Angular Kinematics and Torque

Angular Kinematics Discussion

  • Relating linear and angular stuff to each other

  • Pure rotation: points on an object rotate with the same angular displacement, but different linear displacements based on their distance from the axis of rotation.

Arc Length

  • The relationship between arc length (s)(s), radius (r)(r), and angular displacement θ\theta: s=rθs = r \theta Where:

    • ss is the length of the arc on the circular path.

    • rr is the radius of the circular path.

    • θ\theta is the angular displacement in radians.

  • This equation is a rearrangement of the definition of a radian: radian=sr\text{radian} = \frac{s}{r}

  • Angular kinematic variables ($\theta,,\omega,,\alpha)mustbeinradiansfortheequationstoworkcorrectly.</p></li><li><p>Radiansaretechnicallyunitless,whiledegreesareaunit.Thisiswhyradiansarerequiredtomaintaincorrectunitconsistency.</p></li></ul><h4id="668b93c4−071a−40b0−ae6f−afa9d25543a2"data−toc−id="668b93c4−071a−40b0−ae6f−afa9d25543a2"collapsed="false"seolevelmigrated="true">RelationshipBetweenLinearandAngularQuantities</h4><ul><li><p>) must be in radians for the equations to work correctly.</p></li><li><p>Radians are technically unitless, while degrees are a unit. This is why radians are required to maintain correct unit consistency.</p></li></ul><h4 id="668b93c4-071a-40b0-ae6f-afa9d25543a2" data-toc-id="668b93c4-071a-40b0-ae6f-afa9d25543a2" collapsed="false" seolevelmigrated="true">Relationship Between Linear and Angular Quantities</h4><ul><li><p>s = r \thetawhere:</p><ul><li><p>s=arclength</p></li><li><p>r=radius</p></li><li><p>where:</p><ul><li><p>s = arc length</p></li><li><p>r = radius</p></li><li><p>\theta=angulardisplacement(inradians)</p></li></ul></li><li><p>Theangularkinematicvariablemustbeinradians.</p></li><li><p>Radiansisaunitlessquantity.Ifusingdegrees,degreesareaunitsoyou′dendupwithameterdegrees,whichdoesnotmatchwhatyouwantoutofit.</p></li></ul><h4id="8a02e0cf−e770−4ace−b85b−cf0b030387d2"data−toc−id="8a02e0cf−e770−4ace−b85b−cf0b030387d2"collapsed="false"seolevelmigrated="true">VelocityonaCircularPath</h4><ul><li><p>Instantaneousvelocityistangenttothecircularpath.</p></li><li><p>= angular displacement (in radians)</p></li></ul></li><li><p>The angular kinematic variable must be in radians.</p></li><li><p>Radians is a unitless quantity. If using degrees, degrees are a unit so you'd end up with a meter degrees, which does not match what you want out of it.</p></li></ul><h4 id="8a02e0cf-e770-4ace-b85b-cf0b030387d2" data-toc-id="8a02e0cf-e770-4ace-b85b-cf0b030387d2" collapsed="false" seolevelmigrated="true">Velocity on a Circular Path</h4><ul><li><p>Instantaneous velocity is tangent to the circular path.</p></li><li><p>v_t = r \omegawhere:</p><ul><li><p>where:</p><ul><li><p>v_t=magnitudeofthelinearvelocity(tangential)</p></li><li><p>= magnitude of the linear velocity (tangential)</p></li><li><p>\omega=magnitudeofangularvelocity</p></li><li><p>r=radius</p></li></ul></li><li><p>Usepositivevaluesforangularvelocityand= magnitude of angular velocity</p></li><li><p>r = radius</p></li></ul></li><li><p>Use positive values for angular velocity andr.Onlylookingforthelengthofthatarclength.</p></li></ul><h4id="324dbaa4−cd3a−4a33−bbfc−10d1bf40ce43"data−toc−id="324dbaa4−cd3a−4a33−bbfc−10d1bf40ce43"collapsed="false"seolevelmigrated="true">LinearAccelerations</h4><ul><li><p>Twoperpendicularcomponents:</p><ul><li><p>Tangenttothepath(. Only looking for the length of that arc length.