Torque and Newton's Laws 8B

Announcements and Updates

  • Lecture quiz results were excellent, with no need for in-class review of specific questions. Individual results are available for review.

  • Today's focus is set 8B, covering torque, Newton's laws, and the biceps curl.

  • Exam 3 will follow the format of Exams 1 and 2: multiple-choice and true/false questions based on lecture and practice set material.

Exam 3 Information

  • The theta, omega, and alpha profiles for both the forearm segment angle and the included elbow joint angle during a biceps curl will be provided.

  • PVNA plots of the barbell will not be provided directly, although linear kinetics and energy concepts remain relevant.

  • The instructor plans to cover sets 9A and 9B before the exam, which focus on single-joint systems to understand individual muscle torque production.

  • There will be no lecture quiz due next week to allow focus on exam preparation.

Torpedo Bat Discussion

  • A new "torpedo bat" design was introduced, featuring a thicker cross-section in the primary contact area that tapers towards the end, contrasting traditional bats with uniform width after widening.

  • The modified bat design is legal, with the shape change intended to increase mass in the hitting zone and enlarge the contact surface area.

  • How might this change alter the moment of inertia of the bat about the grip end?

    • The moment of inertia decreases because mass is relocated closer to the axis of rotation (grip end), reducing the r2r^2 terms in the moment of inertia calculation.

    • This should make the bat easier to swing and increase angular acceleration and bat speed.

    • The primary performance enhancement is believed to be a larger hitting area and favorable center of percussion.

Angular Kinetics and Newton's Laws

  • The lecture returns to angular kinetics, focusing on how torque integrates Newton's laws into angular mechanics (kinematics and kinetics).

  • All three of Newton's laws are relevant in this context.

  • Mass moment of inertia is crucial, requiring consideration of mass distribution relative to the axis of rotation.

Mass Distribution in Limbs

  • The mass of human long segments (arms, legs) is typically distributed closer to the proximal end.

  • This distribution makes rotation about the proximal end easier due to lower moment of inertia.

  • In throwing, striking, and kicking, this mass distribution facilitates high angular velocities and accelerations.

