历届IPho试题历届IPho试题iphoProblemsof2ndand9thIPhO.docx
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历届IPho试题历届IPho试题iphoProblemsof2ndand9thIPhO
Problemsofthe2ndand9thInternationalPhysicsOlympiads
(Budapest,Hungary,1968and1976)
PéterVankó
InstituteofPhysics,BudapestUniversityofTechnicalEngineering,Budapest,Hungary
Abstract
Afterashortintroductiontheproblemsofthe2ndandthe9thInternationalPhysicsOlympiad,organizedinBudapest,Hungary,1968and1976,andtheirsolutionsarepresented.
Introduction
FollowingtheinitiativeofDr.WaldemarGorzkowski[1]Ipresenttheproblemsandsolutionsofthe2ndandthe9thInternationalPhysicsOlympiad,organizedbyHungary.IhaveusedProf.RezsőKunfalvi’sproblemcollection[2],itsHungarianversion[3]andinthecaseofthe9thOlympiadtheoriginalHungarianproblemsheetgiventothestudents(myowncopy).Besidesthedigitalizationofthetext,theequationsandthefiguresithasbeenmadeonlysmallcorrectionswhereitwasneeded(typemistakes,smallgrammaticalchanges).Iomittedoldunits,wherebotholdandSIunitsweregiven,andconvertedthemintoSIunits,whereitwasnecessary.
IfwecomparetheproblemsheetsoftheearlyOlympiadswiththelastones,wecanrealizeatoncethedifferenceinlength.Itisnotsoeasytojudgethedifficultyoftheproblems,butthesolutionsaresurelymuchshorter.
Theproblemsofthe2ndOlympiadfollowedthemorethanhundredyearstraditionof**petitionsinHungary.ThetasksofthemostimportantHungariantheoretical**petition(EötvösCompetition),forexample,arealwaysveryshort.Sometimesthesolutionisonlyafewlines,too,buttofindtheideaforthissolutionisratherdifficult.
Ofthe9thOlympiadIhavepersonalmemories;IwastheyoungestmemberoftheHungarianteam.TheproblemsofthisOlympiadwerecollectedandpartlyinventedbyMiklósVermes,alegendaryandfamousHungariansecondaryschoolphysicsteacher.Inthefirstproblemonlythedetailedinvestigationofthestabilitywasunusual,inthesecondproblemonecouldforgettosubtracttheworkoftheatmosphericpressure,butthefully“open”thirdproblemwasreallyunexpectedforus.
Theexperimentalproblemwasdifficultinthesameway:
incontrasttotheOlympiadsoftodaywegotnoinstructionshowtomeasure.(Inthelastyearstheonlysimilarlyopenexperimentalproblemwastheinvestigationof“Themagneticpuck”inLeicester,2000,areallyniceproblembyCyrilIsenberg.)Thechallengewasnottoperformmany-manymeasurementsinashorttime,buttofindoutwhattomeasureandhowtodoit.
Ofcourse,theevaluatingofsuchopenproblemsisverydifficult,especiallyforseveralhundredstudents.Butinthe9thOlympiad,forexample,onlytencountriesparticipatedandthesamepersoncouldread,compare,gradeandmarkallofthesolutions.
2ndIPhO(Budapest,1968)
Theoreticalproblems
Problem1
Onaninclinedplaneof30°ablock,massm2=4kg,isjoinedbyalightcordtoasolidcylinder,massm1=8kg,radiusr=5cm(Fig.1).Findtheaccelerationifthebodiesarereleased.Thecoefficientoffrictionbetweentheblockandtheinclinedplane=0.2.Frictionatthebearingandrollingfrictionarenegligible.
Solution
Ifthecordisstressedthecylinderandtheblockaremovingwiththesameaccelerationa.LetFbethetensioninthecord,Sthefrictionalforcebetweenthecylinderandtheinclinedplane(Fig.2).Theangularaccelerationofthecylinderisa/r.Thenetforcecausingtheaccelerationoftheblock:
andthenetforcecausingtheaccelerationofthecylinder:
.
Theequationofmotionfortherotationofthecylinder:
.
(Iisthemomentofinertiaofthecylinder,Sristhetorqueofthefrictionalforce.)
Solvingthesystemofequationsweget:
(1)
(2)
.(3)
Themomentofinertiaofasolidcylinderis
.Usingthegivennumericalvalues:
.
Discussion(SeeFig.3.)
Theconditionforthesystemtostartmovingisa>0.Insertinga=0into
(1)weobtainthelimitforangle1:
.
Forthecylinderseparately
andfortheblockseparately
.
Ifthecordisnotstretchedthebodiesmoveseparately.WeobtainthelimitbyinsertingF=0into(3):
.
TheconditionforthecylindertoslipisthatthevalueofS(calculatedfrom
(2)takingthesamecoefficientoffriction)exceedsthevalueof
.Thisgivesthesamevaluefor3aswehadfor2.Theaccelerationofthecentersofthecylinderandtheblockisthesame:
thefrictionalforceatthebottomofthecylinderis
theperipheralaccelerationofthecylinderis
.
Problem2
Thereare300cm3tolueneof
temperatureinaglassand110cm3tolueneof
temperatureinanotherglass.(Thesumofthevolumesis410cm3.)Findthefinalvolumeafterthetwoliquidsaremixed.Thecoefficientofvolumeexpansionoftoluene
.Neglectthelossofheat.
