WPCC jx{k,Mn0V8?iV ZƘ#ǂ yP)MgRPZZD( S{?ߢŔB.\3^w 凉 9f|[d%J% d>@ZfQsZEyռ͋Ywa۲Ⲿn-/M)8zz$6_۰?P3!YRS7huq00cAII-fHAɝ._"5@;v7*qcV o+TbAl_Mh7> 3ܜ 05 tBɋyh=)9m\x1P\Qwk#A^N*]q"J'jiT%/]y^Pe^&&]ىY#%x"FY.+UݐZ&:ucwGq^48+p\Z]Y&EWl3E# UN % 0(%U:MU:ddsd.d2w@44'd[3dE5d1dSqd:F dD!K"d%[""d0#$ 0D%d&&)( D+(d()d*B+d+X,d+6-pa.d?.X0d+h01 AY)22 0FH3dz36d65L9d9H:: AS:d:;d ^<g?dz@+B 0DBBd5CdDCPanasonic KX-P2123PANSON24,,,,0nLh(  Z 6Times New Roman RegularX($,@AZ"Arial Regular,@AZ"Arial Regular left(MATRIX{Qdotsub1#Qdotsub2#0}RIGHT)~=~left(MATRIX{1&1&0#0&1&1#m_1&m_2&m_3}right)left(MATRIX{xdotsub1#xdotsub2#xdotsub3}right)XX_hXXjXXjXXjXXGjXXjXXjXXujXX/jXXjXXiXX\_oXX\qXX\qXX\qXX\GqXX\qXX\qXX\uqXX\/qXX\qXX\pXXXXfQ&1XXXXQ~2XXE0XX)XXhXXjXXjXXIjXXjXXjXXwjXX1jXX_iXXoXXqXXqXXIqXXqXXqXXwqXX1qXX_pXXhMXXM1XX M1XXqM0XX0XXXXH1XXq1XXkm1XXmf2XX8m3XXuhXX/jXXjXXjXX]jXXjXXjXXjXXEjXXjXX-iXXuoXX/qXXqXXqXX]qXXqXXqXXqXXEqXXqXX-pXXXXxI?1XXXXxI2XXXXxIM3 T~=~1over2(msub1xdotsub1sup2~+~̀msub2xdotsub2sup2~+~̀msub1xdotsub3sup2)̀XXTXXXXXV1XXF2XXT(XXm*1XXXXmxS21XXXXOm2XXDXX%xS2|2XX7XXm1XXXXxT S24 3XX ) Zleft(MATRIX{xdotsub1#xdotsub2#xdotsub3}right)~=~1overM~left(MATRIX{msub1msub2&msub1&1#̀msub1&msub1&1#̀msub1&msub1+msub2&1}right)~left(MATRIX{Qdotsub1#Qdotsub2#0}right)XXuhXX/jXXjXXjXX]jXXjXXjXXjXXEjXXjXX-iXX&uoXX&/qXX&qXX&qXX&]qXX&qXX&qXX&qXX&EqXX&qXX&-pXXXXx?1XXXXx2XXXXxM3XX3XXX\1XXLMXXuhXX/jXXjXXjXX]jXXjXXjXXjXXEjXXjXX-iXX uoXX /qXX qXX qXX ]qXX qXX qXX qXX EqXX qXX -pXXkXXmv?1XXXX1m?2XXXXvm ?1XXC 1XXNm1XXXXvm 1XXC 1XXNmM1XXm&M1XXiXXmt M2XXC 1XXt _hXXt jXXt jXXt jXXt GjXXt jXXt jXXt ujXXt /jXXt jXXt iXX _oXX qXX qXX qXX GqXX qXX qXX uqXX /qXX qXX pXX+ XX fQv &1XX+ XX Qv ~2XX E0 12345E6U7D8<>; 3|x> *Qsub1~=~xsub2xsub1rsubeXXxQ81XXExXXxxl82XXxXX'xx~81XXxXX9xr8e )Qsub2~=~xsub3xsub2rsubeXXxQ82XXExXXxxl83XXxXX'xx~82XXxXX9xr8e +Qdotsub2~=~xdotsub3xdotsub2XX_XXxQ82XXExXX4xXXxxl83XXxXXFxXX'xx~82 T~=~1over2~(msub1xdotsub1sup2~+~̀msub2xdotsub2sup2~+~̀msub1xdotsub3sup2)̀XXTXXXXXV1XXF2XX(XXmk1XXXXx%S21XXXXm#2XXXXfxS22XXxXXHm1XX= XX x S2u 3XX ) Imsub1xdotsub1+~msub2xdotsub2+~msub3xdotsub3=~0XXxm81XX xXXxxD81XXxXXWxm82XXLxXX-xx82XXxXXxm*83XXxXXmxx83XXxXXx0 <V~=~1over2(k~Qsub1sup2+~k~Qsub2sup2)XXVXXXXXV1XXF2XX[(XXkXXMQS21XXCXXkXXQuS2U2XX) xdotXX6FXXFx T~=~1over2~(msub1xdotsub1sup2+~̀msub2xdotsub2sup2+~̀msub3xdotsub3sup2)XXTXXXXXV1XXF2XX(XXm1XXXXx<S21XXXXOm2XXDXX%xS2|2XXXXmB3XXXXxS23XX? ) T~=1over2~{msub1(msub1+msub2)}overM~̀(Qdotsub1sup2+~Qdotsub2sup2)̀+{msub1sup2}overM~̀Qdotsub1Qdotsub2(3$ !  