ChapterChapter 10.10. UniformUniform CircularCircular MotionMotion AAA PowerPointPowerPointPowerPoint PresentationPresentationPresentation bybyby PaulPaulPaul E.E.E. Tippens,Tippens,Tippens, ProfessorProfessorProfessor ofofof PhysicsPhysicsPhysics SouthernSouthernSouthern PolytechnicPolytechnicPolytechnic StateStateState UniversityUniversityUniversity © 2007 Centripetal forces keep these children moving in a circular path. Objectives:Objectives: AfterAfter completingcompleting thisthis module,module, youyou shouldshould bebe ableable to:to: •• ApplyApply youryour knowledgeknowledge ofof centripetalcentripetal accelerationacceleration andand centripetalcentripetal forceforce toto thethe solutionsolution ofof problemsproblems inin circularcircular motion.motion. •• DefineDefine andand applyapply conceptsconcepts ofof frequencyfrequency andand period,period, andand relaterelate themthem toto linearlinear speed.speed. •• SolveSolve problemsproblems involvinginvolving bankingbanking angles,angles, thethe conicalconical pendulum,pendulum, andand thethe verticalvertical circle.circle. UniformUniform CircularCircular MotionMotion UniformUniform circularcircular motionmotion is motion along a circular path in which there is no change in speed, only a change in direction. v Constant velocity Fc tangent to path. Constant force toward center. Question: Is there an outward force on the ball? UniformUniform CircularCircular MotionMotion (Cont.)(Cont.) TheThe questionquestion ofof anan outwardoutward forceforce cancan bebe resolvedresolved byby askingasking whatwhat happenshappens whenwhen thethe stringstring breaks!breaks! BallBall movesmoves tangenttangent toto v path,path, NOTNOT outwardoutward asas mightmight bebe expected.expected. When central force is removed, ball continues in straight line. CentripetalCentripetal forceforce isis neededneeded toto changechange direction.direction. ExamplesExamples ofof CentripetalCentripetal ForceForce YouYouYou areareare sittingsittingsitting ononon thethethe seatseatseat nextnextnext tototo thethethe outsideoutsideoutside door.door.door. WhatWhatWhat isisis thethethe directiondirectiondirection ofofof thethethe resultantresultantresultant forceforceforce ononon youyouyou asasas youyouyou turn?turn?turn? IsIsIs ititit awayawayaway fromfromfrom centercentercenter ororor towardtowardtoward centercentercenter ofofof thethethe turn?turn?turn? •• CarCar goinggoing aroundaround aa FF c curve.curve. ForceForce ONON youyou isis towardtoward thethe center.center. CarCar ExampleExample ContinuedContinued Reaction F’ TheThe centripetalcentripetal Fc forceforce isis exertedexerted BYBY thethe doordoor ONON you.you. (Centrally)(Centrally) ThereThereThere isisis ananan outwardoutwardoutward force,force,force, butbutbut ititit doesdoesdoes notnotnot actactact ONONON you.you.you. ItItIt isisis thethethe reactionreactionreaction forceforceforce exertedexertedexerted BYBYBY youyouyou ONONON thethethe door.door.door. ItItIt affectsaffectsaffects onlyonlyonly thethethe door.door.door. AnotherAnother ExampleExample DisappearingDisappearing RR platformplatform atat fair.fair. FF c WhatWhat exertsexerts thethe centripetalcentripetal forceforce inin thisthis exampleexample andand onon whatwhat doesdoes itit act?act? TheTheThe centripetalcentripetalcentripetal forceforceforce isisis exertedexertedexerted BYBYBY thethethe wallwallwall ONONON thethethe man.man.man. AAA reactionreactionreaction forceforceforce isisis exertedexertedexerted bybyby thethethe manmanman ononon thethethe wall,wall,wall, butbutbut thatthatthat doesdoesdoes notnotnot determinedeterminedetermine thethethe motionmotionmotion ofofof thethethe man.man.man. SpinSpin CycleCycle onon aa WasherWasher HowHow isis thethe waterwater removedremoved fromfrom clothesclothes duringduring thethe spinspin cyclecycle ofof aa washer?washer? ThinkThink carefullycarefully beforebefore answeringanswering .. .. .. DoesDoes thethe centripetalcentripetal forceforce throwthrow waterwater offoff thethe clothes?clothes? NO.NO.NO. Actually,Actually,Actually, ititit isisis thethethe LACKLACKLACK ofofof aaa forceforceforce thatthatthat allowsallowsallows thethethe waterwaterwater tototo leaveleaveleave thethethe clothesclothesclothes throughthroughthrough holesholesholes ininin thethethe circularcircularcircular wallwallwall ofofof thethethe rotatingrotatingrotating washer.washer.washer. CentripetalCentripetal AccelerationAcceleration ConsiderConsider ballball movingmoving atat constantconstant speedspeed vv inin aa horizontalhorizontal circlecircle ofof radiusradius RR atat endend ofof stringstring tiedtied toto pegpeg onon centercenter ofof table.table. (Assume(Assume zerozero friction.)friction.) n Fc v W R ForceForce FF c andand accelerationacceleration aac towardtoward center.center. WW == nn DerivingDeriving CentralCentral AccelerationAcceleration Consider initial velocity at A and final velocity at B: vf vf B -vo v s v vo o R R A DerivingDeriving AccelerationAcceleration (Cont.)