  • Reducing the moment of inertia II allows more torque τ\tau to contribute to angular acceleration α\alpha ($\$\tau = I \alpha$).</p></li></ul><h3id="bdfca683−2cc6−4db0−985d−b5f1433bca53"data−toc−id="bdfca683−2cc6−4db0−985d−b5f1433bca53"collapsed="false"seolevelmigrated="true">Staticvs.DynamicTorque</h3><ul><li><p>Staticcomponent:Torqueduetolimbweightandmomentarm.</p></li><li><p>Existsevenwhenmoving(dynamically).</p></li><li><p>Reducing\$).</p></li></ul><h3 id="bdfca683-2cc6-4db0-985d-b5f1433bca53" data-toc-id="bdfca683-2cc6-4db0-985d-b5f1433bca53" collapsed="false" seolevelmigrated="true">Static vs. Dynamic Torque</h3><ul><li><p>Static component: Torque due to limb weight and moment arm.</p></li><li><p>Exists even when moving (dynamically).</p></li><li><p>ReducingR_wW(momentarmtimesweight)reducesresistanceagainstmuscleaction.</p></li><li><p>Morenettorquecanbeusedforangularacceleration.</p></li></ul><h3id="b11a0935−3165−4ac4−ae57−9a9a4fc3a9cb"data−toc−id="b11a0935−3165−4ac4−ae57−9a9a4fc3a9cb"collapsed="false"seolevelmigrated="true">Newton′sFirstLaw(AngularInertia)</h3><ul><li><p>Arotatingbodywillcontinueinastateofuniformangularmotionunlessactedonbyanexternaltorque.</p></li><li><p>Mathematicallyrepresentedas:(moment arm times weight) reduces resistance against muscle action.</p></li><li><p>More net torque can be used for angular acceleration.</p></li></ul><h3 id="b11a0935-3165-4ac4-ae57-9a9a4fc3a9cb" data-toc-id="b11a0935-3165-4ac4-ae57-9a9a4fc3a9cb" collapsed="false" seolevelmigrated="true">Newton's First Law (Angular Inertia)</h3><ul><li><p>A rotating body will continue in a state of uniform angular motion unless acted on by an external torque.</p></li><li><p>Mathematically represented as:\sum \tau = 0.</p></li><li><p>Uniformangularmotionimpliesconstantangularmomentum.</p></li><li><p>Angularmomentum(L)iscalculatedas:.</p></li><li><p>Uniform angular motion implies constant angular momentum.</p></li><li><p>Angular momentum (L) is calculated as:L = I \omega,whereIisthemomentofinertiaand, where I is the moment of inertia and\omegaistheangularvelocity.</p></li><li><p>Ifis the angular velocity.</p></li><li><p>IfIchanges,changes,\omegamustadjusttokeepLconstantwhenmust adjust to keep L constant when\sum \tau = 0.</p></li><li><p>Angularaccelerationcanoccurwithoutappliedtorquebyalteringthemomentofinertia.</p></li></ul><h3id="6df84bfb−8d98−4481−92bd−9554098b6227"data−toc−id="6df84bfb−8d98−4481−92bd−9554098b6227"collapsed="false"seolevelmigrated="true">ConservationofAngularMomentumExamples</h3><ul><li><p>Figureskaterspinning:Bydrawingarmsandlegsinward,theskaterreducestheirmomentofinertia(I),causinganincreaseinangularvelocity(.</p></li><li><p>Angular acceleration can occur without applied torque by altering the moment of inertia.</p></li></ul><h3 id="6df84bfb-8d98-4481-92bd-9554098b6227" data-toc-id="6df84bfb-8d98-4481-92bd-9554098b6227" collapsed="false" seolevelmigrated="true">Conservation of Angular Momentum Examples</h3><ul><li><p>Figure skater spinning: By drawing arms and legs inward, the skater reduces their moment of inertia (I), causing an increase in angular velocity (\omega)toconserveangularmomentum(L).</p></li><li><p>Divers:Bytucking(reducingI),diversincreasetheirangularvelocity;extending(increasingI)slowsrotation.Gravitypullsthroughtheircenterofmass,thushavingnoexternaltorque.</p></li></ul><h3id="88ee5757−212f−4022−be0b−20263f85f5bd"data−toc−id="88ee5757−212f−4022−be0b−20263f85f5bd"collapsed="false"seolevelmigrated="true">Newton′sSecondLaw(AngularKinetics)</h3><ul><li><p>AnalogoustoF=MA,theangularformis:) to conserve angular momentum (L).</p></li><li><p>Divers: By tucking (reducing I), divers increase their angular velocity; extending (increasing I) slows rotation. Gravity pulls through their center of mass, thus having no external torque.</p></li></ul><h3 id="88ee5757-212f-4022-be0b-20263f85f5bd" data-toc-id="88ee5757-212f-4022-be0b-20263f85f5bd" collapsed="false" seolevelmigrated="true">Newton's Second Law (Angular Kinetics)</h3><ul><li><p>Analogous to F = MA, the angular form is:\sum \tau = I \alpha,where, where\sum \tauisthesumofexternaltorques,Iisthemomentofinertia,andis the sum of external torques, I is the moment of inertia, and\alphaistheangularacceleration.