</p></li></ul><h4 id="324dbaa4-cd3a-4a33-bbfc-10d1bf40ce43" data-toc-id="324dbaa4-cd3a-4a33-bbfc-10d1bf40ce43" collapsed="false" seolevelmigrated="true">Linear Accelerations</h4><ul><li><p>Two perpendicular components:</p><ul><li><p>Tangent to the path (a_t)</p></li><li><p>Perpendicular/normaltothepath()</p></li><li><p>Perpendicular/normal to the path (a_n)</p></li></ul></li></ul><h5id="dafc08bb−7d92−402f−b72d−d206dcd9949d"data−toc−id="dafc08bb−7d92−402f−b72d−d206dcd9949d"collapsed="false"seolevelmigrated="true">TangentialAcceleration</h5><ul><li><p>)</p></li></ul></li></ul><h5 id="dafc08bb-7d92-402f-b72d-d206dcd9949d" data-toc-id="dafc08bb-7d92-402f-b72d-d206dcd9949d" collapsed="false" seolevelmigrated="true">Tangential Acceleration</h5><ul><li><p>a_t = r \alphawhere:</p><ul><li><p>where:</p><ul><li><p>\alpha=magnitudeofangularacceleration</p></li></ul></li><li><p>Ifangularvelocityisconstant(= magnitude of angular acceleration</p></li></ul></li><li><p>If angular velocity is constant (\alpha = 0),then), thena_t = 0.</p></li></ul><h5id="ae42491f−3ab4−46e5−9465−8ff50cd756aa"data−toc−id="ae42491f−3ab4−46e5−9465−8ff50cd756aa"collapsed="false"seolevelmigrated="true">Normal/Centripetal/RadialAcceleration</h5><ul><li><p>Alwaysdirectedtowardsthecenterofthecircle.</p></li><li><p>.</p></li></ul><h5 id="ae42491f-3ab4-46e5-9465-8ff50cd756aa" data-toc-id="ae42491f-3ab4-46e5-9465-8ff50cd756aa" collapsed="false" seolevelmigrated="true">Normal/Centripetal/Radial Acceleration</h5><ul><li><p>Always directed towards the center of the circle.</p></li><li><p>an = ac = a_r = r \omega^2</p></li><li><p>Alternativeform:</p></li><li><p>Alternative form:a_n = \frac{v^2}{r},wherevisthetangentialvelocity.</p></li><li><p>Findingthemagnitudeoflinearaccelerationis:, where v is the tangential velocity.</p></li><li><p>Finding the magnitude of linear acceleration is:\sqrt{at^2 + an^2}</p></li><li><p>Astheradiusincreases,linearattributesalsoincrease.</p></li><li><p>motion=normal/centripetalacceleration</p></li></ul><h4id="6d95356b−c649−44cf−86af−952434631c16"data−toc−id="6d95356b−c649−44cf−86af−952434631c16"collapsed="false"seolevelmigrated="true">Centripetalvs.CentrifugalForce</h4><ul><li><p>CentripetalForce:</p><ul><li><p>Aforceinwardisathingthatexists.</p></li><li><p>Causescentripetalacceleration.</p></li></ul></li><li><p>CentrifugalForce:</p><ul><li><p>Doesn′texist.</p></li><li><p>Thephenomenonthatweobservebecauseofthelackofacentripetalforcetokeepitonthatcircularpath.</p></li></ul></li><li><p>Ifrestraintsareremoved,theobjectwillwanttogetfurtherawayfromthecenterbecausethevelocityistangenttothepath.</p></li><li><p>Centrifuge:worksoffofthelackofthecentripetalforceonthatobjectthat′ssuspendedinthere.</p></li></ul><h4id="b02298bc−7b1f−4f10−a864−1cf93393f97e"data−toc−id="b02298bc−7b1f−4f10−a864−1cf93393f97e"collapsed="false"seolevelmigrated="true">BicepsCurlExample</h4><ul><li><p>Analysisof</p></li><li><p>As the radius increases, linear attributes also increase.</p></li><li><p>motion = normal/centripetal acceleration</p></li></ul><h4 id="6d95356b-c649-44cf-86af-952434631c16" data-toc-id="6d95356b-c649-44cf-86af-952434631c16" collapsed="false" seolevelmigrated="true">Centripetal vs. Centrifugal Force</h4><ul><li><p>Centripetal Force:</p><ul><li><p>A force inward is a thing that exists.