(Cont.) v v f Definition: ac = -v t o v s vo Similar v s R = Triangles v R mass m v vs vv a = = = c t Rt R Centripetal vm22v Centripetal aFma; acceleration:acceleration: cccR R ExampleExample 1:1: AA 33--kgkg rockrock swingsswings inin aa circlecircle ofof radiusradius 55 mm.. IfIf itsits constantconstant speedspeed isis 88 m/sm/s,, whatwhat isis thethe centripetalcentripetal acceleration?acceleration? v v2 m a m = 3 kg c R R R = 5 m; v = 8 m/s (8 m/s)2 a 12.8 m/s2 c 5 m F = (3 kg)(12.8 m/s2) mv2 Fmacc F = 38.4 N R Fcc = 38.4 N ExampleExample 2:2: AA skaterskater movesmoves withwith 1515 m/sm/s inin aa circlecircle ofof radiusradius 3030 mm.. TheThe iceice exertsexerts aa centralcentral forceforce ofof 450450 NN.. WhatWhat isis thethe massmass ofof thethe skater?skater? DrawDraw andand labellabel sketchsketch mv2 FR Fm; c v = 15 m/s c R v2 (450 N)(30 m) Fc R m 2 450 N 30 m (15 m/s) m=? mm == 60.060.0 kgkg Speed skater ExampleExample 3.3. TheThe wallwall exertsexerts aa 600600 NN forceforce onon anan 8080--kgkg personperson movingmoving atat 44 m/sm/s onon aa circularcircular platform.platform. WhatWhat isis thethe radiusradius ofof thethe circularcircular path?path? DrawDraw andand labellabel sketchsketch NewtonNewton’’ss 2nd2nd lawlaw m = 80 kg; forfor circularcircular motion:motion: v = 4 m/s2 F = 600 N mv22 mv c Fr; rF r = ? (80 kg)(4 m/s)2 r r = 2.13 m 600 N r = 2.13 m CarCar NegotiatingNegotiating aa FlatFlat TurnTurn v FF c R WhatWhat isis thethe directiondirection ofof thethe forceforce ON thethe car?car? Ans.Ans. TowardToward CenterCenter ThisThis centralcentral forceforce isis exertedexerted BYBY thethe roadroad ONON thethe car.car. CarCar NegotiatingNegotiating aa FlatFlat TurnTurn v FF c R IsIs therethere alsoalso anan outwardoutward forceforce actingacting ONON thethe car?car? Ans.Ans. No,No, butbut thethe carcar doesdoes exertexert aa outwardoutward reactionreaction forceforce ONON thethe road.road. CarCar NegotiatingNegotiating aa FlatFlat TurnTurn TheThe centripetalcentripetal forceforce FF c isis thatthat ofof staticstatic frictionfriction ff s :: n F = f Fc R c s m fs v R mg TheThe centralcentral forceforce FF andand thethe frictionfriction forceforce ff The central force FCC and the friction force fss areareare notnotnot twotwotwo differentdifferentdifferent forcesforcesforces thatthatthat areareare equal.equal.equal. ThereThereThere isisis justjustjust oneoneoneforceforce force ononon thethethe car.car.car. TheTheThe naturenaturenature ofofof thisthisthis centralcentralcentral forceforceforce isisis staticstaticstatic friction.friction.friction. FindingFinding thethe maximummaximum speedspeed forfor negotiatingnegotiating aa turnturn withoutwithout slipping.slipping. n Fc = fs fs Fc R R m v mg TheThe carcar isis onon thethe vergeverge ofof slippingslipping whenwhen FF C isis equalequal toto thethe maximummaximum forceforce ofof staticstatic frictionfriction ff s. mv2 f = mg Fc = fs Fc = s s R MaximumMaximum speedspeed withoutwithout slippingslipping (Cont.)(Cont.) F = f n c s fs R mv2 = smg R mg v = sgR F m c R VelocityVelocity vv isis maximummaximum v speedspeed forfor nono slipping.slipping. ExampleExample 4:4: AA carcar negotiatesnegotiates aa turnturn ofof radiusradius 7070 mm whenwhen thethe coefficientcoefficient ofof staticstatic frictionfriction isis 0.70.7.. WhatWhat isis thethe maximummaximum speedspeed toto avoidavoid slipping?slipping? mv2 Fc = fs = s mg m R Fc R From which: v = sgR v s = 0.7 g = 9.8 m/s2; R = 70 m v = 21.9 m/s vgRs (0.7)(9.8)(70m) v = 21.9 m/s OptimumOptimum BankingBanking AngleAngle ByBy bankingbanking aa curvecurve atat thethe optimumoptimum angle,angle, thethe normalnormal F m c R forceforce nn cancan provideprovide thethe necessarynecessary centripetalcentripetal forceforce v withoutwithout thethe needneed forfor aa frictionfriction force.force. f fs = 0 s n n n f w s w w slowslow speedspeed fastfast speedspeed optimumoptimum FreeFree--bodybody DiagramDiagram n AccelerationAcceleration aa isis towardtoward thethe xx center.center. SetSet xx axisaxis alongalong thethe directiondirection ofof aac ,, i.i. e.,e., mg horizontalhorizontal (left(left toto right).right). nn coscos nn nn ++ aac nn sinsin mgmg mgmg OptimumOptimum BankingBanking AngleAngle (Cont.)(Cont.) nn coscos n n nn sinsin mg mgmg ApplyApply mvmv2 FF == mmaa nn sinsin NewtonNewton’’ss 2nd2nd x c RR LawLaw toto xx andand yy axes.axes. FF y == 00 nn coscos == mgmg OptimumOptimum BankingBanking AngleAngle (Cont.)(Cont.) n cos n n nsin tan n sin ncos mg mg mv2