{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Times" 0 18 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 257 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Times" 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 0 12 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 0 12 128 0 128 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2 " -1 257 1 {CSTYLE "" -1 -1 "Courier" 0 11 0 128 128 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "[names deleted]" }}{PARA 0 "" 0 "" {TEXT -1 8 "Group 3B" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 256 13 "Spring Motion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 84 "An investigation of mathematic al models for spring motion and the effect of damping." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with (plots): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(stats):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Part 1: The Un damped Spring" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 113 "Enter your data for the \"undamped\" spring, starting wi th the number of readings and the time step between them. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "n :=100; # Number of distance readings" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "dt:=.05; # Time step between readings" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"$+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dtG$ \"\"&!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "K:=21.7; # Spr ing constant" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "m:=0.550; # Total m ass" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG$\"$<#!\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"mG$\"$]&!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT 267 68 "Enter your distance re adings in the array V, separated by commas. " }{TEXT -1 101 "We calc ulate the average of these data and convert the list to an array so we can index the elements." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 996 "V := [0.811126, 0.773808, 0.729904 , 0.681609, 0.634412, 0.591606, 0.556483, 0.533433, 0.523555, 0.531238 , 0.550995, 0.58502, 0.625632, 0.672828, 0.721123, 0.765027, 0.80015, \+ 0.824297, 0.835273, 0.829785, 0.811126, 0.780393, 0.74088, 0.693683, 0 .646486, 0.602582, 0.566361, 0.542214, 0.529043, 0.531238, 0.547702, 0 .577337, 0.616851, 0.661852, 0.709049, 0.752953, 0.790272, 0.816614, 0 .829785, 0.828688, 0.814419, 0.785881, 0.746368, 0.704659, 0.65856, 0. 61246, 0.575142, 0.5488, 0.532336, 0.532336, 0.546604, 0.571849, 0.608 07, 0.650876, 0.696976, 0.741977, 0.780393, 0.807833, 0.825395, 0.8264 92, 0.815516, 0.791369, 0.757344, 0.714537, 0.669536, 0.626729, 0.5883 13, 0.558678, 0.540019, 0.535628, 0.544409, 0.567459, 0.600387, 0.6420 96, 0.684902, 0.729904, 0.76832, 0.799052, 0.818809, 0.824297, 0.81661 4, 0.79576, 0.763929, 0.724416, 0.680512, 0.637705, 0.598192, 0.566361 , 0.546604, 0.538921, 0.544409, 0.564166, 0.593801, 0.632217, 0.676121 , 0.718928, 0.758441, 0.790272, 0.812224, 0.819907]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "mean1:=describe [mean] (V);" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "a:=convert(V,array):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&mean1G$\"++Vm(y'!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 191 "Now we graph the data versus time. We subtract the mea n so that the data is centered about 0 on the vertical axis. The \"k - 1\" in the plot command reflects the fact that the array numbering " }{TEXT -1 7 "starts " }{TEXT 264 35 "at 1, but our time starts at t = \+ 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "k:='k':" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "graph1:=plot([[(k-1)* dt,a[k]-mean1] $k=1..n],style=line, color=blue, thickness=2):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "display(graph1);" }}{PARA 13 "" 1 " " {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6&7`q7$\"\"!