XX*TXXXxXXV1XXF2XnXXmH1XXZ(XXm0H1XXsXXm~H2XX)XXFMXXs(XXXXQiW2I1XXXXXX|Q/ W2 2XXr )XX X: XXP m 2 R1XXx FMXX XX QH 1XX XX Q 2 M~=~2msub1+~msub2'dxdXXxMXXxXXx2XXKxm71XXxXXxm72 T~=~Bsub{11}(Qdotsub1sup2+̀Qdotsub2sup2)̀~+~Bsub{12}Qdotsub1Qdotsub2XXxTXXxXXxB(811XXx(XX9XXxQ281XXxXXXX_xQ282XXUx)XXxXXxB8812XXXXxQQ81XXXXxQ' 82 8V~=~1over2~k~(Qsub1sup2+Qsub2sup2)XXVXXXXXV1XXF2XXkXXb(XXQXS281XXXXQS22XX ) N2~Bsub{11}Qddotsub1+~Bsub{12}Qddotsub2+~kQsub1~=~0XXx2XXxBK811XXXXxQd81XXxXXwxB812XXXXuxQ82XXKxXXxkXXrxQ81XXxXXpx0 fdover{dt}{partialL}over{partialQdotsubi}~~{partialL}over̀{partialQsubi}̀~=~0X7wXXhdXXMdtX wXXH,XXLXX x,XXXX}xQ8iXXIXwXX,XX$LXX,XXQgiXX+IXXI0 NBsub{12}Qddotsub1+~2~Bsub{11}Qddotsub2+~kQsub2~=~0XXxB812XX]XXxQ81XXxXXx2XXwxB811XXXXuxQ82XXKxXXxkXXrxQ82XXxXXpx0 Qddot_1={{2Bsub{11}k}overBsub{12}Qsub1kQsub2}overleft[Bsub{12}{4Bsub{11}sup2}overBsub{12}right]2!3$M << deUULevel 1Level 2Level 3Level 4Level 5(n$ (  1  ) XXeXXQc1XXXx XXX2XXBE11XXkXXCB12XXQ1XXWXXkXX&Q2XXvXXuxXX/xXXxXXwXXK}XXKuXXK/XXKXXK~XX9B12XX7&XHXX4XX6B%211XXxB|812 Qddotsub2={{kBsub{12}}over{2Bsub{11}}Qsub1kQsub2}  over{left[2Bsub{11}{Bsub{12}sup2}over{2Bsub{11}}right]}XXeXXQc2XXXx&XXXkXXBE12XXC2XXCB11XX*Q1XXXXxkXXQb2XXvXXuxXX/xXXxXXwXXR}XXRuXXR/XXRXXR~XX2XX@B11XX>&XHXX B%212XXx2XX=xB811 +Qdotsub1~=~xdotsub2xdotsub1XX_XXxQ81XXExXX4xXXxxl82XXxXXFxXX'xx~81 >V~=~1over2~(k~Qsub1sup2+~k~Qsub2sup2)!M -, dcXXVXXXXXV1XXF2XX(XXkXXQAS2!1XXXXTkXXQS22XX) xQ~=~left(MATRIX{Qsub{1}#Qdotsub{1}#̀Qsub{2}#Qdotsub{2}}right)XXQXXXXhXXjXXwjXX1jXXjXXjXX_jXXjXXjXXjXXGjXXjXXjXXujXX/jXXjXXiXXoXXqXXwqXX1qXXqXXqXX_qXXqXXqXXqXXGqXXqXXqXXuqXX/qXXqXXpXXAAQ1XXXXAQY1XXA Q2XXXXAxQ82 Qddot_1={{2Bsub{11}k}overBsub{12}Qsub1kQsub2}over  left[Bsub{12}{4Bsub{11}sup2}overBsub{12}right]XXNXXQc1XXXa XXX2XXBvE11XXkXXCBp12XXjQ1XX@XXkXXQ2XXvXXuxXX/xXXxXXwXX4}XX4uXX4/XX4XX4~XX"B12XX &XHXX4XXB%211XXxBe812 !M~=~2msub1+~msub2(3$ !   Qdotsub2XX_XXxQ82XXxMXXxXXx2XXJxm81XX xXXxm82 !  XXXX XX7.XXdd7  THEVIBRATIONSOFMOLECULESII b   THECARBONDIOXIDEMOLECULE#XX :#EXXStudentInstructions   TheseinstructionsaccompanytheMATHCADdiskfileco2run.mcd.#]KE##X.X]K#XXXX.EXXThestudentshouldfirstcompletetheexerciseforthesimpleharmonic W oscillatorinhorun.mcdwithitsinstructions.  ThisprogramillustratesthevibrationsofCO2.Wewillassumethat )  theforceconstantforeachCObondisk.Themoleculeisarrangedalongthexaxiswithcoordinatesofthethreeatomsx1,x2,andx3.The e kineticenergyisthesumofthekineticenergyoftheatoms:8c1 -` 0  `Ex< c88c1-` 0  `Ex dC  c (#(#    (#(#8TheextentQ1thattheleftbondisstretchedorcompressedcompared  toitsequilibriumbondlengthofreis 7  c1-` 0  `EV!x d2U! c (#(#   (#(#SimilarlyforQ2    h W:! `E$22$2W. $M TakingthetimederivativeswegetW:78! `Ev&aav&aW߀andW:! `E&aa&aW߀. &  Ifweassumethatthecenterofmassdoesnotmoveweget̀ -='" c1-` 0  `Ecx dR2b c (#(#   (#(#EEWecansetupthepotentialenergyintermsofthestretchingofeach  Z bond.