UNIT 4

Circular Motion & Gravity

APUnit4:CircularMotion/Gravity-ExamplesandTryThisProblemsEx.1The rotor on a helicopter turns at an angular speed of 3.20 x 102 revolutions per minute. (We sometimes use the abbreviation rpm, but in most cases we use rev/min.) (a) Express this angular speed in radians per second. (b) If the rotor has a radius of 2.00 m, what arc length does the tip of the blade trace out in 3.00 x 102 s? (Ans. a) 33.5 rad/s�b) 2.02 x 104 m )

TryThis1:A waterwheel turns at 1,500 revolutions per hour. Express this rate of rotation in units of radians per second. (Ans. 2.6 rad/s )

Ex.2A wheel rotates with a constant angular acceleration of 3.50 rad/s2. If the angular speed of the wheel is 2.00 rad/s at t= 0, (a) through what angle does the wheel rotate between t= 0 and t= 2.00 s? Give your answer in radians and in revolutions. (b) What is the angular speed of the wheel at t = 2.00 s? (Ans. a) 11.0 rad , 1.75 rev b) 9.00 rad/s )

TryThis2:(a) Find the angle through which the wheel rotates between t= 2.00 s and t= 3.00 s. (b) Find the angular speed when t= 3.00 s. (Ans. (a) 10.8 rad (b) 12.5 rad/s )

Ex.3An airplane propeller slows from an initial angular speed of 12.5 rev/s to a final angular speed of 5.00 rev/s. During this process, the propeller rotates through 21.0 revolutions. Find the angular acceleration of the propeller in radians per second squared, assuming it’s constant. (Ans. 6.25pi rad/s2 )

TryThis3:Suppose, instead, the engine speeds up so that the propeller goes through 28.0 revolutions while the angular speed increases uniformly from 5.00 rev/s to 15.0 rev/s. Find the angular acceleration. (Ans. 7.14pi rad/s2 )

Ex.4A compact disc rotates from rest up to an angular speed of 31.4 rad/s in a time of 0.892 s. (a) What is the angular acceleration of the disc, assuming the angular acceleration is uniform? (b) Through what angle does the disc turn while coming up to speed? (c) If the radius of the disc is 4.45 cm, find the tangential speed of a microbe riding on the rim of the disc when t= 0.892 s. (d) What is the magnitude of the tangential acceleration of the microbe at the given time? (Ans. a) 35.2rad/s2 b) 14.0rad c) 1.40 m/s d) 1.57 m/s2 )

TryThis4:(a) What are the angular speed and angular displacement of the disc 0.300 s after it begins to rotate? (b) Find the tangential speed at the rim at this time. (Ans. a) 10.6 rad/s; 1.58 rad b) 0.472 m/s)

Ex.5A race car accelerates uniformly from a speed of 40.0 m/s to a speed of 60.0 m/s in 5.00 s while traveling counterclockwise around a circular track of radius 4.00 x 102 m. When the car reaches a speed of 50.0 m/s, find (a) the magnitude of the car’s centripetal acceleration, (b) the angular speed, (c) the magnitude of the tangential acceleration, and (d) the magnitude of the total acceleration. (Ans. a) 6.25 m/s2 b) 0.125 rad/s c) 4.00 m/s2 d) 7.42 m/s2 )

TryThis5:Suppose the race car now slows down uniformly from 60.0 m/s to 30.0 m/s in 4.50 s to avoid an accident, while still traversing a circular path 4.00 x 102 m in radius. Find the car’s (a) centripetal acceleration, (b) angular speed, (c) tangential acceleration, and (d) total acceleration when the speed is 40.0 m/s. (Ans. a) 4.00 m/s2 b) 0.100 rad/s c) 6.67 m/s2 d) 7.77 m/s2 )

Ex.6A car travels at a constant speed of 13.4 m/s on a level circular turn of radius 50.0 m. What mini- mum coefficient of static friction, ms, between the tires and roadway will allow the car to make the circular turn without sliding? (Ans. 0.366)

TryThis6:At what maximum speed can a car negotiate a turn on a wet road with coefficient of static friction 0.230 without sliding out of control? The radius of the turn is 25.0 m. (Ans. 7.51 m/s )