</p></li><li><p>Forthisequationtohold,Imustbeconstant.</p></li><li><p>Torquescanalwaysbesummedabouttheobject′scenterofmass.</p></li><li><p>Torquescanbesummedaboutanaxisotherthanthecenterofmassonlyiftheaxisislinearlyfixed(purerotation).</p></li><li><p>Angularaccelerationmustbeinradianspersecondsquared(is the angular acceleration.</p></li><li><p>For this equation to hold, I must be constant.</p></li><li><p>Torques can always be summed about the object's center of mass.</p></li><li><p>Torques can be summed about an axis other than the center of mass only if the axis is linearly fixed (pure rotation).</p></li><li><p>Angular acceleration must be in radians per second squared (rad/s^2)forcalculations.</p></li><li><p>Constantmomentofinertiaimpliesarigidbody(nochangeinsizeorshape).</p></li></ul><h3id="e0abf0e7−c08d−4888−9db9−4a01f52c7e15"data−toc−id="e0abf0e7−c08d−4888−9db9−4a01f52c7e15"collapsed="false"seolevelmigrated="true">ImplicationsofAngularAcceleration</h3><ul><li><p>Anegativevalueof) for calculations.</p></li><li><p>Constant moment of inertia implies a rigid body (no change in size or shape).</p></li></ul><h3 id="e0abf0e7-c08d-4888-9db9-4a01f52c7e15" data-toc-id="e0abf0e7-c08d-4888-9db9-4a01f52c7e15" collapsed="false" seolevelmigrated="true">Implications of Angular Acceleration</h3><ul><li><p>A negative value of\alphaindicatesaccelerationoppositetothedrawndirectioninthediagram.</p></li><li><p>Forarigidobject,angularaccelerationisuniformthroughouttheplane,regardlessofthesummationpoint.</p></li></ul><h3id="be2375cb−9e5c−4b3d−aafe−6ffb89acc6ef"data−toc−id="be2375cb−9e5c−4b3d−aafe−6ffb89acc6ef"collapsed="false"seolevelmigrated="true">FreeBodyDiagramsandEquationsofMotion</h3><ul><li><p>Twoequationsofmotioncanbeapplied:linear(indicates acceleration opposite to the drawn direction in the diagram.</p></li><li><p>For a rigid object, angular acceleration is uniform throughout the plane, regardless of the summation point.</p></li></ul><h3 id="be2375cb-9e5c-4b3d-aafe-6ffb89acc6ef" data-toc-id="be2375cb-9e5c-4b3d-aafe-6ffb89acc6ef" collapsed="false" seolevelmigrated="true">Free Body Diagrams and Equations of Motion</h3><ul><li><p>Two equations of motion can be applied: linear (\sum F = MA)andangular() and angular (\sum \tau = I \alpha).</p></li><li><p>Massmustbeconstantfor).</p></li><li><p>Mass must be constant for\sum F = MAtobevalid,to be valid,\and Imustbeconstantformust be constant for\sum \tau = I \alphatobevalid.</p></li><li><p>EvenifIisnotconstant,thelinearequationofmotioncanstillbeused.</p></li><li><p>Theright−handsideoftheangularequation(to be valid.</p></li><li><p>Even if I is not constant, the linear equation of motion can still be used.</p></li><li><p>The right-hand side of the angular equation (\tau = I \alpha)isreferredtoastheinertialtorque.</p></li></ul><h3id="b03c6f43−64da−4359−a002−707501831dde"data−toc−id="b03c6f43−64da−4359−a002−707501831dde"collapsed="false"seolevelmigrated="true">BicepsCurlExample:SummingTorquesAbouttheCenterofMass</h3><ul><li><p>Whensummingaboutthecenterofmass,forceslikethebone−on−boneforceattheelbowjoint,originallydismissed,nowfactorinduetoamomentarmrelativetothecenterofmass.</p></li><li><p>Theequationbecomescomplexbutremainsmathematicallyvalid.</p></li></ul><h3id="b2602452−ece9−4e2a−b9c6−c2efaa3a1400"data−toc−id="b2602452−ece9−4e2a−b9c6−c2efaa3a1400"collapsed="false"seolevelmigrated="true">BicepsCurlExample:PureRotationattheElbow</h3><ul><li><p>Inpurerotation,torquescanbesummedabouttheelbowjoint:) is referred to as the inertial torque.</p></li></ul><h3 id="b03c6f43-64da-4359-a002-707501831dde" data-toc-id="b03c6f43-64da-4359-a002-707501831dde" collapsed="false" seolevelmigrated="true">Biceps Curl Example: Summing Torques About the Center of Mass</h3><ul><li><p>When summing about the center of mass, forces like the bone-on-bone force at the elbow joint, originally dismissed, now factor in due to a moment arm relative to the center of mass.</p></li><li><p>The equation becomes complex but remains mathematically valid.</p></li></ul><h3 id="b2602452-ece9-4e2a-b9c6-c2efaa3a1400" data-toc-id="b2602452-ece9-4e2a-b9c6-c2efaa3a1400" collapsed="false" seolevelmigrated="true">Biceps Curl Example: Pure Rotation at the Elbow</h3><ul><li><p>In pure rotation, torques can be summed about the elbow joint:\sum \tau{elbow} = I{elbow} \alpha.