历届IPho试题历届IPho试题iphoProblemsof2ndand9thIPhO.docx

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