$\"1++++d fB8!#;7$$\"1+++++++]!#<$\"1,+++q:/&*F/7$$\"1+++++++5F+$\"1++++qv8^F/7$ $\"1+++++++:F+$\"1+++++dUG!#=7$$\"1+++++++?F+$!1++++IWNWF/7$$\"1++++++ +DF+$!1++++I/;()F/7$$\"1+++++++IF+$!1++++V$GA\"F+7$$\"1+++++++NF+$!1++ ++VL`9F+7$$\"1+++++++SF+$!1++++V6_:F+7$$\"1+++++++XF+$!1++++VGv9F+7$$F .F+$!1++++Vrx7F+7$$\"1+++++++bF+$!1,+++Iku$*F/7$$\"1+++++++gF+$!1++++I W8`F/7$$\"1+++++++lF+$!1+++++VQfF<7$$\"1+++++++qF+$\"1++++qlNUF/7$$\"1 +++++++vF+$\"1********p0E')F/7$$\"1+++++++!)F+$\"1++++d$Q@\"F+7$$\"1++ +++++&)F+$\"1++++dIb9F+7$$\"1+++++++!*F+$\"1++++d1l:F+7$$\"1+++++++&*F +$\"1++++d=5:F+7$$\"\"\"F(F)7$$\"1++++++]5!#:$\"1++++dE;5F+7$$\"1+++++ ++6F\\r$\"1++++qN6iF/7$$\"1++++++]6F\\r$\"1++++ql\"\\\"F/7$$\"1+++++++ 7F\\r$!1++++I/GKF/7$$\"1++++++]7F\\r$!1++++IW=wF/7$$\"1+++++++8F\\r$!1 ++++V0C6F+7$$\"1++++++]8F\\r$!1++++V_l8F+7$$\"1+++++++9F\\r$!1++++VB( \\\"F+7$$\"1++++++]9F\\rFY7$$F9F\\r$!1++++Vk58F+7$$\"1++++++]:F\\r$!1+ +++VH95F+7$$\"1+++++++;F\\r$!1++++Ia\">'F/7$$\"1++++++];F\\r$!1++++IW \"p\"F/7$$\"1+++++++F\\rFdq7$$\"1++++++]>F\\r$\"1++++d@*\\\"F+7$$\"\"#F($\"1+ +++d_c8F+7$$\"1++++++]?F\\r$\"1++++d9r5F+7$$\"1+++++++@F\\r$\"1++++q:g nF/7$$\"1++++++]@F\\r$\"1++++qD*e#F/7$$\"1+++++++AF\\r$!1++++Ik??F/7$$ \"1++++++]AF\\r$!1++++IkImF/7$$\"1+++++++BF\\r$!1++++VCO5F+7$$\"1+++++ +]BF\\r$!1++++Vm*H\"F+7$$\"1+++++++CF\\r$!1++++VIk9F+7$$\"1++++++]CF\\ rF_z7$$FDF\\r$!1++++Vi@8F+7$$\"1++++++]DF\\r$!1++++V " 0 "" {MPLTEXT 1 0 7 "t:='t':" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "y0:=(0.835273-.6787664300);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Y:=t->y0*cos(sqrt(K/m)*t + Pi/10);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#y0G$\"++d1l:!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"YGR6#%\"tG6\"6$%)operatorG%&arrowGF(*&%#y0G\"\"\"-%$cosG6#,&*&-%% sqrtG6#*&%\"KGF.%\"mG!\"\"F.9$F.F.%#PiG#F.\"#5F.F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 513 "We add the factor of amplitude of y0 to \+ our model, taking the highest peak from our data and subtracting the m ean from it. We do this, instead of merely taking the value at t=0 bec ause at t=0, we see that the spring is not at its peak. Hence, we adju st for the actual amplitude by taking the highest peak of the data. We then had to adjust for the period (knowing that the data graph does n ot start at its peak at t=0). Thus, we adjusted for this by adding a d elay factor of Pi/10, (which seems to be sufficient)." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 55 "Next we graph the data and the model function together." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "g raph2:=plot(Y(t), t=0..7,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "display(\{graph1,graph2\});" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6&7`q7$\"\"!$\"1++++dfB8!#;7$$\"1++++ +++]!#<$\"1,+++q:/&*F/7$$\"1+++++++5F+$\"1++++qv8^F/7$$\"1+++++++:F+$ \"1+++++dUG!#=7$$\"1+++++++?F+$!1++++IWNWF/7$$\"1+++++++DF+$!1++++I/;( )F/7$$\"1+++++++IF+$!1++++V$GA\"F+7$$\"1+++++++NF+$!1++++VL`9F+7$$\"1+ ++++++SF+$!1++++V6_:F+7$$\"1+++++++XF+$!1++++VGv9F+7$$F.F+$!1++++Vrx7F +7$$\"1+++++++bF+$!1,+++Iku$*F/7$$\"1+++++++gF+$!1++++IW8`F/7$$\"1++++ +++lF+$!1+++++VQfF<7$$\"1+++++++qF+$\"1++++qlNUF/7$$\"1+++++++vF+$\"1* *******p0E')F/7$$\"1+++++++!)F+$\"1++++d$Q@\"F+7$$\"1+++++++&)F+$\"1++ ++dIb9F+7$$\"1+++++++!*F+$\"1++++d1l:F+7$$\"1+++++++&*F+$\"1++++d=5:F+ 7$$\"\"\"F(F)7$$\"1++++++]5!