</p></li><li><p>Causes centripetal acceleration.</p></li></ul></li><li><p>Centrifugal Force:</p><ul><li><p>Doesn't exist.</p></li><li><p>The phenomenon that we observe because of the lack of a centripetal force to keep it on that circular path.</p></li></ul></li><li><p>If restraints are removed, the object will want to get further away from the center because the velocity is tangent to the path.</p></li><li><p>Centrifuge: works off of the lack of the centripetal force on that object that's suspended in there.</p></li></ul><h4 id="b02298bc-7b1f-4f10-a864-1cf93393f97e" data-toc-id="b02298bc-7b1f-4f10-a864-1cf93393f97e" collapsed="false" seolevelmigrated="true">Biceps Curl Example</h4><ul><li><p>Analysis ofanandandatduringabicepscurl.</p></li><li><p>during a biceps curl.</p></li><li><p>anisproportionaltois proportional to\omega^2andandatisproportionaltois proportional to\alpha.</p></li><li><p>Theforearmsegmentangleincreasesasweflexthejoint.</p></li><li><p>Magnitudeofangularvelocitysquaredwillbehillhill.</p></li><li><p>Magnitudeofangularaccelerationbecomesahillhillandahillhillprofile.</p></li><li><p>Angularprofilesoftheforearmsegmentare:</p><ul><li><p>Angle:Increasesduringflexion,decreasesduringextension.</p></li><li><p>AngularVelocity.</p></li><li><p>The forearm segment angle increases as we flex the joint.</p></li><li><p>Magnitude of angular velocity squared will be hill hill.</p></li><li><p>Magnitude of angular acceleration becomes a hill hill and a hill hill profile.</p></li><li><p>Angular profiles of the forearm segment are:</p><ul><li><p>Angle: Increases during flexion, decreases during extension.</p></li><li><p>Angular Velocity\omega:Hill−valleyprofile.</p></li><li><p>AngularAcceleration: Hill-valley profile.</p></li><li><p>Angular Acceleration\alpha:Hill−valley−valley−hillprofile.</p></li></ul></li><li><p>Theequationsfor: Hill-valley-valley-hill profile.</p></li></ul></li><li><p>The equations foranandandat:</p><ul><li><p>:</p><ul><li><p>a_n = r \omega^2</p></li><li><p></p></li><li><p>a_t = r \alpha</p></li></ul></li></ul><h5id="9cf4fd5a−bcff−427b−9e32−83a908fcfd95"data−toc−id="9cf4fd5a−bcff−427b−9e32−83a908fcfd95"collapsed="false"seolevelmigrated="true">Analysisof</p></li></ul></li></ul><h5 id="9cf4fd5a-bcff-427b-9e32-83a908fcfd95" data-toc-id="9cf4fd5a-bcff-427b-9e32-83a908fcfd95" collapsed="false" seolevelmigrated="true">Analysis of\mathbf{a_n}</h5><ul><li><p>Sinceweonlycareaboutthemagnitude:</p><ul><li><p>Angularvelocity,whichhadahill−valleyprofile,becomeshill−hillafterbeingsquared.Largervelocityvaluesbecomemoreaccentuatedwithsquaring.</p></li><li><p></h5><ul><li><p>Since we only care about the magnitude:</p><ul><li><p>Angular velocity, which had a hill-valley profile, becomes hill-hill after being squared. Larger velocity values become more accentuated with squaring.</p></li><li><p>a_nhasahill−hillshape.Astheangularvelocityincreases,morecentripetalforceisneeded.</p></li></ul></li></ul><h5id="3fbb7fc9−277a−4713−907c−80bd7218ff29"data−toc−id="3fbb7fc9−277a−4713−907c−80bd7218ff29"collapsed="false"seolevelmigrated="true">Analysisofhas a hill-hill shape. As the angular velocity increases, more centripetal force is needed.</p></li></ul></li></ul><h5 id="3fbb7fc9-277a-4713-907c-80bd7218ff29" data-toc-id="3fbb7fc9-277a-4713-907c-80bd7218ff29" collapsed="false" seolevelmigrated="true">Analysis of\mathbf{a_t}

