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1、HKIPhOMechanicsAssignmentIVSolution1.Twoidenticalsmallbeadswithmassmareattachedonasmoothcircularringthatstandsverticallyontheground.Initiallythebeadsarerestingonthetopofthering.Theythenstarttoslidedown.Derivetherelationshipbetweenthemassoftheringandthem
2、assofthebeadssuchthattheringcanjumpupfromtheground.Also,findthelocationofthebeadswhentheringjumpsup.Sol:LetthemassoftheringbeMandradiusbeR.Attimet,theanglemmadewiththeverticalisq.Sincethereisnofrictionbetweenbeadsandring,thereforetheforceactingonthebead
3、sarealwayspointingalongN,andNewton’sLawforcircularmotionimplies.Conservationofenergyimplies.Puttingbackinthefirstequationweobtain.NoticethatNbecomesnegativewhenislargeenough.InthiscasethereactionforceN’ofthebeadsontheringwilltrytolifttheringup.Thiscanha
4、ppenwhen.SinceN’=-N,wehavetheconditionforringtojumpup:,or.Forthistohaveasolution,wemusthave15or.2.ThisproblemmodelsthelaunchingofadetectorfromthesurfaceoftheearthtoMars.WeassumethattheearthandMarsaremovingaroundthesunincirclesrestingonthesameplane.Thera
5、diusoftheorbitofMarsisRm,whichis1.5timeslargerthanthatoftheearth.Aneconomicalandsimplewaytoloungethedetectorconsistsof2steps.First,arocketisusedtoacceleratethedetectorontheearth’ssurfacesuchthatitacquiresenoughkineticenergytoovercomethegravitationalforc
6、eoftheearthandbecomesasatellitemovingaroundtheearth.Second,atasuitabletime,anenginethatconnectstothedetectorignitesforashortinstantandacceleratesthedetectoralongitsdirectionofmotion.Afterashort(negligible)time,thespeedofthedetectorincreasestoasuitableva
7、luesuchthatthedetectormovesalonganellipticalorbitthatconnectstheearthandMars,withthetwoplanetslocatedattheendpointsoftheellipse(seeDiagramA)(1)InStepI,whatistheminimumspeedneededforthedetectortobecomeanartificialsatellitethatmovesalongtheearth’sorbit?(2
8、)Afterthedetectorbecomesasatellitemovingaroundtheearth,on1March00.00amofacertainyear,theangulardistancebetweenthedetectorandMarsismeasuredtobe60o(diagramB).Whatisthedatethattheengineofthedetectorshouldbefiredsothatthedetectorcanf
