2010年第六届泛珠三角物理竞赛试题(综合试)

PanPearlRiverDeltaPhysicsOlympiad20102010年泛珠三角及中华名校物理竞赛PanPearlRiverDeltaPhysicsOlympiad20102010年泛珠三角及中华名校物理奥林匹克邀请赛Part-1(Total6Problems)卷-1（共6题）(9:00am–12:00pm,02-19-2010)Mathhints数学提示:111nnxdxxn+=∫+,()32222213xAxdxAx−=−−∫Q1Solvethefollowingshortproblems.(11points)题1解下列各题。（11分）(i)Nucleus-AofmassMAandkineticenergyE0(MAc2)collideswithnucleus-B(massMB)atrest.Aftercollisionnucleus-Amovesat90°relativetoitsinitialvelocitywithkineticenergyE1.Findthekineticenergythathasbeenconvertedtonuclearenergyintheprocess.(Bothnucleicanbeviewedasfreeparticles.)(2points)质量为MA的原子核-A，初始动能E0(MAc2)，与初始静止的原子核-B（质量MB）碰撞。碰撞后原子核-A以与初始速度成90°的方向飞出，动能为E1。求在碰撞过程中由动能转变成核能的能量。（原子核可当作是自由粒子。）（2分）(ii)Findtheresistanceandthecapacitancebetweentwoconcentricsphericalconductorshellsofradiiaandb(a).Thespacebetweentheshellsisfilledwithmediumwithdielectricconstantεandconductivityσ.(2points)两同心圆壳导体，半径分别为a、b(a)。两壳间充满介电常数为ε、导电率为σ的介质。求两导体之间的电阻和电容。（2分）abab(iii)(a)Theelectricfieldofanelectromagneticwaveis()00ikztEExeω−=GG.Whatisitsstateofpolarization?(1point)(b)Writedowntheexpressioninthesamewayasin(a)fortheelectricfieldofaleftcircularlypolarizedelectromagneticwave.(1point)(a)一电磁波的电场为(00ikztEExe)ω−=GG。问电磁波的偏振状态是什么？（1分）(b)以与(a)相同的格式给出左旋圆偏振电磁波的电场。（1分）(iv)Theelectricfieldofanelectromagneticwaveinvacuumis()00ikztEExeω−=GG.FindthemagneticfieldandthePoyntingvector.(2points)真空中一电磁波的电场为()00ikztEExeω−=GG。求电磁波的磁场和Poynting矢量。（2分）(v)UsetheLorentztransformationtoderivetherelativisticDopplerEffectwhentherelativemotionbetweenthesourceandthereceiveris(a)parallelor(b)perpendiculartothewavepropagationdirection.(3points)利用洛伦兹变换，推出下列情形的相对论多普勒效应：(a)波的传播方向与波源、接收器的相对速度平行；(b)波的传播方向与波源、接收器的相对速度垂直。（3分）1PanPearlRiverDeltaPhysicsOlympiad20102010年泛珠三角及中华名校物理竞赛Q2(5points)题2（5分）(a)AspacecraftofmassmmovesinacircularorbitatadistancerfromthecenterofEarth(massME).Whatisitskineticenergy?(1point)一质量为m的飞船，在离地球中心距离为r的圆形轨道上运行。地球质量为ME。求飞船的动能。（1分）(b)Supposethespacecraftfiresarocketforashorttime,speedingitupintheforwarddirection.Afterthisshortfiringtime,itskineticenergyincreasesby30%.WhatisthefurthestdistanceofthespacecraftfromthecenterofEarth?Expressyouranswerasamultipleofr.(4points)飞船启动火箭，在短时间内将飞船向前加速，使它的动能增加了30%。问飞船离地球最远为多少r？（4分）Q3(9points)题3（9分）Asshownbelow,apointlightsourceisinsideaglassblockofrefractiveindexnGatdistanceyfromtheopticalaxisanddistanceTfromthesurface.Aplano-convexlensmadeoftransparentplasticmaterialofrefractiveindexnisplacednearthesource.Thethicknessofthelensisd,andtheradiusofitsconvexsurfaceisR.Thewidthoftheairgapbetweentheglassblockandthelensist.