医学物理学2.docx
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医学物理学2.docx
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医学物理学2
ChapterTwo
TheFundamentallawsofMechanics
Keytermsandphrases
Kinematicsisthestudyofthemotionofobjectsandinvolvesthestudyofdistance,speed,accelerationandtime.Inotherword,kinematicsisthestudyofthemotionofbodieswithoutreferencetotheforcesaffectingthemotion.
Displacement(symbolrors)isaspecifieddistanceinaspecifieddirection.Itisthevectorequivalentofthescalardistance.
Velocityistherateofadisplacementofbody.Itisthespeedofabodyinaspecifieddirection.Velocityisthusavectorquantity,whereasspeedisascalarquantity.Forspeed,wehaveaveragespeedandinstantaneousspeed.Theaveragespeedofanobjectisdeterminedbydividingthedistancethattheobjecttravelsbythetimerequiredtotravelthatdistance.Whiletheinstantaneousspeedisthespeedofanobjectataparticularpointintime.
Accelerationistherateofchangeofvelocityintime.Averageaccelerationisthechangeofvelocitydividedbythetimerequiredforthatchange.Instantaneousaccelerationisthechangeofvelocitythatoccursinaverysmallintervaloftime.
Themassofanobjectisameasureoftheinertiaoftheobject.Inertiaisthetendencyofabodyatresttoremainatrest,andofabodyinmotiontocontinuemovingwithunchangedvelocity.Forseveralcenturies,physicistshavefounditusefultothinkofmassasarepresentationoftheamountoforquantityofmatter.
Momentum:
Thelinearmomentumofabodyistheproductofitsmasstimesitsvelocity,thatis
.
Angularmomentum
isdefinedasthevectorproductofitsdisplacementanditsmomentum,i.e.
.Itisalsodefinedastheproductoftheangularvelocity()ofabodyanditsmomentofinertia(I)abouttheaxisofrotation,i.e.L=I.
Torque()isthemeasureoftheeffectivenessofaforceinproducingrotationofanobjectaboutanaxis.Itismeasuredbytheproductoftheforceandtheperpendiculardistancefromtheaxisofrotationtothelinealongwhichtheforceacts.Itcanbesimplywrittenas
.ItsunitismN(meterNewton).
Momentofinertiaorrotationalinertia:
Justasobjectstendtoresistanychangeofintranslationalmotion,anobjecttendstoresistanychangeinitsrotationalmotion.Thetendencytoresistanychangeintranslationalchangeisreferredtoasinertiaandismeasuredbymeasuringanobject’smassinkg.Momentofinertiaorrotationalinertiaisthemeasureofthetendencyofanobjecttoresistanychangeinitsstateofrotation.
Freefalloccurswhenairresistanceonanobjectisnegligibleandtheonlyforceactingonitisgravity.
Elasticityisthepropertybywhichabodyreturnstoitsoriginalsizeandshapewhentheforcesthatdeformeditareremoved.Thispartcontainssomeconceptslikestress,strainandmodulus.
Theelasticlimitofabodyisthesmalleststressthatwillproduceapermanentdistortioninthebody.Whenastressinexcessofthislimitisapplied,thebodywillnotreturnexactlytoitsoriginalstateafterthestressisremoved.
Contentsinthischapter:
•Theconceptsofdisplacement (位移),velocity,acceleration,angularvelocity,angularaccelerationandangularmomentum(角动量)
•Newton’slawsofmotion,rotationallaws;
•Conservational(守恒的)lawsofmomentum,energyandangularmomentum;
•Elasticityofmaterials:
stress,strainandmodulus.
§2.1ThemotionofPointmasses
2.1.1Displacementandequationofmotion
Whenaparticleismoving,itchangesitspositionwithtime.Generally,themotioncouldbedescribedbyafunctionoftime.
Forexample,aparticlemovesalongx-directioninaconstantspeedv,itspositioncanbewrittenas
x=vt+x0x(t)(2.1)
x0canbedeterminedbytheinitialcondition.
Iftheparticlemovesinathreedimensionalspace,itspathattimetcanbedescribedbythefollowingequation:
r(t)=x(t)i+y(t)j+z(t)k(2.2)
rr(t)(2.3)
whererdenotesvector
andsamefori,j,andk.Quiteafewbooksusedsuchboldformlettersasvectorsforsimplicity.Atthetimeoft+t,r(t)becomesr(t+t).Thechangeofitspathduringthetimeintervaltisgivenby
r=r(t+t)-r(t).
r(t)iscalledthedisplacementoftheparticleint
Periodanditalsohasthreecomponentsinx,yandz
directions,givenasx,yandzrespectively.
r(t)iscalledtheequationofmotionfortheparticle.
