Chapter 14 Angular Kinetics of Human Movement Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D. © 2014 The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin

14-2 Resistance to Angular Acceleration What is moment of inertia? The inertial property for rotating bodies represents resistance to angular acceleration based on both mass and the distance the mass is distributed from the axis of rotation Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D.

14-3 Resistance to Angular Acceleration r r r r m m m m Axis of rotation Moment of inertia is the sum of the products of each particle’s mass (m) and the radius of rotation (r) for that particle squared. I =  mr 2 Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D.

14-4 Resistance to Angular Acceleration Although both bats have the same mass, bat A is harder to swing than bat B because the weight ring on it is positioned farther from the axis of rotation. Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D. A B

14-5 Resistance to Angular Acceleration What is the radius of gyration? Distance from the axis of rotation to a point where the body’s mass could be concentrated without altering its rotational characteristics Used as the index for mass distribution for calculating moment of inertia: I = mk 2 Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D.

14-6 Resistance to Angular Acceleration Knee angle affects the moment of inertia of the swinging leg with respect to the hip because of changes in the radius of gyration for the lower leg (k 2 ) and foot (k 3 ). k1k1 k2k2 k3k3 k1k1 k2k2 k3k3 Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D.

14-7 Resistance to Angular Acceleration During sprinting, extreme flexion at the knee reduces the moment of inertia of the swinging leg. Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D.

14-8 Resistance to Angular Acceleration Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D. The ratio of muscular strength (ability to produce torque at a joint) to segmental moments of inertia (resistance to rotation at a joint) is important for performance in gymnastic events.

14-9 Resistance to Angular Acceleration Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D. Principal moments of inertia of the human body in different positions with respect to different principal axes: (1) principal axis; (2) moment of inertia (kg m ). 2.

14-10 Angular Momentum What is angular momentum? Quantity of angular motion possessed by a body Measured as the product of moment of inertia and angular velocity: H = I  H = mk 2  Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D.

14-11 Angular Momentum Angular momentum is the sum of the local term (I s  s ) and the remote term (mr 2  g ). H = I s  s + mr 2  g CG CG s ss gg Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D.

14-12 Angular Momentum What is the principle of conservation of angular momentum? The total angular momentum of a given system remains constant in the absence of external torques. H 1 = H 2 (mk 2  ) 1 = (mk 2  ) 2 Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D.

14-13 Angular Momentum When angular momentum is conserved, there is a tradeoff between moment of inertia and angular velocity. (Tuck position = small I, large  ) (Extended position = large I, small  ) Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D. H = I 

14-14 Angular Momentum During the airborne execution of a spike in volleyball, a compensatory rotation of the lower extremity offsets the forceful swinging arm so that total body angular momentum is conserved. Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D. H = I 

14-15 Angular Momentum A skillful diver can rotate 180º or more in the air with zero angular momentum because there is a large discrepancy between the radii of gyration for the upper and lower extremities with respect to the longitudinal axes of these two major body segments. Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D. Axis 1 Axis 2

14-16 Angular Momentum What produces change in angular momentum? Angular impulse - the product of torque and the time interval over which the torque acts: T t =  H T t = (I  ) 2 - (I  ) 1 Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D.

14-17 Angular Momentum Springboard reaction force (F) multiplied by its moment arm from the diver’s CG (d ) creates a torque that generates the angular impulse that produces angular momentum at takeoff. Tt =  H CG d F Backward somersault Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D.

14-18 Angular Momentum The arm swing during takeoff contributes significantly to the diver’s angular momentum. Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D.

14-19 Angular Momentum The surface reaction force is used by the dancer to generate angular momentum during the takeoff in the tour jeté. Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D.

14-20 Angular Analogues of Linear Kinematic Quantities What are the angular equivalents of linear kinematic quantities? Linear Angular Mass (m) Moment of inertia (I = mk 2 ) Force (F) Torque (T = Fd ) Momentum (M=mv) Angular momentum (H=mk 2  ) Impulse (Ft) Angular impulse (Fd t) Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D.

14-21 Angular Analogues of Newton’s Laws What is the angular law of inertia? A rotating body will maintain a state of rest or constant rotational motion unless acted on by an external torque that changes the state. Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D.

14-22 Angular Analogues of Newton’s Laws What is the angular law of acceleration? A net torque causes angular acceleration of a body that is: of a magnitude proportional to the torque in the direction of the torque and inversely proportional to the body’s moment of inertia Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D.

14-23 Angular Analogues of Newton’s Laws What is the angular law of acceleration? T = I  T = mk 2  Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D.

14-24 Angular Analogues of Newton’s Laws What is the angular law of reaction? For every angular action, there is an equal and opposite angular reaction. When one body exerts a torque on a second, the second body exerts a reaction torque that is equal in magnitude and opposite in direction on the first body. Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D.

14-25 Centripetal Force What is centripetal force? (Force directed toward the center of rotation for a body in rotational motion) mv 2 F c = r F c = mr  2 FcFc Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D.

14-26 Centripetal Force Cyclists and runners lean into a curve to offset the torque created by centripetal force acting on the base of support (tires). Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D.

14-27 Centripetal Force Free body diagram of a cyclist on a curve. R is centripetal force. When the cyclist is balanced, summing torques at the cyclist’s CG, (R )(d ) = (R )(d ). Basic Biomechanics, 7 th edition By Susan J. Hall, Ph.D. HV RVRV RHRH RVRV RHRH