOn9P56hb5/WB3dojlliLy+2HY5dSKufnk90Nm wC7vuovk94f2wxOT95wnvz/+PEzpXpJNfn+s3jKy9etJ8vtj74SKvyIR+f1h XJEw2zqD/P6QbGqJlEgjvz8C7xbU30wlvz8U6zLdQ/eSPwCXrjsNnrWL/AGo 7jxe/H4r+QOgNl/twFEh+QOgtfjktnVx5A9AeG1F6lj6/VgUACcxx6InYeQP gO5h8Z6kIPIHQO+awMbCn/yBCArbVi69ifyBuCP+yPaJO/kD0Xx2RO8+Z/IH wvfW7+v2juQPxJHT5qkT7ckfiP31d4SNa8kfiJk9w3fmmJM/EF8PvjsTvJr8 PMz0/Vurt5KxNA/VeuNmDTGk18ODqPH93sf6jDV5eGI2SfLoInp9PLjy4w77 6jC24KEhtgiLtOj18hATsevPsDmMeTx8O5Zd8XoGvX4eDqDi8rmpjJN4WH41 Nzd+MvX0P7/fkFtrJzLO4+Hph90NagrUx0PPtd/yfbKMq3jI3yLh/kKaenmw kgi8kzuCcTsPeyQ/ztsuQf1BSN40o8B1MPUH4bNsm5k+/f+HchBGTnzXpt3d xvqD4PG8yqOjjTGCMEE0oS/3I2OLfm5bcD2gjrFzEJrt9x/SfMmYF4QDaVmH uh4x5oJwN7GhqKiUcVIQkhbn/xHeYiwKwp6KtA2m+YzzgnAj8U+dQi7jkiDE fbXmmk8xrgrC3i1jlxYeYVwfhD8jLinu3s+4PQjj3I6OdUlkLBaMoJ1msxZu YSwdjIpsnY2jYxkrB0NxxIPzX0KpPxg+yty4f/2oPxjvov5Nz3an/mAUVSot 2uVI/cGY09vREWBN/cEIG1B4x3oN9fc/vq/57GIj6g/G6KU/z01bTP3B8J+5 7p7MfOoPxiGLhC6xWdTf//zuX3U7VKk/GNv0pPY3KlJ/MG5U2YpXj6H+YBxc tmzno+HUH4IdlQaqZYOoPwQn6h5X3u75yvpDkPjbfe+tTsaaIYh+GOxys4Ux QqA6NXj5rUbGFiEo+vlq0e03jJ1D0Lh+8rLSp4x5IcjeXmxX8YAxF4LIWvW4 F/8wTgrB6Jqayw3XGYtCIFAz7f56mXFeCMZv7zD5e5ZxSQi8PhicHXWScVUI plkWyqtmMK4PweJfD1MW7mPcHoJcrR/yFrsZi4XiYOj5nE0JjKVDMWmd78ot fOoPRdCRM1+Ph1J/KK5Orj12x4/6QzHoU8L69+7UHwpHBTklCSfqD8XxHsnP s22oPxR7et7etDGl/lCcyhI/FL+c+kOhdfFv9Nkl1B8KvS/L3F5pU38oos2N LYfNof5QWHXtMdKbRv2hKJNLXByoRP2hCBt7a+EpOeoPxbagjIU1I6k/DLmP juuNG0r9YZDvXL1sbV8r6w+DV5mMcfJPxpphyLYNtnrcxhhhMHvU5DimmbFF GJTjmr3s6hk7h0G3+nvwkWrGvDD85u2IbapizIVhcJ/SNo1yxklhmOOgtTfm NmNRGGK4pfvLCxnnhUF29500hUuMS8Kw5OaCQ745jKvCsDpM8UDxccb1YTi2 4POesemM28NwOFt5s38qY7FwHEjRCy7bxVg6HG/fXHNQTaD+cKQnySwV8Kk/ HA019xUbQqk/HMky9h0G/tQfjvXHlhSf9qD+cFyQ+5kwYgP1h+PWrIEGobbU Hw6va2+/15hRfzjm5Q4WGa+k/nCYzexeem0p9YcjylP7+bSF1B+OocWLNqZp Un84qi5urRk+g/rDoV570FwwmfrD0TS49Uq3AvVHQEdCXCpYhvojcKNlg12r JPVHIPBZVKrPIOqPQLrE85JPPV9YfwSet7XV+nQxtojAdxmT1tYvjJ0jMHyC W0twE2NeBOxfFFX/qmHMRcBozuMrwheMk/rP506Ll6pkLIqA1shFCw6XMc6L QLxs4lu1EsYlEei4ciygsIBxVQQWCr59WX2RcX0EutWH2NVnM26PwO3tDufC jzMWi8TQ3SFfRqYzlo6E/OpKuexUxsqRGPunbeby3Yw1I/Hpu61aYwL1R2JZ 3paRm2OpPxKb/+l9OyWc+iNxtxmp9wOov59/VWkGeFF/JOrbNa6Odab+SIz8 VDep2J76I/F4ohvPx5L6I6Hdond6nAn1R+JOxZeSUgPqj0S4a3dxhB71R+Km Z8PxmfOpPxJeBTM31c2i/iiIbOxkDkyl/ih8dvx8xEyJ+qMwWspr+LBx1B+F