#:$\"1++++dE;5F+7$$\"1+++++++6F\\r$\"1++++ qN6iF/7$$\"1++++++]6F\\r$\"1++++ql\"\\\"F/7$$\"1+++++++7F\\r$!1++++I/G KF/7$$\"1++++++]7F\\r$!1++++IW=wF/7$$\"1+++++++8F\\r$!1++++V0C6F+7$$\" 1++++++]8F\\r$!1++++V_l8F+7$$\"1+++++++9F\\r$!1++++VB(\\\"F+7$$\"1++++ ++]9F\\rFY7$$F9F\\r$!1++++Vk58F+7$$\"1++++++]:F\\r$!1++++VH95F+7$$\"1+ ++++++;F\\r$!1++++Ia\">'F/7$$\"1++++++];F\\r$!1++++IW\"p\"F/7$$\"1++++ +++F\\r Fdq7$$\"1++++++]>F\\r$\"1++++d@*\\\"F+7$$\"\"#F($\"1++++d_c8F+7$$\"1++ ++++]?F\\r$\"1++++d9r5F+7$$\"1+++++++@F\\r$\"1++++q:gnF/7$$\"1++++++]@ F\\r$\"1++++qD*e#F/7$$\"1+++++++AF\\r$!1++++Ik??F/7$$\"1++++++]AF\\r$! 1++++IkImF/7$$\"1+++++++BF\\r$!1++++VCO5F+7$$\"1++++++]BF\\r$!1++++Vm* H\"F+7$$\"1+++++++CF\\r$!1++++VIk9F+7$$\"1++++++]CF\\rF_z7$$FDF\\r$!1+ +++Vi@8F+7$$\"1++++++]DF\\r$!1++++V]9QF/$\"1KBKTCj)znON'F/7$$\" 1nmm\"z+e_\"F+$\"1R%ya9M()f%F/7$$\"1](o/o\\]0B9lAF<7$$\"1+](oM'f *=#F+$!1s?/pEa`=F/7$$\"1$3xcBXbN#F+$!1c9Y$))\\/Y$F/7$$\"1n\"zW7%\\@DF+ $!1F!4c&>zH]F/7$$\"1]7G8IW(o#F+$!1:f'*)yKXa'F/7$$\"1ML3->R`GF+$!1LtO9! 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With our adjustments for amplitude and our delay (because the dat a did not exactly start at the peak for t=0), we have a model that see ms to show an undamped spring motion. The only difference comes where \+ as t grows larger, the amplitude of the data graph decreases. This is \+ because the data was not ideally undamped (it had a CD attached to the weight), and so air resistance was damping the spring as time went on ." }}{PARA 0 "" 0 "" {TEXT -1 9 "*** good" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Part 2: The Damped Spring" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 263 "" 0 "" {TEXT -1 108 "Enter your data for the damped spr ing, starting with the number of readings and the time step between th em. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "n:=200; # Number of distance readings" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "dt:=0.05; # T ime step between readings" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"$ +#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dtG$\"\"&!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT 256 80 "Enter \+ your distance readings in the array Z, separated by commas. As befor e, w" }{TEXT 268 71 "e calculate the average of these data and convert the list to an array." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1978 "Z := [0.669536, 0.748563, 0.816614, 0.869299 , 0.901129, 0.913203, 0.900032, 0.866006, 0.819907, 0.758441, 0.693683 , 0.619046, 0.545507, 0.488432, 0.451113, 0.452211, 0.451113, 0.452211 , 0.453308, 0.5027, 0.563068, 0.632217, 0.701366, 0.766124, 0.821004, \+ 0.861616, 0.883568, 0.885763, 0.870396, 0.836371, 0.789174, 0.729904, \+ 0.66734, 0.601484, 0.542214, 0.491724, 0.456601, 0.450016, 0.452211, 0 .455504, 0.490627, 0.537824, 0.593801, 0.655267, 0.715635, 0.769417, 0 .812224, 0.841859, 0.856128, 0.853932, 0.835273, 0.801248, 0.755148, 0 .700268, 0.643193, 0.587216, 0.537824, 0.49831, 0.473065, 0.463187, 0. 468675, 0.490627, 0.52575, 0.571849, 0.623436, 0.678316, 0.729904, 0.7 74905, 0.806736, 0.826492, 0.83198, 0.8232, 0.80015, 0.765027, 0.72002 5, 0.671731, 0.621241, 0.572947, 0.532336, 0.5027, 0.487334, 0.484041, 0.496115, 0.52136, 0.55758, 0.602582, 0.649779, 0.699171, 0.743075, 0 .780393, 0.805638, 0.818809, 0.816614, 0.801248, 0.773808, 0.737587, 0 .693683, 0.647584, 0.602582, 0.561971, 0.53014, 0.508188, 0.49831, 0.5 01603, 0.519164, 0.546604, 0.582825, 0.625632, 0.669536, 0.712342, 0.7 48563, 0.778198, 0.79576, 0.802345, 0.79576, 0.7771, 0.750758, 0.71234 2, 0.670633, 0.630022, 0.591606, 0.556483, 0.532336, 0.518067, 0.51477 4, 0.522457, 0.542214, 0.571849, 0.605875, 0.644291, 0.682707, 0.71892 8, 0.748563, 0.769417, 0.781491, 0.780393, 0.770515, 0.74966, 0.72222, 0.687097, 0.649779, 0.61246, 0.58063, 0.554288, 0.536726, 0.529043, 0 .531238, 0.544409, 0.566361, 0.597094, 0.63112, 0.666243, 0.699171, 0. 