1Angular Kinetics of Human Movement (Ch 11)• Torques, Moments, and Levers• Center of Mass/Gravity• Moment of Inertia• Angular Analogues of Newton’s Laws • Angular Momentum and Impulse• Angular Work and Power• Equations of Motion• Segmental Analysis & Inverse DynamicsCopyright © 2017Brian R. Umberger, Ph.D.University of Massachusetts AmherstTorqueA net centric force will cause translationA net eccentric force will cause translation & rotationTorqueForce CoupleTwo equal but oppositely directed forces (F1+ F2= 0), separated by a distance, will form a force “couple”The force couple will produce a torque (T = F1d), causing rotationHowever, there will be no translation because F1and F2will cancel each other out in terms of translationdCalculating TorqueTorque is equal to force times perpendicular distance:T = F  rTorque is also referred to as “moment of force”The perpendicular distance (r) from the axis of rotation to the force is known as the “moment arm”Torque = Force  perp distanceForce (F)perpendicular distance (r)Axis or Center of Rotation}Lever2Calculating TorqueOften the force does not act perpendicular to the lever to which it is appliedHow do you calculate the torque in such cases?Force (F)What is the perpendicular distance (r) in this case?Axis or Center of RotationLeverTwo ways to calculate torque created by a force:Calculating Torqueusing moment arm (r2) of the resultant forceFMUSFMUSr1r1r2T = F  rr2= r1sin , so…T = FMUS (r1 sin )FR= FMUS sin , so…T = (FMUS sin )  r1FRFNRusing moment arm (r1) of the rotary componentMuscle TorqueThe torque produced by a muscle is equal to muscle force (Fmus) times muscle moment arm (r):Tmus= Fmus rThe muscle moment arm is the perpendicular distance from the line of action of the muscle force to the center of rotation of the jointTmus= Fmus rFmusrCenter of Joint RotationMuscle TorqueFR= FMUSFMUSFR(≠ FMUS)FNRIf muscle attaches to bone at an angle, only the component of muscle force perpendicular to the bone (FR: rotary component) contributes to the muscle torque3Example - Tendon RuptureThe lift was a clean and jerk of 175 kg (2.1  his body mass)Rupture occurred as he jerked the weight over-headHe had already won the competition on his prior lift!knee joint torque during the liftZernicke et al. were analyzing weightlifting performance during a competition when a world-class athlete ruptured his patellar tendon during a maximal-effort lift Example - Tendon RuptureThe joint torque (moment) was divided by the estimated patellar tendon moment arm to compute force in the tendon at rupture Given a peak knee extension torque of 560 N m and the approximate moment arm of 0.038 m, the computed tendon force at rupture was > 14,700 NC = joint center of rotation = moment arm distanceM = joint torque (moment)P = patellar tendon force14,700 N  3303 lb  1.7 tons!• Muscle moment arms are generally not constant, but rather change with joint angle• Largest moment arm occurs at an angle of pull ~90• May not correspond to a joint angle of 90• For some muscles(e.g., deltoid) angle ofpull is far less than 90Muscle Moment ArmFMUSMuscle Moment Armbiceps moment arm at its largestrrrmoment arm is smaller with elbow flexed or extended4Net Joint Torque• Net joint torque is the summation of the torques produce by all of the muscles acting simultaneously at a joint• Net torque balances other torques due to:weight of body segments (mg)motion of body segments (ma, Iα)external forces (GRF, wind resistance)• When net joint torque is:same direction as the joint motion = concentricopposite direction of joint motion = eccentricNet Joint Torque• Under normal circumstances, many muscles will be active at a joint simultaneously• Muscles that act as antagonists to the desired movement may also be activee.g., hamstrings active during knee extension• Agonist and antagonist muscle groups:- agonist muscles control direction of movement- antagonist muscle activity may help with fine control and/or enhance movement stabilityNet Joint TorqueKnee extension produced by quadriceps (agonist) activity, with simultaneous hamstring (antagonist) activityExamplequadriceps torque:TQ= 1000 N × 0.05 m = 50 N mhamstring torque:TH= 200 N × 0.06 m = 12 N mnet knee joint torque:TN= T = TQ–THTN= 50 N m – 12 N mTN= 38 N mquadricepstorque (+)hamstringstorque (-)quadricepsforce = 1000 Nmom arm = 0.05 mhamstringsforce = 200 Nmom arm = 0.06 mLevers• Lever - a simple machine that consist of a rigid bar-like body that is free to rotate about an axis• Fulcrum - the point about which the lever can rotate (the axis)• Three classes of levers– First– Second– Third5First Class LeverApplied ForceResistance ForceFulcrumFulcrum is between resistance & applied forceExamples:Second Class LeverApplied ForceResistance ForceFulcrumResistance is between fulcrum & applied forceExamples:Third Class LeverApplied ForceResistance ForceFulcrumApplied force is between resistance & fulcrumExamples:Lever Systemsmechanical advantage = moment arm (force)moment arm (resistance)When force ma > resistance ma:– mechanical advantage > 1.0– low force needed to overcome the resistance– resistance is moved through a limited ROMWhen resistance ma > force ma:– mechanical advantage < 1.0– high force needed to overcome resistance– resistance is moved through a greater ROM6Anatomical Levers• In the human body, most lever systems are third class (mechanical advantage < 1.0)• This arrangement promotes– Large joint range of motion– High joint angular speeds• However, muscle forces must be (greatly!) in excess of the resistance forcesthird class lever systemMuscle (F) moment arms are small, while resistance (R) moments arms tend to be largeStatic EquilibriumA system is in a state of rotational static equilibrium when all of the torques sum to zero (T = 0)How far (d) from the pivot point does the girl on the left have to sit for the system to be in static equilibrium?T = 0(700 N) d  (650 N)(1.5 m)  (825 N)(2.5 m) = 0(700 N) d = 3037.5 Nmd = 4.34 m700 N650 N825 Nd2.5 m1.5 m• The point about which an object’s mass is evenly balanced – The CM of a symmetrical object of uniform density is exactly at the center of the object– When density is not uniform, CM shifts in the direction of greater density• Importance of knowing CM