如下图所示，一点光源在玻璃内，与光轴的距离为y，与玻璃平面的距离为T。玻璃的折射率为nG。一透明塑料平凸透镜折射率为n，厚度为d，凸面的曲率半径为R，与玻璃的距离为t。(a)Determinethewidthoftheairgaptsothatthelightfromthepointsourceformsaparallelbeamafterpassingthroughthelens.(3points)若点光源的光经过透镜后成为平行光束，求t的表达式。（3分）(b)Findthemaximumthicknessofthelensdmaxsuchthatitisstillpossibletoadjustthewidthoftheairgaptoproduceaparallelbeam.(1point)当d大于dmax时，无论怎样调节t都无法得到平行光束，求dmax的表达式。（1分）(c)Findtheangleθbetweentheparallelbeamandtheopticalaxis.(3points)求平行光束与光轴的夹角θ。（3分）(d)Ifdisallowedtochangebyasmallamountδd,andRisallowedtochangebyδR,findtheangularchangeδθ.(2points)若d改变一小值δd，R改变一小值δR，求夹角的变化δθ。（2分）dytTnGn2PanPearlRiverDeltaPhysicsOlympiad20102010年泛珠三角及中华名校物理竞赛3Q4(5points)题4（5分）TheEarth’smagneticfieldisapproximatelyadipolefield.OnecanimaginethereisamagneticdipolematthecenteroftheEarth.Forsimplicity,assumethatliesontherotationaxisoftheEarth,pointingfromnorthtosouth,asshowninthefigure.HongKongisatthelatitudeofGmG22α=°.ThemagneticfieldatdistancerGfromadipoleis()0534rmBrr⎡⎤=−⎢⎥⎣⎦3mrμπ⋅GGGGG.如图所示，地球的磁场可当作为一个磁偶极子的场，偶极子mG在地球中心，与地球自转轴同轴，由北指向南。香港在北纬22α=°。离磁偶极子rG处的磁场为()05334mrrmBrrμπ⋅⎡⎤=−αHKmGrGαHKmGrGGGGG⎢⎥⎣⎦G。(a)FindthedirectionofthemagneticfieldinHongKongintermsofeast,west,north,south,andtheangletothehorizontalplain.(2points)求在香港的地球磁场方向（以东、南、西、北和水平面为基准）。（2分）(b)Ahorizontalelectricwireof10metersinlengthcarrying100Aofcurrentisinthenorth-southdirection.Findthemagneticforceonthewire.(YouneedtorecallroughlythestrengthofEarthmagneticfieldonEarthsurface.)(3points)一段南北走向的10米长的电缆，载有100A的电流。求地球磁场对电缆的力。（你需自己给出地球磁场的大约强度。）（3分）Q5(8points)题5（8分）2341PVP2P4P1V2V3V12341PVP2P4P1V2V3V1ShowninthefigureistheDieselideal-gascycle.Allprocessesarequasi-staticandthegasismonatomic.Thetwoprocesses1Æ2and3Æ4arebothadiabatic.Findtheefficiencyofthecycleintermsof32VVα=and12rVV=.右图所示的是一个单原子理想气体的Diesel循环。循环中的每个过程都是准稳态的。1Æ2、3Æ4为绝热过程。求循环的效率（用32VVα=、12rVV=来表达）。Q6(12points)题6（12分）AuniformhalfdiskoffinitethicknesswithradiusRandmassMisrestingonaroughhorizontalfloor.(a)Findthemomentofinertiarelativetotheaxisperpendiculartothehalfdiskandthroughthecontactpointonthefloor.(6points)PP(b)AnimpulsePissuddenlyappliedtotheedgeofthehalfdiskasshown.Thehalfdiskrollswithoutslippingafterwardsandmaintainscontactwiththefloor.Findtheminimumimpulsetoflipthehalfdisk.(6points)一个有一定厚度的均匀半圆盘，半径R，质量M，放在粗糙水平地面上。(a)一轴线与盘面垂直，幷穿过半圆盘在平衡时与地面的接触点。求半圆盘相对于该轴线的转动惯量。（6分）(b)一冲量P突然作用于盘的边缘。之后半圆盘作纯滚动，幷一直保持与地面接触。求能使半圆盘翻转的最小冲量。（6分）《THEEND完》PanPearlRiverDeltaPhysicsOlympiad20102010年泛珠三角及中华名校物理竞赛PanPearlRiverDeltaPhysicsOl