729904, 0.752953, 0.765027, 0.769417, 0.763929, 0.748563, 0.725513, 0. 696976, 0.66295, 0.630022, 0.599289, 0.571849, 0.552092, 0.541116, 0.5 38921, 0.546604, 0.564166, 0.588313, 0.617948, 0.650876, 0.682707, 0.7 1344, 0.737587, 0.754051, 0.762832, 0.761734, 0.750758, 0.732099, 0.70 7952, 0.678316, 0.646486, 0.615753, 0.589411, 0.566361, 0.552092, 0.54 6604, 0.549897, 0.561971, 0.581728, 0.606972, 0.63551, 0.665145, 0.694 78, 0.720025, 0.738684, 0.750758, 0.755148]:" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "mean2:=describe [mean] (Z);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "b:=convert(Z,array):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&mean2G$\"+]_`!f'!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 259 28 "Here is a plot of this data." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "k:='k':" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 57 "graph3:=plot([[(k-1)*dt,b[k]-mean2] $k=1..n] ,style=line):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "display(graph3);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6%7d w7$\"\"!$\"1++++vC[5!#<7$$\"1+++++++]F+$\"1,+++v%4&*)F+7$$\"1+++++++5! #;$\"1+++]Zgv:F47$$\"1+++++++:F4$\"1+++]ZX-@F47$$\"1+++++++?F4$\"1+++] Zv?CF47$$\"1+++++++DF4$\"1+++]Z\\TDF47$$\"1+++++++IF4$\"1+++]Zy4CF47$$ \"1+++++++NF4$\"1+++]Z_p?F47$$\"1+++++++SF4$\"1+++]Z`3;F47$$\"1+++++++ XF4$\"1++++vuQ**F+7$$F.F4$\"1++++v%HY$F+7$$\"1+++++++bF4$!1++++Dv+SF+7 $$\"1+++++++gF4$!1+++]_YN6F47$$\"1+++++++lF4$!1+++]_@1;F47$$\"1++++++]7Fiq$\"1+++]ZiD?F47$$\"1+++++++8Fiq$\"1+++] Z9XAF47$$\"1++++++]8Fiq$\"1+++]Z4nAF47$$\"1+++++++9Fiq$\"1+++]ZU8@F47$ $\"1++++++]9Fiq$\"1+++]ZFiqF_p7$$\"1++++++]>Fiq$!1+++]_\\N?F47$$\"\"#F($!1+++]_E%o\"F47$$\" 1++++++]?Fiq$!1+++]_H77F47$$\"1+++++++@Fiq$!1,+++DDDlF+7$$\"1++++++]@F iq$!1++++]_'y$Fbu7$$\"1+++++++AFiq$\"1++++v9ecF+7$$\"1++++++]AFiq$\"1+ ++]Zj.6F47$$\"1+++++++BFiq$\"1+++]ZqJ:F47$$\"1++++++]BFiq$\"1+++]Z0G=F 47$$\"1+++++++CFiq$\"1+++]Zuq>F47$$\"1++++++]CFiq$\"1+++]Zy[>F47$$FCFi q$\"1+++]Z>iU\"F47$$\"1+++++++EFiq$\"1+ +++vW4'*F+7$$\"1++++++]EFiq$\"1++++vW@TF+7$$\"1+++++++FFiq$!1++++D0'e \"F+7$$\"1++++++]FFiq$!1++++Dv$=(F+7$$\"1+++++++GFiqF\\x7$$\"1++++++]G Fiq$!1+++]_V2;F47$$\"1+++++++HFiq$!1+++]_))f=F47$$\"1++++++]HFiq$!1+++ ]_me>F47$$\"\"$F($!1+++]_y.>F47$$\"1++++++]IFiqFgw7$$\"1+++++++JFiq$!1 +++]_.L8F47$$\"1++++++]JFiq$!1++++DX?()F+7$$\"1+++++++KFiq$!1++++DvhNF +7$$\"1++++++]KFiq$\"1++++vCE>F+7$$\"1+++++++LFiqF[u7$$\"1++++++]LFiq$ \"1+++]Z^e6F47$$\"1+++++++MFiq$\"1+++]Z#oZ\"F47$$\"1++++++]MFiq$\"1+++ ]ZQu;F47$$FMFiq$\"1+++]ZEH< \"F47$$\"1+++++++cFiq$\"1+++]Z1n8F47$$\"1++++++]cFiq$\"1+++]Z\"HV\"F47 $$\"1+++++++dFiqFe\\m7$$\"1++++++]dFiq$\"1+++]ZY!=\"F47$$\"1+++++++eFi q$\"1*******\\Z/<*F+7$$\"1++++++]eFiqFi[m7$$\"1+++++++fFiq$\"1++++v%z: \"F+7$$\"1++++++]fFiq$!1++++D:.HF+7$$\"\"'F($!1*******\\_Zu'F+7$$\"1++ ++++]gFiq$!1+++]_qD5F47$$\"1+++++++hFiqF]cl7$$\"1++++++]hFiq$!1+++]_') 49F47$$\"1+++++++iFiq$!1+++]_zU9F47$$\"1++++++]iFiq$!1+++]_'fO\"F47$$ \"1+++++++jFiqF[v7$$\"1++++++]jFiqFe^l7$$\"1+++++++kFiq$!1++++D&yJ&F+7 $$\"1++++++]kFiq$!1++++DDw9F+7$$FdoFiq$\"1++++vMlBF+7$$\"1++++++]lFiq$ \"1++++vW()fF+7$$\"1+++++++mFiqF/7$$\"1++++++]mFiqF`y7$$\"1+++++++nFiq $\"1+++]ZPC7F47$$\"1++++++]nFiqFbfl7$$\"1+++++++oFiq$\"1+++]Zh96F47$$ \"1++++++]oFiq$\"1,+++vkg!