Lecture 16
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1) Angular kinetics is the study of forces that cause rotation or torques. Torque is a measure of how much a force causes an object to rotate and depends on the force magnitude and its moment arm. 2) The moment arm is the distance from the axis of rotation to where the force is applied. Torque is calculated by multiplying the force by the moment arm. 3) Resultant joint torque is the single torque that has the same rotational effect as all the individual torques acting on a joint. It provides a simplified view of which muscle groups are most active at a joint.
Lecture 16
- 1. Angular Kinetics Torque Kinetics • study of the relationship between the forces acting on a system and the motion of the system Angular Motion (Rotation) Objectives: • All points in an object or system move in a circle • Define angular kinetics about a single axis of rotation. All points move • Define and learn to compute moment through the same angle in the same time arms, torque, and resultant torques Linear Kinetics • Introduction to resultant joint torques, • The kinetics of particles, objects, or systems anatomical torque descriptions, and force undergoing rotation couples Torque (or Moment) Line of Action • A measure of the extent to which a force will cause • The line of action of a force is the imaginary line an object to rotate about a specific axis that extends from the force vector in both directions • A net force applied through the center of mass produces translation • A net force applied away from the center of mass line of action of F (i.e. an eccentric force) produces both translation and rotation F F F 1
- 2. Moment Arm Computing a Moment Arm • Shortest distance from a force’s line of action to the • Need to know: axis of rotation – distance (d) from axis of rotation to point at which • Moment arm is always perpendicular to the line of force is applied action and passes through the axis of rotation – angle (θ) at which force is applied • Use trigonometry to compute moment arm (d⊥) line of action of F d⊥ = d sin θ 90° moment arm of F F axis of rotation F θ axis of rotation d Moment Arm Examples Computing Torque • Torque has: axis of rotation d⊥ = d d – a magnitude – a direction (+ or –) θ – a specific axis of rotation d⊥= d sin θ F F • The magnitude of the torque (T) produced by a force is the product of the force’s magnitude (F) times the force’s moment arm (d⊥): axis of T = F d⊥ T = F d⊥ rotation d⊥ = d sin θ d⊥ = 0 θ F • Units: F d F d – English: foot-pounds (ft-lb) d⊥ – SI : Newton-meters (Nm) 2
- 3. Direction of a Torque Example Problem #1 • Positive torque : acts counterclockwise about the Shown below are 4 muscles acting across a joint. axis of rotation. Which muscles have the largest and smallest • Negative torque : acts clockwise about the axis force? moment arm? torque magnitude? • Determine direction using the right hand rule: 3 100 N 2 100 N – Place right hand on force vector, fingers towards arrow tip joint 50° – Curl fingers around axis of rotation 20° – Torque acts in direction that fingers are curled 60° 1 150 N 0.01 m T>0 T<0 limb segment 0.02 m 90° axis of F F 0.04 m rotation 4 35 N Torque Composition Resultant Joint Torque • Process of determining a single resultant (or net) torque from two or more torques. • The effects of all forces acting about a joint can be • Performed by adding the torques together, taking the duplicated exactly by the combination of: sign (direction) of the torque into account – A resultant joint force acting at the joint center • Resultant torque has same effect on rotation as the – A resultant joint torque acting about the axis of individual torques acting together rotation through the joint center T3 T net = |T 1| – |T 2| + |T 3| • Resultant joint force is the vector composition of all F3 forces acting across a joint. T1 • Resultant joint torque is the composition of the axis of torques produced about the joint axis by these forces. rotation T2 • Note: Forces that do not act across the joint (e.g. F1 F2 weight) are not included in the resultant joint force or torque. Note: |T| = magnitude of torque T (≥ 0) 3
- 4. Example Use of Resultant Joint Torque • Typically, joint contact force, muscle forces, ligament Fcontact forces, etc. cannot be determined individually Tresultant Fresultant • We can compute resultant joint forces and torques Facl d ⊥acl based on data measured external to the body knee joint center • Except near the limits of the anatomical range of d ⊥quads Fquads Fquads Fcontact motion, the main contributors to the resultant joint d ⊥hams Fhams torque are the muscles tibia • The resultant joint torque provides a simplified picture Facl Fhams of which muscle groups are most active about a joint Tresultant = (Fquads d⊥quads) + (F acl d⊥acl) – (Fhams d⊥hams) Example Problem #2 Force Couple Shown is a forearm with 2 elbow flexors and 1 • For pure rotation about the center of mass, the center elbow extensor. Find the resultant joint torque for of mass must remain stationary from Newton’s 1st the 3 combinations of forces shown in the table: law, the net force on the object must equal zero • Force couple : Two forces of equal magnitude, Ft 0 0 32 Ft Fcontact Fbi applied in opposite directions. Produce pure rotation Fbr about the center of mass. Fcontact 8 3.2 46.4 T = F (d⊥1 + d ⊥2 ) 30° F Fbi 16 10 20 0.025 m 0.05 m d⊥2 0.10 m W=8N d⊥1 Fbr 0 2.4 4.8 ΣF=0 0.25 m F RJT Force Couple Net Effect 4
- 5. Anatomical Torques • Positive & negative torques depend on frame of reference chosen: y y Fquad Fquad knee knee T>0 T<0 x x • To avoid this problem, joint torques are typically described by the joint motion that would occur if the segment moved in the direction of the torque (e.g. Fquad produces a knee extension torque) 5