Ex.7The Daytona International Speedway in Daytona Beach, Florida, is famous for its races, especially the Daytona 500, held every spring. Both of its courses feature four-story, 31.0 o banked curves, with maximum radius of 316 m. If a car negotiates the curve too slowly, it tends to slip down the incline of the turn, whereas if it’s going too fast, it may begin to slide up the incline. (a) Find the necessary centripetal acceleration on this banked curve so the car won’t slip down or slide up the incline. (Neglect friction.) (b) Calculate the speed of the race car. (Ans. a) 5.89 m/s2 b) 43.1 m/s )

TryThis7:A racetrack is to have a banked curve with radius of 245 m. What should be the angle of the bank if the normal force alone is to allow safe travel around the curve at 58.0 m/s? (Ans. 54.5o)

Ex.8(a) Three 0.300-kg billiard balls are placed on a table at the corners of a right triangle, as shown. Find the net gravitational force on the cue ball (designated as m1) resulting from the forces exerted by the other two balls. (b) Find the components of the gravitational force of m2 on m3. (Ans. a) 7.6 x1011N at 29.3o b) x-component 1.44 x 10 11 N , y-component 1.92 x 10 11 N )

TryThis8:Find magnitude and direction of the force exerted by m1 and m3 on m2. (Ans. 5.85 10 11

N, -75.8o )

Ex.9An astronaut standing on the surface of Ceres, the largest asteroid, drops a rock from a height of 10.0 m. It takes 8.06 s to hit the ground. (a) Calculate the acceleration of gravity on Ceres. (b) Find the mass of Ceres, given that the radius of Ceres is RC = 5.10 x 102 km. (c) Calculate the gravitational acceleration 50.0 km from the surface of Ceres. (Ans. a) 0.308 m/s2 b) 1.20 x 1021 kg c) 0.255 m/s2 )

TryThis9:An object takes 2.40 s to fall 5.00 m on a certain planet. (a) Find the acceleration due to gravity on the planet. (b) Find the planet’s mass if its radius is 5,250 km. (Ans. a) 1.74 m/s2 (b) 7.19 x 1023 kg )

Unit4Problems:CircularMotionandGravity

Video4.1-Radians,AngularDisplacement/Velocity/Acceleration1.Awheelhasaradiusof4.1m.Howfar(pathlength)doesapointonthecircumferencetravelifthewheelisrotatedthroughanglesof30o,30rad,and30rev,respectively?(Ans.2.1m,120m,770m)2.Apotter’swheelmovesuniformlyfromresttoanangularspeedof1.00rev/sin30.0s.(a)Finditsangularaccelerationinradianspersecondpersecond.(b)Woulddoublingtheangularaccelerationduringthegivenperiodhavedoubledfinalangularspeed?(Ans.a).209rad/s2b)Yes.)3.Adentist’sdrillstartsfromrest.After3.20sofconstantangularacceleration,itturnsatarateof2.51x104rev/min.(a)Findthedrill’sangularacceleration.(b)Determinetheangle(inradians)throughwhichthedrillrotatesduringthisperiod.(Ans.a)821rad/s2b)4,210rad)4.Acentrifugeinamedicallaboratoryrotatesatanangularspeedof3,600rev/min.Whenswitchedoff,itrotatesthrough50.0revolutionsbeforecomingtorest.Findtheconstantangularaccelerationofthecentrifuge.(Ans.-226rad/s2)5.Acarinitiallytravelingat29.0m/sundergoesaconstantnegativeaccelerationofmagnitude1.75m/s2afteritsbrakesareapplied.(a)Howmanyrevolutionsdoeseachtiremakebeforethecarcomestoastop,assumingthecardoesnotskidandthetireshaveradiiof.33m?(b)Whatistheangularspeedofthewheelswhenthecarhastraveledhalfthetotaldistance?(Ans.a)116revb)62.1rad/sec)