Kfcs1t8bRf1RCBysvnOzBPX3s0/UYcMB1B+Fh/nFO8V/t7D+KJiXC9aXf2PM RaEm03DY3hbGSVHwcd5xwPY9Y1EUvhu9HKBcwzgvCrEXLpq0PGdcEoVfk47w Ch4xroqCpY5SyLYyxvVRsL1UaGlfwrg9CtdCuoarFzIWi0ZP1NrMvouMpaMR 9d5h9IscxsrRMPuW4pB7grFmNOwGxsVvz2CMaKilHue772dsEY2BhlutDPYw do7Gw8pcMZVt1B8NqXHHtw7iqD8aDWM/NH6IpP5ofGsWk38YRP3RCHwYqXLZ h/qjMXjCs4EZbtQfjadWhwq2OlJ/NPyGmxuG2FB/NO70xRx3MaP+aEwflvXc ciX1xwDKdm8NQf0x6HMck79Al/pjkPHD1XW2FvXHYKdhRf0UdeqPQXDhK02l KdQfg+mto6wUJlJ/DI6n/DSUk6P+GHTz5YeMHUX9MbA6+vvgWAnqj0HT0Gm/ xg2g/hi0fFKZMeH3Z9Yfg0OpUdNVOhmXxEA+5FLnjC+Mq2IQN9Y2cV4T4/oY 3JvW2qlfy7g9Bv8EiU1f85KxGB+T6+WnOVQxlubDNrq43becsTIfGV6xW+Pu MNbk41Prmw8p1xmDj+Yg99HZVxhb8NEeLhK/fZ6xMx+9RxX+eX2aMY+PrJx1 xj+OMeb4GB/1+eCYQ4yT+BglYZurlcJYxMfdkGm71+6ifj6+/2jSDE+gfj4e f1bKOBxL/Xx0lTpUlIRTPx8uk6ffbQ6kfj4ic4dzMpuoPxZfsU9M35X6Y3Gv fPVyXwfqj8UK+5umh62pPxY/FUIVHphSfyz2r+/J6l1B/bEYnlDbPRfUHwtf pezh3rrUH4umyg91x7SoPxa3HVXDXqlTfyxssz7/KzuV+mMxIO9+vaUS9cdC eb9xftI46o/FsNRfJo+lqT8WAwd7HB0jSf2x0HWWz7MbRP2xuLFEFH/kzyfW H4dQix0jmr4zlo7DeB9n2zltjJXjoBl9yTGqmbFmHHoSR0wqbWCMOERV9WTI vmFsEQeHnVIv3J8xdo5D7+TK8vwKxrw4OCk+ipAsY8zFQU4+rW5jCeOkONi+ LBK7VshYFIeX2ndfj7rMOK+fZxv4+5xjXBIHmZd3rpWeYlwVhwLPZ4VTjjGu j4O3hnhIQhrj9jhMy2x815RM/fGwNv4rYbyL+uNx7knj+/MJ1B8PHXHVcNk4 6o+HQanJDX4E9cfDRf5z/gce9cfDQTHJw8qH+uMxc8TDsmI36o9Hu9HsmjlO 1B+PzsWGp47ZUn88UrYfmSRjQf3x0H/1YPVWY+qPR1iv04w/BtQfj2v1jddC FlN/PIS7Wn980ab+eKxUHf7BS4P64+H5+WHCezXq56CKf564qjCW4HBcFFf5 TpHeDw5vT+6Ndh/LWJ5D+sBtTz+OpPeHQ9Du76/8JBircbgw1y+xcwC9Xxyc /8a0xvQ0/5cXcoBi/i/x74zBIb/i7MXkr4xXcXgY8VV2UjNjCw7T0kap5jYw tufQWy56o/+GsTOHcffm6T9+xtibwy2n9SYejxjzOBQlnO77XcY4ksPqlTlO KbcZcxyUKn+6qV9nvJ2DietkmdIrjJM4GLkVernmMk7rfz8srNzFzjAWcdAq jRsqymR8hsOl6McWy9IZ53EYblCz9P0+xgX9PSO0n23fw7ik//1v8pXS2M74 PgfT77ItLwSMqzjICf7x42IYV3MY9UkyRT2McT2Huuy49dUBjJs56DteurfV m3E7h9Yruk91XBl3c8hNOCNsdmAsJoB+QcrTdBvGEgKY2u++Z2HOWFqA8B0z 7IcaM5YXQHf3353FBoyVBei6ccwpajFjNQHepGZWztdhrClAx8m0+g4Nun8B BogUkvJm0P0LcMCisIanSvcvwI8jI8q1JtL9CzBu1xnLH3J0/wJk6KhEXJem +xfgweuRukJJun8BtlUMOmgymO5fABPuQJLs34/s/gUYzzlOrf/JmOt/fFiL 9fkOxtsFEE15pcRvYZwkQKvfdeGaJsZpAmwyVuOU6hiLBIiuuir3rZrxGQE+ zfu9vOwJ4zwBakp2SR19yLhAgMd1Yv7hpYxLBEiSbXezKGF8X4DZf9s61IsY VwnQY3tQbtgVxtUCbO3iP/54nnG9AIllC1TuZzFu7n/8mw0SOZmM2wWotd+x