Mechanics: Describing Motion and Solving Problems...
Transcript of Mechanics: Describing Motion and Solving Problems...
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Mechanics:DescribingMotionandSolvingProblems(ReferenceChapter#1and#2Knight)
OverallAim:TousewhatyouknowaboutNewtonianMechanicstodescribewhatyouseeinwords,images,diagrams,andmathematicalrepresentations.
DONOW:Witnessthemotioninthedemoand/orvideoandtellmeeverythingyoucanfromthestartofthemotiontotheend
TheParticleModelandMotionDiagrams
LessonAim:Toreviewmotionterminologyandconceptsandapplytomotiondiagrams
TheParticleModelofMotion-Simplificationwherewetreatamovingobjectasifallofitsmasswereconcentratedatasinglepoint.Theparticle(akapoint)hasmassbutnosize,shape,ordistinctionbtwtopandbottomorleftandright.
• Goodforanalyzingtranslationalmotion(motionofanobjectalongatrajectory)butdoesn’tworkforobjectsundergoingarotation-exampleanalyzingthemotionofeachtoothonagear.
TheMotionDiagram
Usedotstorepresenttheparticleasitmovesinitstrajectoryand0,1,2,3,etc.tolocatethepositionoftheparticleatapointintimeorthe“framenumber”wherethetimeintervalbetweenpositionchangesisconstant
Q:Howisthemotionofthecarchangedifthemotiondiagramlookslikethis?
0 1 2 3
0 1 2 3
0 1 2 3
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2
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TimeInterval(ChangeinTime),∆tistf-timeasurestheelapsedtimeasanobjectmovesfromaninitialposition,ritoafinalpositionrf.Thevalueof∆tisindependentofthespecificclockusedtomeasurethetimes.
Specificityoftermsisimportantforthiscourse:Changeintime,position,changeinposition,displacement,velocity,instantaneousvsaveragevelocityoracceleration,etc.
REVIEW:Scalarsarequantitiesthatcanbedescribedwithamagnitude(akanumber)onlywhilevectorsrequiremagnitudeanddirection!
Howisachangeinpositiondescribed?(asadisplacement)∆r(BOLDTEXTDENOTESVECTORQUANITYinthesenotes)
Displacement,∆risr1-r0,orthepositionvectoratlocation1minusthepositionvectoratlocation0.
NOTEONVARIABLES&SUBSCRIPTS:Variablesinphysicsrepresentaphysicalquantitylikedistant,speed,time,etc.,soyouwillusesymbolsotherthanx,y,orzthatyouhaveusedinmathclass.Inthiscasepositionisgiventhesymbol“d”.TheGreeksymboldelta,Δ,means...whichequals_________________-________________always!Ifasubscriptisused,thenumberindicatestheinstantandtellsyourightawaythatthatisanistantaneousvalue.Forexample,positionattime=0is0cmandisdenotedasd1andthepositionattime=3sis12cmanddenotedasd2.
Δd=df,i=df-di
So,ifIwantedtocalculatethechangeinpositionfromposition1toposition2,Iwouldwrite
d2,1=d2-d1=12cm-0cm=12cm
sinceinitialisposition1andfinalisposition2
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Lookatthediagrambelowanddistinguishbetweenthepositionvectorsandthedisplacementvector!
end?
QUESTION:Howdothepositionvectorschangeiftheoriginorreferencepointwasat(50ft,0)?Howaboutthedisplacementvector?
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Question:Howwouldyoufindthedisplacementvectoroftheparticleasitgoesfromitsinitialtofinalposition?
Describingthetimerateofchangeofdistancetraveled(speed)andtimerateofchangeoftheposition(velocity)
SCALER:
AverageSpeed,vavgis!"#$%&'( !"#$%&%'
!"#$ !"#$%&'( !"#$% !"#$%&'()Speedisnotavectorandisnotbasedon
displacementbutontheactualdistancetraveled!So,vavg=!"#$%&'( !"!"#$#%
∆ !
VECTOR:
AverageVelocity,vavg=∆!∆!or𝒅𝒊𝒔𝒑𝒍𝒂𝒄𝒆𝒎𝒆𝒏𝒕
∆!and
InstantaneousVelocity,v=lim∆!→!∆𝒓∆!or!𝒓
!"iswhenyouarelookingatthevelocityatan
instantintime
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MotionDiagramswithVelocityVectorsThelengthofthevelocityvectorrepresentstheaveragespeedoftheobjectasitmovesbetweenthetwopoints.Example:TortoisevstheHareWhoismovingfaster?Howdoyouknow?
