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Chapter 5A. Torque

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Chapter 5A. Torque ChapterChapter 5A.5A. TorqueTorque AAA PowerPointPowerPointPowerPoint PresentationPresentationPresentation bybyby PaulPaulPaul E.E.E. Tippens,Tippens,Tippens, ProfessorProfessorProfessor ofofof PhysicsPhysicsPhysics SouthernSouthernSouthern PolytechnicPolytechnicPolytechnic StateStateState UniversityUniversityUniversity © 2007 TorqueTorque isis aa twisttwist oror turnturn thatthat tendstends toto produceproduce rotation.rotation. ** ** ** ApplicationsApplications areare foundfound inin manymany commoncommon toolstools aroundaround thethe homehome oror industryindustry wherewhere itit isis necessarynecessary toto turn,turn, tightentighten oror loosenloosen devices.devices. Objectives:Objectives: AfterAfter completingcompleting thisthis module,module, youyou shouldshould bebe ableable to:to: •• DefineDefine andand givegive examplesexamples ofof thethe termsterms torque,torque, momentmoment arm,arm, axis,axis, andand lineline ofof actionaction ofof aa force.force. •• Draw,Draw, labellabel andand calculatecalculate thethe momentmoment armsarms forfor aa varietyvariety ofof appliedapplied forcesforces givengiven anan axisaxis ofof rotation.rotation. •• CalculateCalculate thethe resultantresultant torquetorque aboutabout anyany axisaxis givengiven thethe magnitudemagnitude andand locationslocations ofof forcesforces onon anan extendedextended objectobject.. •• Optional:Optional: DefineDefine andand applyapply thethe vectorvector crosscross productproduct toto calculatecalculate torque.torque. DefinitionDefinition ofof TorqueTorque TorqueTorqueis is defineddefined asas thethe tendencytendency toto produceproduce aa changechange inin rotationalrotational motion.motion. Examples: TorqueTorque isis DeterminedDetermined byby ThreeThree Factors:Factors: ••• TheTheThe magnitudemagnitudemagnitude ofofof thethethe appliedappliedapplied force.force.force. ••• TheTheThe directiondirectiondirection ofofof thethethe appliedappliedapplied force.force.force. ••• TheTheThe locationlocationlocation ofofof thethethe appliedappliedapplied force.force.force. DirectionMagnitudeLocation of of ofForce force force TheEachThe forces 40-Nof thenearer force 20-N the forcesproducesend of has the atwice wrenchdifferent the 20 N 2020 N N torquetorque as due does to the the 20 N have greater torques. 2040 N N direction20-N force. of force. 20 N20 N UnitsUnits forfor TorqueTorque TorqueTorque isis proportionalproportional toto thethe magnitudemagnitude ofof FF andand toto thethe distancedistance rr fromfrom thethe axis.axis. Thus,Thus, aa tentativetentativeformula formula mightmight be:be: == FrFr Units: Nm or lbft = (40 N)(0.60 m) = 24.0 Nm, cw 6 cm == 24.024.0 NNm,m, cwcw 40 N DirectionDirection ofof TorqueTorque TorqueTorque isis aa vectorvector quantityquantity thatthat hashas directiondirection asas wellwell asas magnitude.magnitude. Turning the handle of a screwdriver clockwise and then counterclockwise will advance the screw first inward and then outward. SignSign ConventionConvention forfor TorqueTorque By convention, counterclockwise torques are positive and clockwise torques are negative. Positive torque: ccw Counter-clockwise, out of page cw Negative torque: clockwise, into page LineLine ofof ActionAction ofof aa ForceForce TheThe lineline ofof actionactionofof aa forceforce isis anan imaginaryimaginary lineline ofof indefiniteindefinite lengthlength drawndrawn alongalong thethe directiondirection ofof thethe force.force. F2 F1 Line of F3 action TheThe MomentMoment ArmArm TheThe momentmoment armarmofof aa forceforce isis thethe perpendicularperpendicular distancedistance fromfrom thethe lineline ofof actionaction ofof aa forceforce toto thethe axisaxis ofof rotation.rotation. F1 r F r 2 F r 3 CalculatingCalculating TorqueTorque ••• ReadReadRead problemproblemproblem andandand drawdrawdraw aaa roughroughrough figure.figure.figure. ••• ExtendExtendExtend linelineline ofofof actionactionaction ofofof thethethe force.force.force. ••• DrawDrawDraw andandand labellabellabel momentmomentmoment arm.arm.arm. ••• CalculateCalculateCalculate thethethe momentmomentmoment armarmarm ififif necessary.necessary.necessary. ••• ApplyApplyApply definitiondefinitiondefinition ofofof torque:torque:torque: == FrFr Torque = force x moment arm ExampleExample 1:1: AnAn 8080--NN forceforce actsacts atat thethe endend ofof aa 1212--cmcm wrenchwrench asas shown.shown. FindFind thethe torque.torque. • Extend line of action, draw, calculate r. rr == 1212 cmcm sinsin 606000 == (80(80 N)(0.104N)(0.104 m)m) == 10.410.4 cmcm == 8.318.31 NN mm Alternate:Alternate: AnAn 8080--NN forceforce actsacts atat thethe endend ofof aa 1212--cmcm wrenchwrench asas shown.shown. FindFind thethe torque.torque. positive 12 cm Resolve 