1EXS 387 - BiomechanicsDr. MoranThursday April 12, 2007Spring 2007Moment of InertiaEXS 387 2Lecture Objectives• Moment of Inertia (MOI)• Radius of Gyration• Determining• Human Body MOI• Angular Momentum• Conservation• Transfer2EXS 387 3Review QuestionsFor Further Study• What does choking up on a baseball bat accomplish? Answer in terms of moment of inertia and axis of rotation.• Explain, in terms of MOI, why adolescents gymnasts are typically at an advantage in terms of rotations as compared to when these same athletes are fully mature.• A volleyball player jumps into the air to spike the ball. She leaves the ground without any angular momentum (H=0). As she swings her arm forward to spike the ball, what happens to her legs if her trunk and non-hitting arm do not rotate?• What factors affect angular momentum? Which has the greatest effect on H?• P478 #5• P479 #2 (good for PE teachers), 3EXS 387 4What do we remember?• Which of the following is not an advantage of the human body utilizing 3rdclass levers?(a) streamline design(b) angular speed(c) large range of motion possible(d) increased mechanical advantage3EXS 387 5Moment of InertiaResistance To Rotate• Def: the body’s tendency to resist acceleration• Remember that inertia is proportional to its _______• Closer MASS is distributed to axis of rotation the easier it is to ________• Rotation Resistance dependent on ______ and its ___________________Example: the ease of swinging the leg (rotating at hip) during walking is dependent on whether the knee is flexed or extendedEXS 387 6Moment of Inertia(determination)mkmrI =Σ=2k = radius of gyrationSI unit = kg m2Moment of inertia is DEPENDENT on the4EXS 387 7Radius of GyrationRepresents the object’s mass distribution with respect to a given axis of rotation to a point at which the mass of the body can theoretically be concentratedIt is always longer than the distance to the CGEXS 387 8Moment of InertiaHuman Body• Rotation can occur about different axes•5EXS 387 9• Rotation occurs about three principle axes•• Segment MOI change with Moment of InertiaWhole BodyEXS 387 10Linear vs Angular• Comparisons between LINEAR and ANGULAR• MASS  MOMENT OF INERTIA• FORCE  TORQUE (MOMENT)• MOMENTUM  ANGULAR MOMENTUM• NEWTON’S LAWS  NEWTON’S LAW (angular version)LINEAR ANGULAR6EXS 387 11Angular Momentum• Def: the quantity of ANGULAR motion that a body possesses» Represented as _____• LINEAR  M = mv• ANGULAR H = or H = EXS 387 12Angular MomentumFactors Affecting• Mass• Mass Distribution with respect to axis of rotation••7EXS 387 13Conservation of Angular Momentum• In the of external torques (off the ground)• The total angular momentum of a given system remains constant inthe absence of external torques.• Useful in any analysis of airborne athletes (divers, gymnasts, etc.)•EXS 387 14Diving ExampleConservation of H• A 20 kg diver is positioned so that his radius of gyration is 0.5 m as he leaves the board with an angular velocity of 4 rad/s. What is the diver’s angular velocity when he assumes a tuck position, altering his radius of gyration to 0.25 m?Position 2Position 18EXS 387 15Transfer of Angular Momentum• Can one transfer angular velocity from one principal axis to another while airborne?• Think about a diver – how do they incorporate a twist(s) into their dives?• TECHNIQUES– Asymmetrical positioning of the arms–– Cat