Question:CanyoutelliftheHareorTortoiseisacceleratingormovingataconstantvelocity
Describingthetimerateofchangeofthevelocityvector
AverageAcceleration,aavg=∆𝒗∆!or𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒗𝒆𝒍𝒐𝒄𝒊𝒕𝒚
∆!and
InstantaneousAcceleration,a=lim∆!→!∆𝒗∆!or!𝒗
!"iswhenyouarelookingattheaccelerationataninstantin
time
Anobject’saverageaccelerationvectorpointsinthesamedirectionasthedirectionof∆v
Note:Anobject’saveragevelocityvectorpointsinthesamedirectionofthedisplacementvectorandthisisthedirectionofmotion
Whatwouldthemotiondiagramlooklikeforaballrollingdownanincline?(Justsketchit)
Note:Whendrawingacompletemotiondiagramwithvelocityvectors,youaredeterminingtheaverageacceleration!(Justas∆rwillgiveyouVavg)
Contextisimportantthesubscriptavgisoftendroppedwhendescribingaverageaccelerationorvelocity,soitisimportanttounderstandtheproblemyouareworkingwith
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TheCompleteMotionDiagram-FindingtheAccelerationVector-justuseheadtotailmethod
LEARNINGCHECK:Describeaphysicalscenarioforeachofthefollowingtwomotiondiagrams–what’sgoingonandhowdoyouknow?
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Motiondiagramsforthiscoursepurposesarejustatooltohelpyouvisualizethemotion.Iwillnotaskyoutodrawthesetoscaleandwon’taskyoudrawtheseonanexam,butsketchingthemcanbeveryhelpfulandyoushouldbeabletointerpretthem.
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Determiningsignsforposition,velocity,andacceleration(Traditionalconventions+xistotheright,+yisup,and+zisoutofthepage)
Assessment for Understanding: Two balls are on tracks A and B. Ball A is released from rest and rolls down an incline while ball B rolls horizontally at constant speed. Ball B overtakes ball A near the beginning, as the motion diagram shows, but later ball A overtakes ball B. Identify the time or times (if any) at which the two balls have the same speed.
WHY?EXPLAINTHIS!!!
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PositionTimePlots:Allowyoutotrackthepositionofaparticleasafunctionoftime(YouneedtoknowhowtodrawandinterprettheseforHW,Quizzes,Tests,etc)
Examplefora1-Dmotionofastudentwalking
LearningCheck:Whatdoes1-Dmean?
Howdothetwopositiontimeplotsdifferintermsofinformation?
meters meters
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PuttingEverythingTogethertoSolveProblems:
YOUWILLNEEDGOODPHYSICALINSIGHT,THEABILITYTOEXPRESSYOURSELFINWORDSANDEQUATIONS+ABILITYTOSOLVEPROBLEMSMATHEMATICALLY!
Aim:Tobeabletoplotposition,velocity,andaccelerationvstimegraphs
VisualizationoftheSituationExample-Motiontype,Forces,Torques,Constraints,etc.
IFYOUCAN’T“SEE”THEPROBLEM,YOUWON’TBEABLETOSOLVEIT!
*THISTAKESALOTOFEXPERIENCEANDPRACTICETODOWELL
• Identifyknowns&unknowns(couldbepositions,velocities,andaccelerationsasafunctionoftimeorforces,torques,physicaldataonthebody(beyondparticlemodel)
• Representthesituationusingsketches,diagrams,graphs,etc.
Understandhowtoapproachproblemphysically-applyamodel(Ex:Newton’sLaws,ConservationofEnergy,Gauss’sLaw,Coulomb’sLaw,etc))tosetupequations
Mathematicalsolutioninvariablestoarriveatarelationship&/ornumericalvaluewithinagivenuncertainty(sigfigs)
THEPHYSICS
THEMATH
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IDENTIFYINGTHECOMPONENTSOFTHEPROBLEMSOLVINGPROCESSASSHOWNINTHISPICTORALREPRESENTATIONOFTHEFOLLOWINGPROBLEM-GUESSSSYSTEMSTILLWORKS!
THISPROBLEMISANEXAMPLEFORHOWPROBLEMSWILLGETABITMORECOMPLEX:
Asmallrocketislaunchedverticallywithanaccelerationof30m/s2suchthatitrunsoutoffuelafter30seconds.Whatisitsmaximumaltitude?
WORKINPAIRSTOPUTTOGETHERYOURBESTPRESENTATIONOFASOLUTION-DOtheBESTYOUCAN!