</p></li><li><p>Muscletorqueispositive(counterclockwise),weighttorque(WRW)isnegative,andthehandforcetorqueisnegative.</p></li></ul><h3id="7d1796a9−36a3−4030−8d59−4abf1be794a9"data−toc−id="7d1796a9−36a3−4030−8d59−4abf1be794a9"collapsed="false"seolevelmigrated="true">SimplifiedBicepsCurlAnalysis</h3><ul><li><p>Thebarbellweightcanbemergedintotheforearmandhandsystemforsimplification.</p></li><li><p>Thetotalsystemweightisthesumofforearm/handandbarbellweights.</p></li><li><p>Thetotalmomentofinertiaisthesumofforearm/handandbarbellmomentsofinertia.</p></li></ul><h3id="f41be092−9f74−4dfb−bd13−5a2d60262071"data−toc−id="f41be092−9f74−4dfb−bd13−5a2d60262071"collapsed="false"seolevelmigrated="true">BenefitsofSimplification</h3><ul><li><p>Thesimplifiedequationhighlightsthatmuscletorqueoffsetsthetorqueduetotheweightofthesystem,withanyremainingtorquecontributingtoangularacceleration.</p></li><li><p>Theforceatthehandisnowinternaltothesystemandnotconsideredatorqueproducer.</p></li></ul><h3id="738a7c62−2a8f−4c12−889c−3593d8a55675"data−toc−id="738a7c62−2a8f−4c12−889c−3593d8a55675"collapsed="false"seolevelmigrated="true">LimitationsoftheSimplifiedModel</h3><ul><li><p>Themodelassumesonlyonemuscleisresponsiblefortorquegeneration,whichisn′trealistic.</p></li><li><p>Multiplemuscles(biceps,brachialis,brachioradialis,pronatorteres)contributetoelbowflexion.</p></li><li><p>Passiveelements(ligaments,fascia)andantagonistscanalsoproducetorques.</p></li></ul><h3id="02154de7−d90a−4c24−aa7c−4dd34626ecf8"data−toc−id="02154de7−d90a−4c24−aa7c−4dd34626ecf8"collapsed="false"seolevelmigrated="true">GeneralizedFreeBodyDiagram</h3><ul><li><p>Atechniquetosimplifythefreebodydiagramwhileaccountingformultipleforces.</p></li><li><p>Forcescrossingthejointaregroupedintoasinglejointreactionforce.</p></li><li><p>Thetorqueproducedbytheseforcesabouttheelbowjointisaccountedforbyanetmuscularmoment.</p></li></ul><h3id="8700ab11−3955−43b4−b685−52d9ec62cd04"data−toc−id="8700ab11−3955−43b4−b685−52d9ec62cd04"collapsed="false"seolevelmigrated="true">NetMuscularMoment</h3><ul><li><p>Thesumofalltorque−producingelements,includingmuscles,bone−on−boneforce,ligaments,andfascia,attheelbowjoint.</p></li><li><p>Mathematicallyrepresentedas:</p><ul><li><p>.</p></li><li><p>Muscle torque is positive (counterclockwise), weight torque (WRW) is negative, and the hand force torque is negative.</p></li></ul><h3 id="7d1796a9-36a3-4030-8d59-4abf1be794a9" data-toc-id="7d1796a9-36a3-4030-8d59-4abf1be794a9" collapsed="false" seolevelmigrated="true">Simplified Biceps Curl Analysis</h3><ul><li><p>The barbell weight can be merged into the forearm and hand system for simplification.</p></li><li><p>The total system weight is the sum of forearm/hand and barbell weights.</p></li><li><p>The total moment of inertia is the sum of forearm/hand and barbell moments of inertia.</p></li></ul><h3 id="f41be092-9f74-4dfb-bd13-5a2d60262071" data-toc-id="f41be092-9f74-4dfb-bd13-5a2d60262071" collapsed="false" seolevelmigrated="true">Benefits of Simplification</h3><ul><li><p>The simplified equation highlights that muscle torque offsets the torque due to the weight of the system, with any remaining torque contributing to angular acceleration.</p></li><li><p>The force at the hand is now internal to the system and not considered a torque producer.</p></li></ul><h3 id="738a7c62-2a8f-4c12-889c-3593d8a55675" data-toc-id="738a7c62-2a8f-4c12-889c-3593d8a55675" collapsed="false" seolevelmigrated="true">Limitations of the Simplified Model</h3><ul><li><p>The model assumes only one muscle is responsible for torque generation, which isn't realistic.</p></li><li><p>Multiple muscles (biceps, brachialis, brachioradialis, pronator teres) contribute to elbow flexion.</p></li><li><p>Passive elements (ligaments, fascia) and antagonists can also produce torques.</p></li></ul><h3 id="02154de7-d90a-4c24-aa7c-4dd34626ecf8" data-toc-id="02154de7-d90a-4c24-aa7c-4dd34626ecf8" collapsed="false" seolevelmigrated="true">Generalized Free Body Diagram</h3><ul><li><p>A technique to simplify the free body diagram while accounting for multiple forces.</p></li><li><p>Forces crossing the joint are grouped into a single joint reaction force.