Video4.2-CentripetalAcceleration/Force,VerticalCircles

6.A55.0-kgiceskaterismovingat4.00m/swhenshegrabsthelooseendofarope,theoppositeendofwhichistiedtoapole.Shethenmovesinacircleofradius.8maroundthepole.Determinetheforceexertedbythehorizontalropeonherarms.(Ans.1100N)7.Acertainlighttruckcangoaroundaflatcurvehavingaradiusof150mwithamaximumspeedof32.0m/s.Withwhatmaximumspeedcanitgoaroundacurvehavingaradiusof75.0m?(Ans.22.6m/s)8.A50.0-kgchildstandsattherimofamerry-go-roundofradius2.00m,rotatingwithanangularspeedof3.00rad/s.(a)Whatisthechild’scentripetalacceleration?(b)Whatistheminimumforcebetweenherfeetandthefloorofthecarouselthatisrequiredtokeepherinthecircularpath?(c)Whatminimumcoefficientofstaticfrictionisrequired?(Ans.a)18m/s2b)900Nc)1.84)9.Whatisthetangentialaccelerationofabugontherimofa10-in.-diameterdiskifthediskmovesfromresttoanangularspeedof78rev/minin3.0s?Whenthediskisatitsfinalspeed,whatisthetangentialvelocityofthebug?(Ans..35m/s2,1m/s)10.Anairpuckofmass0.25kgistiedtoastringandallowedtorevolveinacircleofradius1.0monafrictionlesshorizontaltable.Theotherendofthestringpassesthroughaholeinthecenterofthetable,andamassof1.0kgistiedtoit.Thesuspendedmassremainsinequilibriumwhilethepuckonthetabletoprevolves.(a)Whatisthetensioninthestring?(b)Whatisthehorizontalforceactingonthepuck?(c)Whatisthespeedofthepuck?(Ans.a)9.8Nb)9.8Nc)6.3m/s)

11.Aroller-coastervehiclehasamassof500kgwhenfullyloadedwithpassengers.(a)Ifthevehiclehasaspeedof20.0m/satpointA,whatistheforceofthetrackonthevehicleatthispoint?(b)Whatisthemaxi-mumspeedthevehiclecanhaveatpointBforgravitytoholditonthetrack?(Ans.a)25,000Nb)12m/s)

Video4.3-LawofUniversalGravitation

12.Acoordinatesystem(inmeters)isconstructedonthesurfaceofapooltable,andthreeobjectsareplacedonthetableasfollows:a2-kgobjectattheoriginofthecoordinatesystem,a3.0-kgobjectat(0,2),anda4-kgobjectat(4,0).Findtheresultantgravitationalforceexertedbytheothertwoobjectsontheobjectattheorigin.(Ans.1.1x10-10Nat72oNofE)13.TheaveragedistanceseparatingEarthandtheMoonis384,000km.FindthenetgravitationalforceexertedbytheEarthandtheMoonona3x104kgspaceshiplocatedhalfwaybetweenthem.(MassofEarth:5.97x1024kg;MassofMoon:7.35x1022kg)(Ans.321Ndirectedtowardtheearth)14.Twoobjectsattracteachotherwithagravitationalforceofmagnitude1x108Nwhenseparatedby20cm.Ifthetotalmassoftheobjectsis5.00kg,whatisthemassofeachobject?(Ans.2kgand3kg)15.Objectswithmassesof200kgand500kgareseparatedby.4m.Atwhatpositionbetweenthemcana50.0-kgobjectbeplacedsoastoexperienceanetforceofzero?(Ans..245mfromthe500kgmass.)

ConceptualQuestions:CircularMotion

1.Correctthefollowingstatement:“Theracecarroundstheturnataconstantvelocityof90milesperhour.”2.Isitpossibleforacartomoveinacircularpathinsuchawaythatithasatangentialaccelerationbutnocentripetalacceleration?3.Apailofwatercanbewhirledinaverticalcircularpathsuchthatnowaterisspilled.Whydoesthewaterremaininthepail,evenwhenthepailisupsidedownaboveyourhead?4.Describethepathofamovingobjectintheeventthattheobject’saccelerationisconstantinmagnitudeatalltimesand(a)perpendiculartoitsvelocity;(b)paralleltoitsvelocity.5.Ithasbeensuggestedthatrotatingcylindersabout10mileslongand5milesindiameterbeplacedinspaceforcolonies.Thepurposeoftheirrotationistosimulategravityfortheinhabitants.Explaintheconceptbehindthisproposal.