GIVENS&UNKNOWNS
SimplifyingAssumption/(s)
PictorialRepresentation
1. TypeofPictorialRepresentation:2. Knowns:3. Unknowns:4. CoordinateSystem:5. SymbolUse:
EQUATION/SOLVE/SUBSTITUTION
IDPROBLEMTYPE(KINEMATICS/NEWTON’SLAWS/CONERVATIONLAWS-ENERGYORMOMENTUM)
MATHEMATICALREPRESENTATION
DOESANSWER/SOLUTIONMAKESENSE?
NOWYOUSOLVEANDMAKESUREYOURANSWERMAKESSENSE&YOUEXPRESSYOURANSWERUSING3SIGNIFICANTFIGURES
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Anotherlookattheballfallingwhendroppedfromrestproblemusingitsmotiondiagram:Sketchtheposition-timeplotandvelocitytimeplotforthetwomotiondiagrambelow.Follow-up:Wheredoesthelargestaccelerationoccur?
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DESCRIBETHEMOTIONDEPICTEDASCOMPLETELYASPOSSIBLEFORTHEPOSITION-TIMEPLOTSHOWNBELOWINCORPORATINGTHEVALUESGIVEN.ASSUMETHATTHEINITIALVELOCITYIS1.0km/mintotheleftatx=10km.WHATWOULDTHEVELOCITY-TIMEPLOTLOOKLIKE?PLOTandLABEL-DONOTSKETCH!
Whycanyouuseaveragevelocitytocalculateallvaluesalongthepositiontimecurve?
Theterm“plot”meansuserealvaluesandtoscaleforAP
VERBALDESCRIPTIONUSINGVALUES,ANDDIRECTIONOFMOTION:
Forthep-tcurve:
Apositivepositionmeans…
Anegativepositionmeans…
Azeroslopemeans…
Apositiveslopemeans…
Anegativeslopemeans…
Thesteepertheslopethe…
FortheV-tcurve:
Apositivevelocitymeans…
Anegativevelocitymeans…
Azeroslopemeans…
Apositiveslopemeans…
Anegativeslopemeans…
Thesteepertheslopethe….
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Learning Check 1: Match the Displacement- time curve with the proper Velocity- time curve
Learning Check 2: Remember what you learned last year Match the Velocity Curve with its Displacement Curve and describe the 1-D motion (stories are fine)
(C4 CHAPTER #2) Learning Check 3: Below are two displacement- time graphs for the motion of objects A and B as they move along the same axis. What can you tell me about the motions depicted in each? Do objects A and B ever have the same speed? If so, at what time or times?
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Aim:Howarethevarious1-Dmotionsdescribedinwords,math,diagrams,andplots?
Possibilitiesfor1-DMotionallmovingNOMO(Nochangeinposition)
ConstantVelocity(V=const;a=0)
ConstantAcceleration(a=const)*IncludesFreeFall
Non-ConstantAcceleration(aincreasesordecreaseswithtimeorposition)
Bestfriendsfromlastyear
NEW
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ConstantVelocityMotion(UniformMotion)-Straight-linemotioninwhichequaldisplacementsoccurduringany
successiveequaltimeintervalsAverageVelocity,vavg=∆𝒓∆!or𝒅𝒊𝒔𝒑𝒍𝒂𝒄𝒆𝒎𝒆𝒏𝒕
∆!ifworkingalongthexaxis,vavg=
∆𝒙∆!orifalong
theyaxis,vavg=∆𝒚∆!.Sincevelocitydoesn’tchangetheaveragevalueistheinstantaneousvalue.
AcceleratedMotion(WhenVelocityisnotConstant)-InstantaneousVelocityandtheDerivativeofPositionwithRespecttoTime
TryingtofindV2forthisverticalacceleration
Thelimitingcase:InstantaneousVelocity,v=lim∆!→!∆𝒚∆!or!𝒚
!"
*Thisyearitwillbeimportanttoconsiderinitialpositionsandconditions!
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Theinstantaneousvelocityataninstantistheslopeatapointoftheposition-timecurve.
