講義案例講稿

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UNIVERSITYOFCAMBRIDGEINTERNATIONALEXAMINATIONS

GeneralCertificateofEducation

AdvancedSubsidiaryLevelandAdvancedLevel

MATHEMATICS 9709/41

Paper4Mechanics1(M1) October/November2010

1hour15minutes

AdditionalMaterials: AnswerBooklet/Paper

GraphPaper

ListofFormulae(MF9)

READTHESEINSTRUCTIONSFIRST

IfyouhavebeengivenanAnswerBooklet,followtheinstructionsonthefrontcoveroftheBooklet.WriteyourCentrenumber,candidatenumberandnameonalltheworkyouhandin.

Writeindarkblueorblackpen.

Youmayuseasoftpencilforanydiagramsorgraphs.

Donotusestaples,paperclips,highlighters,glueorcorrectionfluid.

Answerallthequestions.

Givenon-exactnumericalanswerscorrectto3significantfigures,or1decimalceinthecaseofanglesindegrees,unlessadifferentlevelofaccuracyisspecifiedinthequestion.

Whereanumericalvaluefortheaccelerationduetogravityisneeded,use10ms?2.Theuseofanelectroniccalculatoripected,whereappropriate.

Youareremindedoftheneedforclearpresentationinyouranswers.

Atoftheexamination,fastenallyourworksecurelytogether.

Thenumberofmarksisgiveninbrackets[]atofeachquestionorpartquestion.Thetotalnumberofmarksforthispaperis50.

Questionscarryingsmallernumbersofmarksareprintedearlierinthepaper,andquestionscarryinglargernumbersofmarkslaterinthepaper.

Thisconsistsof4printedpages.

?UCLES2010 [Turnover

PAGE

2

1

v(ms1)

V

P

O

2

4

Q

t(s)

TwoparticlesPandQmoveverticallyundergravity.Thegraphsshowtheupwardvelocityvms?1oftheparticlesattimets,for0≤t≤4.PstartswithvelocityVms?1andQstartsfromrest.

FindthevalueofV. [2]

GiventhatQreachesthehorizontalgroundwhent=4,find

thespeedwithwhichQreachestheground, [1]

theheightofQabovethegroundwhent=0. [2]

2Acarofmass600kgtravelsalongahorizontalstraightroad,withigineworkingatarateof40kW.Thetomotionofthecarisconstantandequalto800N.ThecarpassesthroughthepointAontheroadwithspeed25ms?1.Thecar’saccelerationatthepointBontheroadishalfitsaccelerationatA.FindthespeedofthecaratB. [5]

P1

P2

X

A

C

B

3

ThediagramshowsthreeparticlesA,BandChanginglyinequilibrium,eachbeingattachedtoofastring.TheotherendsofthethreestringsaretiedtogetherandareatthepointX.ThestringscarryingAandCpassoversmoothfixedhorizontalpegsP1andP2respectively.Theweights

ofA,BandCare5.5N,7.3NandWNrespectively,andtheangleP1XP2isarightangle.Findthe

angleAP1XandthevalueofW. [5]

?UCLES2010 9709/41/O/N/10

AparticlePstartsfromafixedpointOattimet=0,wheretisinseconds,andmoveswithconstantaccelerationinastraightline.TheinitialvelocityofPis1.5ms?1anditsvelocitywhent=10is

3.5ms?1.

FindthediscementofPfromOwhent=10. [2]

AnotherparticleQalsostartsfromOwhent=0andmovesalongthesamestraightlineasP.TheaccelerationofQattimetis0.03tms?2.

GiventhatQhasthesamevelocityasPwhent=10,showthatitalsohasthesamediscementfromOasPwhent=10. [5]

Aparticleofmass0.8kgslidesdownaroughinclinednealongalineofgreatestslopeAB.ThedistanceABis8m.TheparticlestartsatAwithspeed3ms?1andmoveswithconstantacceleration

2.5ms?2.

FindthespeedoftheparticleattheinstantitreachesB. [2]

GiventhattheworkdoneagainstthefrictionalastheparticlemovesfromAtoBis7J,findtheangleofinclinationofthene. [4]

WhentheparticleisatthepointXitsspeedisthesameastheaveragespeedforthemotionfromA

toB.

Findtheworkdonebythefrictionalfortheparticle’smotionfromAtoX. [3]

A

X

B

3m

Hm

hm

6

AsmoothslideABisfixedsothatitshighestpointAis3mabovehorizontalground.Bishmabovetheground.AparticlePofmass0.2kgisreleasedfromrestatapointontheslide.Theparticlemovesdowntheslideand,afterpassingB,continuesmovinguntilithitstheground(seediagram).ThespeedofPatBisvBandthespeedatwhichPhitsthegroundisvG.

InthecasethatPisreleasedatA,itisgiventhatthekineticenergyofPatBis1.6J.Find

thevalueofh, [3]

thekineticenergyoftheparticleimmediaybeforeitreachestheground, [1]

theratiovG:vB. [2]

InthecasethatPisreleasedatthepointXoftheslide,whichisHmabovetheground(seediagram),itisgiventhatvG:vB=2.55.FindthevalueofHcorrectto2significantfigures.[3]

?UCLES2010 9709/41/O/N/10 [Turnover

7

Q

30°

3.2N

P

ParticlesPandQ,ofmasses0.2kgand0.5kgrespectively,areconnectedbyalightinextensiblestring.Thestringpassesoverasmoothpulleyattheedgeofaroughhorizontaltable.PhangslyandQisincontactwiththetable.Aofmagnitude3.2NactsonQ,upwardsandawayfromthepulley,atanangleof30?tothehorizontal(seediagram).

ThesystemisinlimitingequilibriumwithPabouttomoveupwards.FindthecoefficientoffrictionbetweenQandthetable. [6]

Theofmagnitude3.2NisnowremovedandPstartstomovedownwards.

Findtheaccelerationoftheparticlesandthetensioninthestring. [4]

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