Unit4LabActivityInthislabactivityyourgroupwilluseaWhirligigtomeasurethetangentialvelocityofanobjectrotatinginUniformCircularMotion(UCM).WiththedatayouobtainyouwillverifytheformulaforCentripetalForce.Materials:1)Whirligigapparatus(String,rubberstoppers,tube,alligatorclip)2)Variousmasses3)Timer4)ScaleProblemStatement:YourtaskistodemonstratethatthetangentialvelocityofanobjectinUCMisrelatedtotheCentripetalForcebeingappliedtotheobjectaccordingtotheformula:Fc=(mv2)/r.Inordertodothis,youwilluseadevicecalledaWhirligig(showntotheright.)Firstmeasurearadiusof15cmonthestring(showninthepictureasL).Youwillusefivedifferentmasses(M)atthebottomoftheWhirligig.Foreachmassyouwilldeterminevtbytiminghowlongittakesthestoppertorotate20timesandthendividingby20.Becertainthatthealligatorclipremainsstationaryaboutaninchbelowthetubewhileyouaretiming.Onceyouhaveobtainedthesemeasurements,youwillmovethealligatorclipandcreateanewradiusforthestopper,20cm.Repeattheprocedureformeasuringvtusingfivedifferentmasses(M).Finally,movethealligatorcliptocreatearadiusof30cmandmeasurevtasbefore,usingfivedifferentmasses.AnalysisQuestions:1)Reportalldatafromyourtrialsintableform.Besuretoshowyourunitsofmeasurement.2)Youwillcreatetwographsforeachofyourthreesetsofmeasurements.(Sixgraphstotal.)FirstcreateagraphusingFcforthey-axisandVtforthex-axis.Connectthepointsofthisgraphusingasmoothcurve.(Itshouldcomeoutfairlyparabolic.)Next,createasecondgraphusingFcforthey-axisand(Vt)2forthex-axis.ConnectthesepointsusingaLineofBestFitprocess.Determinetheslopeofthisline.3)Viewthevideofromourclasswebsitethatexplainstheprocessofcreatingandinterpretingthemeaningofthegraphs.Inyourreport,calculatethevaluesform/rinyourexperimentsandcomparethemtothevaluesobtainedfromyourgraphs.Discusstheresults,rememberingtotalkaboutthevarioussourcesoferror.

PhysicsLabReportChecklistParticipantNames:____________________________________________________________Pd.______LabTitle:_____________________________________________DueDate________________OverallReportFormat

1.TitlePagecontainsGroupnames,ExperimentTitle,DateDue,andPeriod2.Allsectionsreportarepresent,inproperorder,andareclearlylabeledinBoldUnderlineprint.

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Introduction 1.Thereisaclearstatementofthepurpose/objectiveofthelab.2.Thereisabriefdiscussionoftherelevantbackgroundinformationandtheoryinvolvedinthelab.

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ExperimentalProcedure

1.Thereisaclearwrittendescriptionoftheproceduresfollowedintheexperiment.2.Thereisadiagramorpictureoftheexperimentset-upwithcomponentslabeled.

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RawData 1.Allcollecteddatafromtheexperimentispresentedintableformorotherappropriateformat.2.Unitsofmeasurementsareclearlyindicated.

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01Analysis 1.Asampleofeachtypeofcalculationneededin

theexperimentisprovidedandisdonecorrectly.2.ResultsofcalculationsarepresentedforALLdataintableform.3.Graphsofdataareincluded,areneat,clearlytitled,anddisplayappropriateunitsandscales.4.Sourcesofmeasurementerrorareindicatedanddiscussed.

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Conclusions 1.Finalconclusionofthelabisclearlystatedandshowsunderstandingoftheobjectiveofthelab.2.Suggestionsforhowthelabcouldbeimprovedareincluded.(Whatwouldyouhavedonedifferentlyifyouweredoingthelabagain?)

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Note:Labreportsmustbetyped,nothandwritten,withthefollowingexceptions:1)Graphsmaybedonebyhandongraphpaper.2)Samplecalculationsmaybeneatlyhandwritten.

TotalScore:________/25Comments:__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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