80-N force into components as shown. Note from figure: rx = 0 and ry = 12 cm = (69.3 N)(0.12 m) == 8.318.31 NN mm asas beforebefore CalculatingCalculating ResultantResultant TorqueTorque ••• Read,Read,Read, draw,draw,draw, andandand labellabellabel aaa roughroughrough figure.figure.figure. ••• DrawDrawDraw freefreefree-body--bodybody diagramdiagramdiagram showingshowingshowing allallall forces,forces,forces, distances,distances,distances, andandand axisaxisaxis ofofof rotation.rotation.rotation. ••• ExtendExtendExtend lineslineslines ofofof actionactionaction forforfor eacheacheach force.force.force. ••• CalculateCalculateCalculate momentmomentmoment armsarmsarms ififif necessary.necessary.necessary. ••• CalculateCalculateCalculate torquestorquestorques dueduedue tototo EACHEACHEACH individualindividualindividual forceforceforce affixingaffixingaffixing properproperproper sign.sign.sign. CCWCCWCCW (+)(+)(+) andandand CWCWCW (((-).--).). ••• ResultantResultantResultant torquetorquetorque isisis sumsumsum ofofof individualindividualindividual torques.torques.torques. ExampleExample 2:2: FindFind resultantresultant torquetorque aboutabout axisaxis AA forfor thethe arrangementarrangement shownshown below:below: FindFind duedue toto 30 N negative 20 N each force. 0 r each force. 30 300 Consider 20-N Consider 20-N 6 m 2 m A 4 m forceforce first:first: 40 N r = (4 m) sin 300 The torque about A is = 2.00 m clockwise and negative. = Fr = (20 N)(2 m) 20 == --4040 NN mm = 40 N m, cw 20 ExampleExample 22 (Cont.):(Cont.): NextNext wewe findfind torquetorque duedue toto 3030--NN forceforce aboutabout samesame axisaxis AA.. r 20 N FindFind duedue toto 30 N negative each force. 0 each force. 30 300 Consider 30-N Consider 30-N 6 m 2 m A 4 m forceforce next.next. 40 N r = (8 m) sin 300 The torque about A is = 4.00 m clockwise and negative. = Fr = (30 N)(4 m) 30 == -120-120 NN mm = 120 N m, cw 30 ExampleExample 22 (Cont.):(Cont.): Finally,Finally, wewe considerconsider thethe torquetorque duedue toto thethe 4040--NN force.force. FindFind duedue toto 30 N positive 20 N each force. 0 each force. 30 r 300 Consider 40-N Consider 40-N 6 m 2 m A 4 m forceforce next:next: 40 N r = (2 m) sin 900 The torque about A is = 2.00 m CCW and positive. = Fr = (40 N)(2 m) = +80 N m = 80 N m, ccw 4040 = +80 N m ExampleExample 22 (Conclusion):(Conclusion): FindFind resultantresultant torquetorque aboutabout axisaxis AA forfor thethe arrangementarrangement shownshown below:below: 30 N 20 N ResultantResultant torquetorque isis thethe sumsum ofof 300 300 6 m 2 m individualindividual torques.torques. A 4 m 40 N R = 20 + 20 + 20 = -40 N m -120 N m + 80 N m = - 80 N m Clockwise RR = - 80 N m PartPart II:II: TorqueTorque andand thethe CrossCross ProductProduct oror VectorVector Product.Product. Optional Discussion This concludes the general treatment of torque. Part II details the use of the vector product in calculating resultant torque. Check with your instructor before studying this section. TheThe VectorVector ProductProduct Torque can also be found by using the vector product of force F and position vector r. For example, consider the figure below. F Sin Torque F The effect of the force F at angle (torque) r is to advance the bolt out of the page. Magnitude: Direction = Out of page (+). (FSin)r DefinitionDefinition ofof aa VectorVector ProductProduct The magnitude of the vector (cross) product of two vectors A and B is defined as follows: AxB= l A l l B l Sin In our example, the cross product of F and r is: F x r = l F l l r l Sin MagnitudeMagnitude onlyonly F Sin F In effect, this becomes simply: r (F Sin ) r or F (r Sin ) Example:Example: FindFind thethe magnitudemagnitude ofof thethe crosscross productproduct ofof thethe vectorsvectors rr andand FF drawndrawn below:below: 12 lb Torque r x F = l r l l F l Sin r x F = (6 in.)(12 lb) Sin 600 6 in. r x F = 62.4 lb in. 6 in. r x F = l r l l F l Sin 600 r x F= (6 in.)(12 lb) Sin 120 Torque 12 lb r x F = 62.4 lb in. ExplainExplain differencedifference.. Also,Also, whatwhat aboutabout F x r?? DirectionDirection ofof thethe VectorVector Product.Product. C TheThe directiondirection ofof aa vectorvector productproduct isis determineddetermined byby thethe B rightright handhand rule.rule. B A A -C AA xx BB == CC (up)(up) CurlCurl fingersfingers ofof rightright handhand BB xx AA == --CC (Down)(Down) inin directiondirection ofof crosscross propro-- ductduct ((AA toto BB)) oror ((BB toto AA).). WhatWhat isis directiondirection ThumbThumb willwill pointpoint inin thethe ofof AA xx C?C? directiondirection ofof productproduct CC.. Example:Example: WhatWhat areare thethe magnitudemagnitude andand directiondirection ofof thethe crosscross product,product, rr xx F?F? 10 lb Torque r x F = l r l l F l Sin r x F = (6 in.)(10 lb) Sin 500 6 in. r x F = 38.3
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