2.1.2VelocityandAcceleration
Fig.2.1Thedisplacements
1.Instantaneousvelocityandacceleration
Theaveragevelocityoftheparticleduringtwithdisplacementofrisr/t.Itisalsoavector,itsdirectionisthesameastheincrementr.Theinstantaneousvelocityoftheparticleattimetis
(2.4)
Theinstantaneousvelocityisnotnecessarilyaconstant.Itcanbeafunctionoftime.Atthemomentt,
willgivetheexactpositionofthatparticle.Usingequation(2.4),wecouldgettheexactvalueofvelocityatthatparticularmoment.Generallyspeaking,themagnitudeofthevelocityiscalledspeed,denotedby
wherevrepresentsinstantaneousspeed,dsistheabsolutepathlengthdisplacedatthetimeintervaldt.IntheCartesiancoordinatesystem,asi,jandkdonotchangewithtime,sowehave
.(2.5)
Thethreecomponentsontheaxesofx,yandzareexpressedas
(2.6)
Andthemagnitudeofthevelocityis
(2.7)
Example2.1Thepositionofaparticlemovinginx-yplaneisdescribedbythefollowingparametricequationsgivenby
(E.1.1)
where
.Find
(1)thepath(ororbit)functionf(x,y)=0;
(2)velocityatanytime;
(3)positionvectorsatt=0andt=6s,thedisplacement
andpathlengthsduringthistimeinterval.
Solution
(1)rearrangetheequation(E.1.1)as
(E.1.2)
Squareonbothsidesanddeletet,thenwehave
(E.1.3)
Theaboveequationdescribesthepathofthatparticle.Itisaequationofacirclewithradiusranditscentrelocatesat(r,0).i.e.x=randy=0.
(2)
(E.1.4)
andthemagnitudeofthevelocityis
(E.1.5)
Whichmeansthatthemotionisacircularmotionwithconstantspeed.Theanglebetween
andx-directioncanbesimplyfoundas
(E.1.6)
Byinspectionofthesignvxandvyataparticulartime,youcandeterminewhichquadranttheangleisin.
(3)atfirst,wecouldgetthetwopositionvectorsatt=0andt=6srespectively,
t=0
(E.1.7)
t=6s
(E.1.8)
thedisplacementduring
is
Itsshortcutlengthis
Whilethepathlengthduringthetimeintervalshouldbecalculatedbythefollowing:
=angularvelocity×timeinterval×r
Inabovecalculations,theconceptthatthearclengthisequaltotheangleinradianmultipliedbyradiuswasused.
2.Acceleration
Thepathofaparticlemovingintwoorthreedimensionsisacurveingeneral,itsvelocitychangesinbothmagnitudeanddirection.Themagnitudeofthevelocitychangeswhentheparticlespeedsuporslowsdown.Thedirectionofthevelocitychangesbecausethevelocityistangenttothepathandthepathbendscontinuously.Todescribetheaveragevelocityrateofchangeinvelocityduringthetimeintervalt,theaverageaccelerationisdefinedas
.(2.8)
Theinstantaneousaccelerationofaparticleisdefinedinthesamewayasinstantaneousvelocityas
(2.9)
Example2.2Supposethatthemotionofaparticleisdescribedbytheequationofmotionx=20+4t2.Findthespeedandtheaccelerationoftheparticleatt=2s.
Solution:
(a)findspeed
usingthedifferentiationformula
wehave
(1)thespeedatanymomentt
(2)thespeedatt=2seconds
v(t)|t=2s=8t|t=2s=16(m/s)
(b)Acceleration
Example2.3Thecoordinateofaparticlemovinginthethreedimensionalspacegivenasfunctionsoftimeby
where
x(t)=1+2t2(m),y(t)=2t+t3(m)andz(t)=3t+4t2(m).
Findtheparticle’svectorsandmagnitudesofitsposition,velocityandaccelerationatt=2s.
Solution:
(1)positionatt=2s,thisis
Itsmagnitudeis
(2)findthevelocityattimet
.
Att=2s,wehave
themagnitudeofthevelocityis
(3)Acceleration
Att=2s,theaccelerationis
Themagnitudeoftheaccelerationis
Example2.4Apersonstandingonacliffpullsaboatbyapulley,asshowninFig.2.2.Supposethattheheightofthecliffish,therateoftheropepulledisu.Find:
(1)thevelocityoftheboat;
(2)itsacceleration.
Solution:
Astheboatmovesonthesurfaceofwater,itsmotionisinonedimension.Setthex-axispointtotheright,choosetheoriginatthefootpointofthepulley,andletrepresentthevariablelengthoftheropeatanytime.Sotherelationamongh,andxcanbefoundeasilybyPythagoreanTheorem(勾股定理)
andtheequationofmotionfortheboatisFig.2.2diagramforexample2.4
Takethetimederivativeofx,wehave
where
astheboatmovestowardsnegativedirectionofx-axis.Therefore,thespeedoftheboatis
Obviously,thevelocityisexpressedasafunctionofcoordinatex.Theaccelerationoftheboatisthen
Soaslongasweknowwhereitis,i.e.positionx,wecouldsimplycalculatetheacceleration.Theresultindicatesthataccelerationandvelocityareinthesamedirection,sothespeedoftheboatwillbecomelargerandlargerwiththevalueofxdecreased.
§2.2Newton’sLawofMotion
Weknowfromexperiencethatanobjectatrestneverstartstomovebyitself;inordertomoveabody,apushorapullmustbeexertedonitbysomeotherbody.Similarly,aforceisrequiredtoslowdownortostopabodyalreadyinmotion,andt
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