OTGdcbcAbpxZavB+xmJCPLywfMG6vYwlhPDbOCRw2Q7G0kJkqKgtUt/MWF6I wiyDNLlYxspCKBS0bh8UwVhNCPGFt4d/4zHWFOLKNBvldz6MFwqx7Oiy0qfu jCFE1lOl3tINjFcJcU1uy93r9owthJB9MmLCJSvG9kLM3GU2KGcNY2ch1N69 izqxgrG3EN3JrvFHwZgnhB7Pc0z6Irp/IfT9t2kfmk/3L8TP1A1NaXPo/oX4 PWyLymE1un8hwkft+5KhQvcvxOy+KYaZE+j+hfA1ezw1S47uX4iBiVr7cqXp /vt7Pd4lXpOk+xfia3q0zJ3BdP9CQHr/5Ed/Pyz9D1H7ViI= "]]}, {}}, {}}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->{True, True}, AxesLabel->{None, None}, AxesOrigin->{0., 0.6913651044560836}, DisplayFunction->Identity, Frame->{{False, False}, {False, False}}, FrameLabel->{{None, None}, {None, None}}, FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}}, GridLines->{None, None}, GridLinesStyle->Directive[ GrayLevel[0.5, 0.4]], ImagePadding->All, Method->{"CoordinatesToolOptions" -> {"DisplayFunction" -> ({ (Part[{{Identity, Identity}, {Identity, Identity}}, 1, 2][#]& )[ Part[#, 1]], (Part[{{Identity, Identity}, {Identity, Identity}}, 2, 2][#]& )[ Part[#, 2]]}& ), "CopiedValueFunction" -> ({ (Part[{{Identity, Identity}, {Identity, Identity}}, 1, 2][#]& )[ Part[#, 1]], (Part[{{Identity, Identity}, {Identity, Identity}}, 2, 2][#]& )[ Part[#, 2]]}& )}}, PlotRange->{{0., 600.}, {0.7186041503814593, 1.2633850688889738`}}, PlotRangeClipping->True, PlotRangePadding->{{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.05], Scaled[0.05]}}, Ticks->{Automatic, Automatic}]], "Output", CellChangeTimes->{3.707005332211391*^9, 3.707005363394676*^9, 3.707005598694912*^9}] }, Open ]] }, WindowSize->{916, 651}, WindowMargins->{{133, Automatic}, {32, Automatic}}, FrontEndVersion->"10.4 for Mac OS X x86 (32-bit, 64-bit Kernel) (April 11, \ 2016)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[558, 20, 1753, 34, 114, "Input"], Cell[CellGroupData[{ Cell[2336, 58, 5693, 133, 488, "Input"], Cell[8032, 193, 193, 3, 28, "Output"], Cell[8228, 198, 9666, 170, 232, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[17931, 373, 5676, 134, 471, "Input"], Cell[23610, 509, 171, 3, 28, "Output"], Cell[23784, 514, 9247, 164, 243, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[33068, 683, 5539, 131, 471, "Input"], Cell[38610, 816, 119, 2, 28, "Output"], Cell[38732, 820, 9186, 162, 280, "Output"] }, Open ]] } ] *)