*F+7$$\"1+++++++pFiq$\"1++++vk;jF+7$$\"1++++ ++]pFiq$\"1++++vM/GF+7$$\"\"(F(Fcel7$$\"1++++++]qFiq$!1++++DNfYF+7$$\" 1+++++++rFiq$!1*******\\_B%yF+7$$\"1++++++]rFiq$!1+++]_lZ5F47$$\"1++++ +++sFiq$!1+++]_FB7F47$$\"1++++++]sFiq$!1+++]_5+8F47$$\"1+++++++tFiq$!1 +++]_:y7F47$$\"1++++++]tFiq$!1+++]_WY6F47$$\"1+++++++uFiq$!1++++DDp#*F +7$$\"1++++++]uFiq$!1++++D&f>'F+7$$F^pFiq$!1++++DN$z#F+7$$\"1++++++]vF iq$\"1++++]Z*=(Fbu7$$\"1+++++++wFiqFhel7$$\"1++++++]wFiqF[u7$$\"1+++++ ++xFiq$\"1++++v%**Q*F+7$$\"1++++++]xFiqFdal7$$\"1+++++++yFiqF`y7$$\"1+ +++++]yFiq$\"1+++]Zv[5F47$$\"1+++++++zFiqF/7$$\"1++++++]zFiq$\"1++++v% fk'F+7$$\"\")F($\"1++++vC#z$F+7$$\"1,+++++]!)Fiq$\"1++++]Z'*QFbu7$$\"1 +++++++\")FiqFd^m7$$\"1++++++]\")Fiq$!1++++DXwfF+7$$\"1*************>) FiqFe^l7$$\"1++++++]#)Fiq$!1+++]_hp5F47$$\"1,++++++$)Fiq$!1+++]_Pz6F47 $$\"1++++++]$)Fiq$!1+++]_K,7F47$$\"1+++++++%)FiqFgjl7$$\"1************ \\%)Fiq$!1++++Dv)[*F+7$$FfpFiq$!1++++D0uqF+7$$\"1,+++++]&)Fiq$!1++++Db 5TF+7$$\"1+++++++')Fiq$!1++++]_x\")Fbu7$$\"1++++++]')FiqFdam7$$\"1**** *********p)Fiq$\"1++++vkQaF+7$$\"1++++++]()FiqF[hl7$$\"1,++++++))Fiq$ \"1++++vu*\\*F+7$$\"1++++++]))Fiq$\"1+++]ZyP5F47$$\"1+++++++*)Fiq$\"1+ ++]Z!o-\"F47$$\"1************\\*)FiqFg]m7$$\"\"*F($\"1++++va/tF+7$$\"1 ,+++++]!*Fiq$\"1++++v%)*)[F+7$$\"1+++++++\"*FiqF__l7$$\"1++++++]\"*Fiq $!1++++Dvc7F+7$$\"1*************>*Fiq$!1++++D0IVF+7$$\"1++++++]#*Fiq$! 1++++DDkpF+7$$\"1,++++++$*FiqFffm7$$\"1++++++]$*FiqF\\[n7$$\"1+++++++% *FiqFgjl7$$\"1************\\%*Fiq$!1+++]_c\"4\"F47$$F^qFiqF[il7$$\"1,+ ++++]&*Fiq$!1,+++DbKxF+7$$\"1+++++++'*Fiq$!1++++D:3_F+7$$\"1++++++]'*F iq$!1++++DNaBF+7$$\"1*************p*Fiq$\"1++++]Z\"4'Fbu7$$\"1++++++]( *Fiq$\"1++++vksNF+7$$\"1,++++++)*FiqFial7$$\"1++++++])*Fiq$\"1,+++v/jz F+7$$\"1+++++++**FiqFg]m7$$\"1************\\**FiqFb[l-%'COLOURG6&%$RGB G$\"#5!\"\"F(F(-%&STYLEG6#%%LINEG-%+AXESLABELSG6$%!GF\\en-%%VIEWG6$%(D EFAULTGF`en" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 " Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "R:=0.913203 - . 6590535250;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG$\"+]Z\\TD!#5" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 150 "The highest peak of the data is \+ 0.913203, subtracting from this the mean, 0.6590535250, we find a valu e for R, the original amplitude. R = .2541494750" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 52 "Enter the compone nts of your approximating function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "t:='t':" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "R:=.2541494750;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " L:=0.1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "theta:=1.82*Pi;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "delta:= Pi/2.8;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 41 "y := t -> R*exp(-L*t)*cos(theta*t-delta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG$\"+]Z\\TD!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG$\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %&thetaG,$%#PiG$\"$#=!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&deltaG ,$%#PiG$\"+r&G9d$!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGR6#%\"tG 6\"6$%)operatorG%&arrowGF(*(%\"RG\"\"\"-%$expG6#,$*&%\"LGF.9$F.!\"\"F. -%$cosG6#,&*&%&thetaGF.F5F.F.%&deltaGF6F.F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 93 "Graph y(t) and the \+ data together. Adjust the constants until you are satisfied with the \+ fit." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "graph4:=plot(y(t), t=0..7,color=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "display(\{graph3,graph4\});" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6%7dw7$\"\"!$\"1++++vC[ 5!#<7$$\"1+++++++]F+$\"1,+++v%4&*)F+7$$\"1+++++++5!