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`€EJÞ$‹‹JÞ$‹W߀is€the€time€derivative€of€the€distance€x€or€the€velocity.€€TheÐ ¯$M Ðpotential€energy€V€is€given€byñrñÌñrññsñ€ñsñwhere€k€is€the€force€constant€and€x€theÏdisplacement€from€the€equilibrium€position.€€To€obtain€the€differentialÏequation€of€motion€we€use€the€Lagrange€function€LÌßc€1-` 0  `€E°()x؆dl è') ¡cßâ °±°±°(#°(#â°â °±°±°±°±â°â °±°±°±°±â°â °±°±°±°±â°â °(#°(#°±°±âwhich€is€then€substituted€into€Lagrangeððs€equation€of€motionÐ Ÿ-='" Ðßc€1 -` 0  `€E°cxØÔ dÏ èb ¡cßâ °±°±°(#°(#â°â °±°±°±°±â°â °±°±°±°±â°â °±°±°±°±â°â °(#°(#°±°±âgives€the€differential€equation€€€€ßW€:! `€Eû› ]]û› ]W߀or€€ßW€:"#! `€Eý{  P Pý{  PW߀€€€€€€€€Ð Ú x Ѐ€€€€€€€Ìßc€1-` 0  `€E° x¿ dd÷è  ˆcßâ °±°±°(#°(#â°â °±°±°±°±â°â °±°±°±°±â°â °(#°(#°±°±âEXERCISE€1Ìà  àSubstitute€L€in€equation€(3)€into€Lagrangeððs€equation€(4)€to€obtainÏthe€differential€equation€(5).€€Remember€that€when€you€take€a€partialÏderivative€you€assume€that€all€other€variables€are€constants.ÌÌEXERCISE€2Ìà  àIn€the€computational€document€set€k=3,€m=1,€and€in€the€initialÏvalues€matrix€set€x=.2€and€dx/dt€=€0.€€Run€the€program€(press€F9)€andÏnote€the€graph€on€page€2.€€The€red€line€represents€x€as€a€function€ofÏtime€and€the€green€dash€line€represents€dx/dt.€€When€x=0€what€is€theÏvelocity€dx/dt?€€When€the€dx/dt=0€what€is€the€value€of€x?ÌÌEXERCISE€3Ìà  àNow€change€k€by€a€factor€of€10.€€How€has€the€frequencyÏchanged?€€Return€k€to€its€original€value€and€raise€m€by€a€factor€of€10.€ÏWhat€happens€to€the€frequency?ÌÌEXERCISE€4Ìà  àChange€k€to€300€and€the€mass€m€to€1.€€Now€observe€the€graph.€ÏIs€this€the€expected€behavior?€€Is€there€an€explanation€for€this€graph?ÌÌTHE€DIATOMIC€MOLECULEÌÌà  àThe€same€type€of€differential€equation€applies€to€the€diatomicÏmolecule€with€modest€adjustments€of€the€above€equations.€€WeÏdescribe€the€motions€in€terms€of€an€òòinternalóó€coordinate€q€which€is€theÐ ^,ü%" Ðamount€that€the€distance€between€the€atoms€has€been€distorted€fromÐ |-'# Ðthe€equilibrium€bond€distance.€€For€the€mass€we€must€use€the€reducedÏmass€which€is€calculated€from€the€òòindividualñiññhñ€atom€massesñhññiññjñóóñjññkñ€atom€massesñkññjññgñóóñgññjñ€as€follows:Ð € Ðßc€1 -` 0  `€E°Ÿx<r d“ tèž cßâ °±°±°(#°(#â°â °±°±°±°±â°â °±°±°±°±â°â °±°±°±°±â°â °(#°(#°±°±âwhere€mòò1óó€and€mòò2óó€are€the€masses€in€kg€of€the€òòindividualóó€atoms.€€ThisÐ  ´ Ðquantity€can€be€calculated€using€gram€molecular€weights€Mòò1óó€and€Mòò2€òò€Ð 4Ò Ðóóóóßc€1-` 0  `€E°Sx<v dŒ tèR  cßâ °±°±°(#°(#â°â °±°±°±°±â°â °±°±°±°±â°â °±°±°±°±â°â °(#°(#°±°±âwhere€Nòòavóó€is€Avagadroððs€number.ñlñ€€The€kinetic€energyñlñÐ Êh  Ðñtñßc€1- ` 0  `€E°éxæ<óèè ¯ cßñtññtñßc€1- ` 0  `€E°éxؼ dÿèè ¡ cßâ °±°±°(#°(#â°â °±°±°±°±â°â °±°±°±°±â°â °±°±°±°±â°â °(#°(#°±°±âñtñThe€potential€energy€q€is€written€€€€€€€€€€€Ìñvñßc€1'(- ` 0  `€E°xæ<óè~ ¯ cßñvññvñßc€1/0- ` 0  `€E°xØÆ dìè~ ¡ cßâ °±°±°(#°(#â°â °±°±°±°±â°â °±°±°±°±â°â °±°±°±°±â°â °(#°(#°±°±âñvñThe€differential€equation€of€motion€then€becomesÌñxñßc€1 - ` 0  `€E° xæ<óè  ¯ cßñxññxñßc€145- ` 0  `€E° xØ dWè  ¡ cßâ °±°±°(#°(#â°â °±°±°±°±â°â °±°±°±°±â°â °±°±°±°±â°â °(#°(#°±°±âñxñwhere€ßW€:)*!  `€E-$PP-$P Wß.Ð Œ$* ÐÌA€graph€suitable€for€animation€is€given€at€the€end€of€the€computationalÏdocument.€€Place€the€cursor€just€to€the€right€of€this€diagram€and€thenÏclick€on€ððWindowðð€then€ððanimationðð,€the€ððcreateðð€and€a€dialog€box€shouldÏappear.€€Move€the€cursor€to€the€upper„left€side€of€the€diagram€and€holdÏdown€the€mouse€button€to€mark€off€the€region€you€wish€to€animate.ÌNow€press€the€light€bulb€button€which€establishes€automatic€modeÏcalculation.€€Now€ñmñcñmñlick€on€animate€and€the€computer€will€prepare€theÐ |-'# Ðframes€of€the€atoms€at€different€positions.€€A€small€playback€pictureÏappears€at€the€left.€€Click€on€the€arrow€at€the€left€bottom€and€theÏanimation€will€play€back.ÌÌÌÌÌÌÌÌÌÌÌÌ