Solution
Ifthevolumeattemperaturet1isV1,thenthevolumeattemperature
is
.Inthesamewayifthevolumeatt2temperatureisV2,at
wehave
.Furthermoreifthedensityoftheliquidat
isd,thenthemassesare
and
respectively.Aftermixingtheliquidsthetemperatureis
.
Thevolumesatthistemperatureare
and
.
Thesumofthevolumesaftermixing:
Thesumofthevolumesisconstant.Inourcaseitis410cm3.Theresultisvalidforanynumberofquantitiesoftoluene,asthemixingcanbedonesuccessivelyaddingalwaysonemoreglassofliquidtothemixture.
Problem3
Parallellightraysarefallingontheplanesurfaceofasemi-cylindermadeofglass,atanangleof45,insuchaplanewhichisperpendiculartotheaxisofthesemi-cylinder(Fig. 4).(Indexofrefractionis
.)Wherearetheraysemergingoutofthecylindricalsurface?
Solution
Letususeangletodescribethepositionoftheraysintheglass(Fig.5).Accordingtothelawofrefraction
.Therefractedangleis30foralloftheincomingrays.Wehavetoinvestigatewhathappensifchangesfrom0to180.
Itiseasytoseethatcannotbelessthan60(
).Thecriticalangleisgivenby
;hence
.Inthecaseoftotalinternalreflection
hence
.Ifismorethan75therayscanemergethecylinder.Increasingtheanglewereachthecriticalangleagainif
.Thustheraysareleavingtheglasscylinderif:
CE,arcoftheemergingrays,subtendsacentralangleof90.
Experimentalproblem
Threeclosedboxes(blackboxes)withtwoplugsocketsoneacharepresentforinvestigation.Theparticipantshavetofindout,withoutopeningtheboxes,whatkindofelementsareinthemandmeasuretheircharacteristicproperties.ACandDCmeters(theirinternalresistanceandaccuracyaregiven)andAC(5OHz)andDCsourcesareputattheparticipants’disposal.
Solution
Novoltageisobservedatanyoftheplugsocketsthereforenoneoftheboxescontainsasource.
MeasuringtheresistancesusingfirstACthenDC,oneoftheboxesgivesthesameresult.Conclusion:
theboxcontainsasimpleresistor.Itsresistanceisdeterminedbymeasurement.
OneoftheboxeshasaverygreatresistanceforDCbutconductsACwell.Itcontainsacapacitor,thevaluecanbecomputedas
.
ThethirdboxconductsbothACandDC,itsresistanceforACisgreater.Itcontainsaresistorandaninductorconnectedinseries.Thevaluesoftheresistanceandtheinductancecanbecomputedfromthemeasurements.
9thIPhO(Budapest,1976)
Theoreticalproblems
Problem1
AhollowsphereofradiusR=0.5mrotatesaboutaverticalaxisthroughitscentrewithanangularvelocityof=5s-1.InsidethesphereasmallblockismovingtogetherwiththesphereattheheightofR/2(Fig.6).(g=10m/s2.)
a)Whatshouldbeatleastthecoefficientoffrictiontofulfillthiscondition?
b)Findtheminimalcoefficientoffrictionalsoforthecaseof=8s-1.
c)Investigatetheproblemofstabilityinbothcases,
)forasmallchangeofthepositionoftheblock,
)forasmallchangeoftheangularvelocityofthesphere.
Solution
a)Theblockmovesalongahorizontalcircleofradius
.Thenetforceactingontheblockispointedtothecentreofthiscircle(Fig.7).ThevectorsumofthenormalforceexertedbythewallN,thefrictionalforceSandtheweightmgisequaltotheresultant:
.
Theconnectionsbetweenthehorizontaland**ponents:
.
Thesolutionofthesystemofequations:
.
Theblockdoesnotslipdownif
.
Inthiscasetheremustbeatleastthisfrictiontopreventslipping,i.e.slidingdown.
b)Ifontheotherhand
somefrictionisnecessarytopreventtheblocktoslipupwards.
mustbeequaltotheresultantofforcesS,Nandmg.Conditionfortheminimalcoefficientoffrictionis(Fig.8):
.
c)Wehavetoinvestigateaandbasfunctionsofandinthecasesa)andb)(see Fig.9/aand9/b):
Incasea):
iftheblockslipsupwards,itcomesback;ifitslipsdownitdoesnotreturn.Ifincreases,theblockremainsinequilibrium,ifdecreasesitslipsdownwards.
Incaseb):
iftheblockslipsupwardsitstaysthere;iftheblockslipsdownwardsitreturns.Ifincreasestheblockclimbsupwards-,ifdecreasestheblockremainsinequilibrium.
Problem2
Thewallsofacylinderofbase1dm2,thepistonandtheinnerdividingwallareperfectheatinsulators(Fig.10).Thevalveinthedividingwallopensifthepressureontherightsideisgreaterthanontheleftside.Initiallythereis12gheliumintheleftsideand2 gheliumintherightside.Thelengthsofbothsidesare11.2dmeachandthetemperatureis
.Outsidewehaveapressureof100 kPa.Thespecificheatatconstantvolumeiscv = 3.15 J/gK,atconstan