    • The angular acceleration ($\alpha) profile, which is hill-valley-valley-hill, becomes hill-hill-hill-hill after taking the magnitude.

    Halfway Point of Up Phase

    • Vertical acceleration is zero.

    • Tangential acceleration is zero.

    • Horizontal acceleration is large and negative.

    • Normal/centripetal/radial acceleration is large.

    Effect of Repetition Speed

    • Changing repetition speed alters ω\omega by changing Δt\Delta t.

    • Acceleration has a squared effect.

    • Forearm and hand length have linear effects, not squared effects like changes in time.

    Rolling Ball Example

    • Relationship between linear and angular velocity:

      • v<em>cm=rωv<em>{cm} = r \omega or ω=v</em>cmr\omega = \frac{v</em>{cm}}{r}

    • At any instant, the point on the ball touching the ground has zero linear velocity if the ball is not slipping.

    Torque

    • Main construct of angular kinetics.

    • Analogous to force in linear kinetics.

    • Capital T is the variable, and it is a vector quantity.

    Definition

    • The rotational analog to force.

    • The effect of a force that tends to cause a change in a body's state of angular position or motion.

    • Units: Newton-meter (N·m) or foot-pound (ft·lb) in the English system

      • 1 N·m = 0.736 ft·lb

    • Biomechanists prefer the term "moment" or "moment of force".

    Calculation

    • Torque is the product of the magnitude of a force and its moment arm.

    • A vector about an axis perpendicular to the plane of the force and moment arm.

    Convention

    • Counterclockwise rotation: positive torque

    • Clockwise rotation: negative torque

    • Magnitude of torque = (magnitude of force) * (length of moment arm).

    • The moment arm is the perpendicular distance from the line of action of the force to the axis of rotation.

    Subscripts

    • Use a subscript on TT to connect it with the force (e.g., TFT_F).

    • Sometimes a subscript is associated with the axis of rotation (e.g., TAT_A).

    Example: Free Body Diagram of Forearm and Hand

    • Forces:

      • Bone-on-bone contact force

      • Muscle force (FmF_m)

      • Weight of forearm and hand

      • Force from barbell

    Torque Calculations

    • Torque from muscle: T<em>m=∣F</em>m∣∗rmT<em>m = |F</em>m| * r_m

    • Torque from forearm and hand (weight): T<em>w=−∣W∣∗r</em>wT<em>w = -|W| * r</em>w

    • Torque from force in hand: T<em>H=−∣F</em>H∣∗rHT<em>H = -|F</em>H| * r_H

    • Torque from bone on bone is zero because the line of action goes through the axis of rotation, so arm is 0.

    • These torques have different signs because some promote rotation in different directions.

    Lever Arm vs. Moment Arm

    • Avoid the term "lever arm" because it has two definitions:

      • Synonymous with moment arm (perpendicular distance).

      • Straight-line distance from the axis to the point of force application.

    • Leverage: conveys the notion of increasing the moment arm.

    • The moment arm is the perpendicular distance from the line of action of the force to the axis of rotation.

    Torque as a Vector Cross Product

    • Avoid this approach.

    • The way Torque is a vector cross product is T=F×ΔpT = F \times \Delta p

    • T=F×ΔPsin⁡(θ)T = F \times \Delta P \sin(\theta)

    • ΔP\Delta P: A displacement to the point of application from the axis which is just ends up being the length.

    • But instead the preferred way is T=F×RT = F \times R