Equations:ConstantAccelerationandUniform(ConstantVelocityMotion)in1-D
*THESEEQUATIONSAPPLYONLYFORUNIFORM(AKACONSTANT)ACCELERATION
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PRACTICE-MORECOMPLEX(FROMMITOPENCOURSEWARE)SOLVEONANOTHERSHEETOFPAPERAbusleavesastopatMITandacceleratesataconstantratefor5seconds.Duringthistimethebustraveled25meters.Thenthebustraveledataconstantspeedfor15seconds.Thenthedrivernoticedaredlight18metersaheadandslamsonthebrakes.Assumethebusdeceleratesataconstantrateandcomestoastopsometimelaterjustatthelight.
a) Whatwastheinitialaccelerationofthebus?
b)Whatwasthevelocityatthebusafter5seconds?
c)Whatwasthebrakingaccelerationofthebus?Isitpositiveornegative?
d)Howlongdidthebusbrake?
e)Whatwasthedistancefromthebusstoptothelight?
f)Makeagraphofthepositionvs.timefortheentiretrip.
g)Makeagraphofthevelocityvs.timefortheentiretrip.
h)Makeagraphoftheaccelerationvs.timefortheentiretrip.
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FreeFall-Motionwheretheonlyforceactingonanobjectistheforceduetogravity,FgRememberGalileoandthegravitydiluterakatheinclinedplaneWhenworkingonmotionslidingupordowninclines,youjustplaceyourcoordinateaxessothattheyalignwiththemotion.Thex-axisparalleltotheplaneandtheyaxisperpendicular!Thesigndependsonyou-Isuptheplaneordowntheplaneisconsideredpositive?Youdecide!Justbesticktoyourcoordinatesystemandsignconventions.
When the angle of the incline= 90 degrees, then you have free fall!
Use the angle of incline to slow down the motion so you can measure it!
MotiononanInclinedPlane
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FreeFallPracticeProblem:
Twostonesarereleasedfromrestatacertainheight,oneaftertheother.
a)Willthedifferencebetweentheirspeedsincrease,decrease,orstaythesame?
b)Willtheirseparationdistanceincrease,decrease,orstaythesame?
c)Willthetimeintervalbetweentheinstantsatwhichtheyhitthegroundbesmallerthan,equalto,orlargerthanthetimeintervalbetweentheinstantsoftheirrelease?d)Plotthespeedvs.timeforbothballsinthesameplot.e)Plotthepositionvs.timeofthetwoballsinthesameplot.(DOTHISONGRAPHPAPERANDMARKALLSIGNIFICANTPOINTSOFINTEREST)
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ANOLDQUIZPROBLEM:Trythisone!
Directions:Youmayusereferencetablesandcalculators.Showallworkandenjoy.Youhave15minutes.
Arubberballisthrownupfromthegroundwithspeedvo.Simultaneously,asecondrubberballatheighthdirectlyabovetheballisreleasedfromrest.A. Atwhatheightabovethegrounddotheballscollide?Expressyouranswerintermsoffundamental
constants(inthiscaseg)andthevariablesgiven.
B. Whatisthemaximumvalueofhforwhichacollisionoccursbeforethefirstballfallsbacktotheground?
C. Forwhatvalueofhdoesthecollisionoccurwhenthefirstballisatitshighestheight?
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Thisyear,accelerationsmaynotalwaysbeconstantandyour“oldfriends”equationsonlyapplytoconstantacceleration.So,thisisonereasonyouwillneedcalculus!Now,youcanalsohaveaninstantaneousaccelerationthatvaries.
Inotherwords,youtakethederivativeofthevelocitywithrespecttotimetofindtheinstantaneousacceleration.Tofindinstantaneousvelocity,youtakethederivateofthedisplacementwithrespecttotime.
Instantaneous acceleration as at a specific instant of time t is given by the
derivative of the velocity
Application: Rank in order, from largest to smallest, the accelerations aA– a
C at points A – C.
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TableofCommonDerivatives
Youneedtoknowtheseandbeabletoapplytofindrateofchangeoffunctions
1. 0.)( =constdxd
2. 1)( =xdxd
3. 1)( −= nn nxxdxd
4. [ ] [ ])()( xfdxdcxcf
dxd
=
5. [ ] [ ] [ ])()()()( xgdxdxf
dxdxgxf
dxd
+=+
6. [ ] [ ] [ ])()()()( xgdxdxf
dxdxgxf
dxd
−=−
7. [ ] [ ] [ ])()()()()()( xfdxdxgxg
dxdxfxgxf
dxd
+= (The Product Rule)
8. [ ] [ ]
[ ]2)(
)()()()(
)()(
xg
xgdxdxfxf
dxdxg
xgxf
dxd −
=⎥⎦
⎤⎢⎣
⎡ (The Quotient Rule)
9. xxdxd cos)(sin =
10. xxdxd sin)(cos −=
11. xxdxd 2sec)(tan =
12. x
xdxd 1)(ln =
13. xx eedxd
=