</p></li><li><p>The torque produced by these forces about the elbow joint is accounted for by a net muscular moment.</p></li></ul><h3 id="8700ab11-3955-43b4-b685-52d9ec62cd04" data-toc-id="8700ab11-3955-43b4-b685-52d9ec62cd04" collapsed="false" seolevelmigrated="true">Net Muscular Moment</h3><ul><li><p>The sum of all torque-producing elements, including muscles, bone-on-bone force, ligaments, and fascia, at the elbow joint.</p></li><li><p>Mathematically represented as:</p><ul><li><p>\sum (Elbow \, Flexors) - \sum (Elbow \, Extensors) + \sum (Passive \, Components)</p></li></ul></li><li><p>Canrepresentflexororextensortorqueswithoutfocusingonindividualmuscles.</p></li></ul><h3id="9e19ce99−57af−4fd5−b4a6−42de807c9f13"data−toc−id="9e19ce99−57af−4fd5−b4a6−42de807c9f13"collapsed="false"seolevelmigrated="true">AngularEquationofMotionwithGeneralizedFreeBodyDiagram</h3><ul><li><p>Theequationbecomes:</p></li></ul></li><li><p>Can represent flexor or extensor torques without focusing on individual muscles.</p></li></ul><h3 id="9e19ce99-57af-4fd5-b4a6-42de807c9f13" data-toc-id="9e19ce99-57af-4fd5-b4a6-42de807c9f13" collapsed="false" seolevelmigrated="true">Angular Equation of Motion with Generalized Free Body Diagram</h3><ul><li><p>The equation becomes:NMM - WRW = I \alpha</p></li><li><p>NMM:NetMuscularMoment(torqueproducedbymuscles)</p></li><li><p>WRW:Torqueduetoweight</p></li><li><p>I</p></li><li><p>NMM: Net Muscular Moment (torque produced by muscles)</p></li><li><p>WRW: Torque due to weight</p></li><li><p>I\alpha:Inertialtorque</p></li><li><p>Thisequationincludesanycombinationofflexor,extensor,andpassivetissuescreatingtorquesatthejoint.</p></li><li><p>Itstillaccountsforweightandinertialtorque.</p></li></ul><h3id="46c18d3b−23b5−48dd−9b1d−342b38743ab3"data−toc−id="46c18d3b−23b5−48dd−9b1d−342b38743ab3"collapsed="false"seolevelmigrated="true">LinearEquationofMotionwithGeneralizedFreeBodyDiagram</h3><ul><li><p>Incorporatesforcearrowswithoutaddressingthetorquearrow.</p></li><li><p>Addressestheforcesinhorizontalandverticaldirections.</p></li></ul><h3id="a717d683−e751−45cd−afc0−94cbb5eed2ec"data−toc−id="a717d683−e751−45cd−afc0−94cbb5eed2ec"collapsed="false"seolevelmigrated="true">TorqueProfilesUnderDifferentConditions</h3><ul><li><p>Netmuscularmomentattheelbowduringabicepscurlunderquasi−staticconditions:</p><ul><li><p>Linearandangularacclerationsarezero.</p></li><li><p>EquationreducestoNMME=WRW.</p></li><li><p>Torqueisgreatestwhenweightisfurthesthorizontallyfromtheelbowjoint.</p></li><li><p>Uniphasicactivationpattern,withflexortorqueneededthroughouttheentireupanddownphases.</p></li></ul></li><li><p>Netmuscularmomentattheelbowduringabicepscurlunderdynamicconditions:</p><ul><li><p>I: Inertial torque</p></li><li><p>This equation includes any combination of flexor, extensor, and passive tissues creating torques at the joint.</p></li><li><p>It still accounts for weight and inertial torque.</p></li></ul><h3 id="46c18d3b-23b5-48dd-9b1d-342b38743ab3" data-toc-id="46c18d3b-23b5-48dd-9b1d-342b38743ab3" collapsed="false" seolevelmigrated="true">Linear Equation of Motion with Generalized Free Body Diagram</h3><ul><li><p>Incorporates force arrows without addressing the torque arrow.</p></li><li><p>Addresses the forces in horizontal and vertical directions.</p></li></ul><h3 id="a717d683-e751-45cd-afc0-94cbb5eed2ec" data-toc-id="a717d683-e751-45cd-afc0-94cbb5eed2ec" collapsed="false" seolevelmigrated="true">Torque Profiles Under Different Conditions</h3><ul><li><p>Net muscular moment at the elbow during a biceps curl under quasi-static conditions:</p><ul><li><p>Linear and angular acclerations are zero.</p></li><li><p>Equation reduces to NMME = WRW.</p></li><li><p>Torque is greatest when weight is furthest horizontally from the elbow joint.</p></li><li><p>Uniphasic activation pattern, with flexor torque needed throughout the entire up and down phases.</p></li></ul></li><li><p>Net muscular moment at the elbow during a biceps curl under dynamic conditions:</p><ul><li><p>I\alphaisnotzero,butissmallerthanWRW.</p></li><li><p>AddingIis not zero, but is smaller than WRW.</p></li><li><p>Adding I\alpha$$ and WRW terms together.

  • Constant I, thus I can be constant only here.

  • Hill valley valley hill profile.