#;$\"1+++]Zgv:F47$$ \"1+++++++:F4$\"1+++]ZX-@F47$$\"1+++++++?F4$\"1+++]Zv?CF47$$\"1+++++++ DF4$\"1+++]Z\\TDF47$$\"1+++++++IF4$\"1+++]Zy4CF47$$\"1+++++++NF4$\"1++ +]Z_p?F47$$\"1+++++++SF4$\"1+++]Z`3;F47$$\"1+++++++XF4$\"1++++vuQ**F+7 $$F.F4$\"1++++v%HY$F+7$$\"1+++++++bF4$!1++++Dv+SF+7$$\"1+++++++gF4$!1+ ++]_YN6F47$$\"1+++++++lF4$!1+++]_@1;F47$$ \"1++++++]7Fiq$\"1+++]ZiD?F47$$\"1+++++++8Fiq$\"1+++]Z9XAF47$$\"1+++++ +]8Fiq$\"1+++]Z4nAF47$$\"1+++++++9Fiq$\"1+++]ZU8@F47$$\"1++++++]9Fiq$ \"1+++]ZFiqF_p7$$\"1++ ++++]>Fiq$!1+++]_\\N?F47$$\"\"#F($!1+++]_E%o\"F47$$\"1++++++]?Fiq$!1++ 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{TEXT -1 462 "Look above for our values of R, L, theta, and \+ delta. Our approximation is pretty good. It fits the period and the sa me damping and the same amplitude. One difference is in the first vall ey, where we see our model goes down farther than the data. However, l ooking at the graph of the data, we see that this seems to be a data e rror, since the graph of the data is not very smooth here. The device \+ did not seem to collect enough data here to make the graph smooth." }} {PARA 0 "" 0 "" {TEXT -1 133 "*** You're right about the first valley, but there seems to be a systematic drift in your fit -- could the fre quency be off a little?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 261 49 "Now enter your formula for L in terms of c and m." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 95 "L:='L': K:='K': theta:='theta': delta:='delta': R:= 'R': t:='t': m:='m': c:='c': p:='p': b:='b':" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 76 "Use your value of L to estimate the value of c in the differential equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "y := t -> R*exp(-L*t)*cos(theta*t-delta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"yGR6#%\"tG6\"6$%)operatorG%&arrowGF(*(%\"RG\"\"\"-%$expG6#,$*&%\" LGF.9$F.!\"\"F.-%$cosG6#,&*&%&thetaGF.F5F.F.%&deltaGF6F.F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "diff(y(t),t,t) + p*diff(y(t) ,t) + b*y(t): simplify (%): factor (%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*(%\"RG\"\"\"-%$expG6#,$*&%\"LGF%%\"tGF%!\"\"F%,.*&)F+\"\"#F%-%$ cosG6#,&*&%&thetaGF%F,F%F%%&deltaGF-F%F%*(F+F%-%$sinGF4F%F7F%F1*&F2F%) F7F1F%F-*(%\"pGF%F+F%F2F%F-*(F?F%F:F%F7F%F-*&%\"bGF%F2F%F%F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "r:=L->L^2 - theta^2 - p*L + \+ b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGR6#%\"LG6\"6$%)operatorG%& arrowGF(,**$)9$\"\"#\"\"\"F1*$)%&thetaGF0F1!\"\"*&%\"pGF1F/F1F5%\"bGF1 F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "s:=L->2*L - p*the ta;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sGR6#%\"LG6\"6$%)operatorG% &arrowGF(,&9$\"\"#*&%\"pG\"\"\"%&thetaGF1!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(s(L)=0,L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"pG\"\"\"%&thetaGF&#F&\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "p = c/m;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% \"pG*&%\"cG\"\"\"%\"mG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "L = (c*theta)/(2*m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"LG,$*& *&%\"cG\"\"\"%&thetaGF)F)%\"mG!\"\"#F)\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve(r(L)=0,theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*$-%%sqrtG6#,(*$)%\"LG\"\"#\"\"\"F,*&F*F,%\"pGF,!\"\"%\"bGF,F,,$ F#F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "We know the spring is dam ped... so we know L must be positive for the R*e^(-Lt) term to decay.. ." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "L=s olve (L=c*sqrt(L^2-L*(c/m)+b)/(2*m), L);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%\"LG6$,$*&*&,&*$)%\"cG\"\"#\"\"\"F.*$-%%sqrtG6#,(*$)F,\"\"%F.F .*(%\"bGF.)%\"mGF-F.F+F.!\"%*&F8F.)F:F6F.\"#;F.F.F.F,F.F.,&*&F:F.F+F.F .*$)F:\"\"$F.F;!\"\"#F.F-,$*&*&,&F*F.F/FDF.F,F.F.F?FDFE" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "*** This is two formulas for L. Which do you want?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "b:=K/m; b:=21 .7/.550;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG*&%\"KG\"\"\"%\"mG! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG$\"+XXXXR!