<e1/` p m `Exex< ve<<e19;/` p m `Exex dd he<#EE # (#(#     (#(#UnfortunatelywehavedescribedthekineticenergyTintermsofxcoordinates,andthepotentialenergyVintermsofQcoordinates.InordertoobtaintheequationsofmotionwemustdescribeTandVinthesamecoordinates.WechoosetousetheQcoordinates,sowemustobtainbothTandVintermsofQs.TheQsintermsofxsarec1-` 0  `Ex= d u~ c (#(#      (#(#ToobtainthexcoordinatesintermsoftheQsweinvertthematrix:c1 -` 0  `EQ"x=d? uP" c (#(#      (#(#WhereW:@D!  `E3)223)2 W1.1 )" ThekineticenergycanbecalculatedsubstitutingtheW:!  `EQ*Q* Wsaboveinto "*#  c1- ` 0  `EA+xr d @+  c (#(#   |-'# andafterconsiderablealgebraweget5566c1 "- ` 0  `ExFde ~  c5wherethetotalmassofCO5 (#(#    (#(#5255is53W:#%!  `EJ XJ  W343W:#%!  `E 22 2 W345.577CollectingtheconstantsinB11andB12wecanrewritethekinetic   energyc1&'- ` 0  `E5x:{ d r4  c (#(#   (#(#e1()/` p  `Exx d]  hewhichwiththepotentialenergy (#(#     (#(#canbesubstitutedintoLagrangesdifferentialequationfirstforQ1then B forQ2 ` c1,--` 0  `Ex9 dcq~ c (#(#    (#(#togivetheresultingequationsofmotion.c1*+-` 0  `E x) da  c (#(#      (#(#c1./-` 0 [ `Eo#x) da" c߰UnfortunatelytheseequationscannotbesolvedbyMATHCADinthisform.Wemustsolvethemforeachofthesecondderivatives: "*#  >c104-` 0  `Ecx<b c>>c1>?-` 0  `Ecx* d$?b c (#(#       (#(#>c156-` 0  `E;xU d?; c (#(#       (#(#ThevaluesofKandthemassesoftheatomsarenowincludedinthecomputationaldocumentwherethedifferentialequationsaresolved.EXERCISE1  Theinputtothedifferentialequationsroutinehassomeadjustableconstants.Wemustsetvaluesofthemassesoftheatomsm1andm2 : whichcanbetakenas16and12forOandC.Thevalueoftheforceconstantshouldbesetto200.ThecolumnmatrixQgivestheinitialvaluesoftheQsandtheirinitialvelocities:e1<=/` p rk `E" xj dZ  e b!(#b!(#(#(# (#(#b!(#b!(#  Seteachinitialvelocityequalto3andeachinitialpositionequaltozero.Calculateallfourpagesofthedocument.Whichvibrationdoesthissimulate?Fortheanimationpartplacethecursor(redcross)justtotherightofthetopoftheanimationplotandclickonGraphicsandAnimationandcreate.Thenwiththemouseblockouttheportionof |-'# thepictureyouwishtoanimate.Thenpressthelightbulbbuttontoactivatetheautomaticmode.Theautomationwillnowberecorded.Apicturewillappearattheleftfortheplaybackfeature.Clickingonthelowerleftbuttonwillactivatetheplayback.  NotetheFouriertransformofthemotionofQ1onthelastpage.   Thisisaplotinfrequencyspaceofthevibrationalfrequenciescontainedinthismotion.EXERCISE2  GototheQmatrixandchangeoneofthederivativesfrom3to3.Repeatthecalculations.Hasthefrequencybecomelargerorsmaller?Whichvibrationdoesthisrepresent?EXERCISE3  GoagaintotheQmatrixandchangetheinitialPOSITIONQ1to2 $ andchangetheinitialVELICITYEEW:BC! `EBaaBaW#EE$#Ԁto3.Repeatthecalculation.Note B thatthemotionbecomesamixtureofthetwomodesandappearstobehaphazard.ButtheFouriertransformcandisplaytwofrequencies(eventhoughthephasingofthemaximaisnotgreat).