\")" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "solve(.5*(c^2+sqrt(c^4-4*39. 45*.55^2*c^2+16*39.45*.55^4))*c/(.55*c^2-4*.55^3)=0.1,c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+_A/^ " 0 "" {MPLTEXT 1 0 85 "solve(.5*(c^2-sqrt(c^4-4*39.45*.55^2*c^2+16*39.45*.55 ^4))*c/(.55*c^2-4*.55^3)=0.1,c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$ \"+TN=^ " 0 "" {MPLTEXT 1 0 10 "c:=.21703;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG$\"&.<#!\"&" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Account for the switch from gram s to kg." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 24 "Part 3: Critical Damping" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 298 "Experiment with t he solutions of the initial value problem as you increase c in the dif ferential equation. In the definition of Eq below, enter your final va lue for c (in place of 300) and plot the solution of the resulting ini tial value problem. Increase c in steps of 500 and record what happens ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with (DEtools):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "y:='y':" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "m:=0.55; K:=21.7;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG$\"#b!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG$\"$<#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "c:=8.009;\nEq:= diff(y(t),t$2)+(c/m)*diff(y(t),t)+(K/ m)*y(t)=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "DEplot(Eq, y(t), t=0. .3, [[y(0)=-0.2,D(y)(0)=0]], y= -0.25..0.25, stepsize=0.05, linecolor= blue);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG$\"%4!)!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#EqG/,(-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F-\" \"#\"\"\"-F(6$F*F-$\"+===c9!\")F*$\"+XXXXRF7\"\"!" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6%7in7$\"\"!$!\"#!\"\"7 $$\"\"&F*$!192,[dv@>!#;7$$\"#5F*$!1+_(euB0v\"F17$$\"#:F*$!12->9QDY:F17 $$\"#?F*$!1\")*Q]J/-M\"F17$$\"#DF*$!1v,Of6gZ6F17$$\"#IF*$!16nb()4n[(*! #<7$$\"#NF*$!1&fTuPdsB)FK7$$\"#SF*$!1mU)eN1^$pFK7$$\"#XF*$!1)oW#eF K7$$\"#]F*$!16ZWoCZ$)[FK7$$\"#bF*$!1\\vN-:x*3%FK7$$\"#gF*$!105BLGMAMFK 7$$\"#lF*$!1%[#4\"3iA'GFK7$$\"#qF*$!1)p]!zF$HR#FK7$$\"#vF*$!1%=MW[N++# FK7$$\"#!)F*$!1Y`%p4X8n\"FK7$$\"#&)F*$!1mzD&=(\\'R\"FK7$$\"#!*F*$!1kGK ,fum6FK7$$\"#&*F*$!1)*)fXQYtu*!#=7$$\"$+\"F*$!1L#[()p()G9)Fgq7$$\"$0\" F*$!1^I)fXMB!oFgq7$$\"$5\"F*$!1&)zhm9O#o&Fgq7$$\"$:\"F*$!1)e@GK@nu%Fgq 7$$\"$?\"F*$!1psT(*G5lRFgq7$$\"$D\"F*$!1nUtsw;7LFgq7$$\"$I\"F*$!1ync&* )Qnw#Fgq7$$\"$N\"F*$!1y+!p'376BFgq7$$\"$S\"F*$!1?-%yOG0$>Fgq7$$\"$X\"F *$!1!e+F\"*3Eh\"Fgq7$$\"$]\"F*$!1n\\S@I/Z8Fgq7$$\"$b\"F*$!1#yJ*R*4_7\" Fgq7$$\"$g\"F*$!1!>@%4>3*R*!#>7$$\"$l\"F*$!16#\\sF*$!1&f\\8.pH>$Fiu7$$\"$&>F*$!1oQn9>9nEFiu7$$ \"$+#F*$!1r9RE%4zA#Fiu7$$\"$0#F*$!1yo^a4,h=Fiu7$$\"$5#F*$!1M^U[V`a:Fiu 7$$\"$:#F*$!1@exu\"H&)H\"Fiu7$$\"$?#F*$!1xL[gOo%3\"Fiu7$$\"$D#F*$!1yNn e$[01*!#?7$$\"$I#F*$!1mPvv6VovF[z7$$\"$N#F*$!1_wVK,/AjF[z7$$\"$S#F*$!1 vjK[$34G&F[z7$$\"$X#F*$!1m\"H=9L7T%F[z7$$\"$]#F*$!1c*oAbyZo$F[z7$$\"$b #F*$!1&oWrfez2$F[z7$$\"$g#F*$!1%HX3]r5d#F[z7$$\"$l#F*$!1gScK+mZ@F[z7$$ \"$q#F*$!1O1[&=xRz\"F[z7$$\"$v#F*$!1M;O\\)R&)\\\"F[z7$$\"$!GF*$!1/7g&) fv^7F[z7$$\"$&GF*$!1Qh7+KhX5F[z7$$\"$!HF*$!1\"3')*Hg=M()!#@7$$\"$&HF*$ !1qM\"o3:eH(F]^l7$$\"$+$F*$!1],X=#>V4'F]^l-%&STYLEG6#%%LINEG-%*THICKNE SSG6#\"\"$-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%%VIEWG6$;$!+++++:!#5$ \"++++]J!\"*;$!++++]FF]`l$\"++++]FF]`l-%+AXESLABELSG6$Q\"t6\"Q%y(t)Fj` l" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" } }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 400 "The first graph with c=0.217 l ooked very much like our graph above, meaning this damping constant is a good approximation. When we increased this by 0.5 to 0.717, the dam ping became more severe and the amplitude dropped much faster than bef ore. As we continue to increase the constant by 0.5, the amplitude dec ays more and more, until at c=1.717, the amplitude is very close to ze ro after just 3 sec." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "c:= sqrt(4*21.7*0.55);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG$\"+oQT4p! \"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 310 " The value of c=sqrt( 4K*m) is the critical damping constant. This means that it is right on the border of being underdamped and ovedamped so that it no longer os cillates. In the phase portrait, this is along a straight line which g oes straight to the origin. The mass will come to rest without oscilla tion." }}{PARA 0 "" 0 "" {TEXT -1 403 " If we start at c=0, and ap proach this value of c, the solution gets increasingly damped, and the solution decays to zero faster and faster. When we reach this critica lly damped value of c, the solution stops oscillating and the mass com es to a rest. If we take c to be greater than this value (6.909 in our case), then the system is overdamped. The mass will again come to res t without oscillation." }}{PARA 0 "" 0 "" {TEXT -1 399 " Our equati on is y'' + (c/m)y' + (K/m)y = 0. The characteristic equation for this is s^2 + (c/m)s + k/m = 0. The roots produce -c/m +/- sqrt((c/m)^2 - \+ 4mk)/2. So if c^2 - 4km < 0 then the roots are complex and we have und erdamping. If this quantity = 0, then it is critically damped (we have repeated roots). If it is greater than zero, then we have two real ro ots and the solution is overdamped." }}{PARA 0 "" 0 "" {TEXT -1 47 "** * good -- but what happens with overdamping?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Part 4: Summary" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 265 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 266 49 "Enter your answers to the summary questions her e." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 173 " \+ The equation y'' + (c/m)y' + (K/m)y = 0, can have a solution of the form y(t) = Re^(-Lt)cos(theta*t - delta), because the derivatives of \+ y(t) are proportional to y(t). " }}{PARA 0 "" 0 "" {TEXT -1 103 "*** N ot with the cosine as a factor. But second-order combinations of sine and cosine can add up to 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 167 "This is because the functions sin and cos can \+ be interchanged by tweaking the delay factor in each function. Sin can be thought of as cos with a delay, and vise versa." }}{PARA 0 "" 0 " " {TEXT -1 400 " Our equation is y'' + (c/m)y' + (K/m)y = 0. The c haracteristic equation for this is s^2 + (c/m)s + k/m = 0. The roots p roduce -c/m +/- sqrt((c/m)^2 - 4mk)/2. So if c^2 - 4km < 0 then the ro ots are complex and we have underdamping. If this quantity = 0, then i t is critically damped (we have repeated roots). If it is greater than zero, then we have two real roots and the solution is overdamped." }} {PARA 0 "" 0 "" {TEXT -1 572 " If we start at c=0, and approach th is value of c, the solution gets increasingly damped, and the solution decays to zero faster and faster. When we reach this critically dampe d value of c, the solution stops oscillating and the mass comes to a r est. This is the border to overdamping and at this critically damped v alue, the spring will merely come back to equilibrium from the distanc e it was stretched to begin with. If we take c to be greater than this value (6.909 in our case), then the system is overdamped. The mass wi ll again come to rest without oscillation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "*** A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 15 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }