CHP Center for Health and Performance DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Part II Angular Kinetics Why do objects move? [email protected] DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Course Objectives • Define moment of inertia • Explain how the human body's moment of inertia may be manipulated • Explain Newton's first law of motion as it applies to angular motion • Explain Newton's second law of motion as it applies to angular motion • Explain Newton's third law of motion as it applies to angular motion • Define angular impulse • Define angular momentum • Explain the relationship between angular impulse and angular momentum To create rotation 1 Linear movement – no rotation 2. Linear movement and rotation 3. Rotation DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Moment of Inertia • Moment of Inertia (I) is the angular equivalent of mass • Affected by the mass of an object & how it’s distributed relative to the axis of rotation • To calculate Inertia we must do a certain amount of assumptions • A heavier bat is harder to swing than a lighter bat • A longer bat is harder to swing than a shorter one DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Mathematical Definition of Moment of Inertia Moment of inertia - (I) = moment of inertia about axis a through the center of gravity Σ = summation symbol = mass of particle i = radius (distance) from particle i to axis of rotation through the center of gravity Formula (7.1) Formula (7.2) DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Mathematical Definition of Angular Inertia A segment’s radius of gyration (k) represents the distance at which all of its mass can be said to act Radius of gyration (k) is the angular equivalent of center of mass Used to calculate with for example body segments Formula (7.2) DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Different Axes Rotation can occur about different axes When mass is distributed closer to axis the moment of inertia is lower (see middle image) There are three main axes: 1. Frontal 2. Transverse 3. Longitudinal DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Moments of Inertia about Eccentric Axes • If an object is free and unconstrained to rotate about any axis, it will rotate about an axis through it’s centre of gravity • Eccentric axis does not pass through the implements’s center of gravity • Radius is the distance between the cg to the eccentric axis (where it rotates) • Formula (7.3) DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Calculation example A 0.5 kg lacrosse stick has a moment of inertia about a transverse axis through its center of gravity of 0.10 kg·m². When a player swings the stick at an opponent, the axis of rotation is through the end of the stick, 0.8 m from the center of gravity of the stick. What is the moment of inertia relative to this swing axis? Solution: Step 1: Identify the known quantities. m = 0.5 kg Icg = 0.10 kg·m² r = 0.80 m Step 2: Identify the unknown variable to solve for. Iswing = ? Step 3: Search for equations with the known and unknown variables (equation 7.3). Ib = Icg + mr² Iswing = Icg + mr² Step 4: Substitute the known quantities and solve for the unknown variable. Iswing = 0.10 kg·m² + (0.5 kg)(0.8 m)² = 0.42 kg·m² DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Moments of Inertia about Different Axes An object can only have one linear inertia (mass), but it can have more than one angular inertia because it can rotate about many different axes. Spin a book about an axis of rotation parallel to its spine. Is it easier or harder to spin about this axis than about an axis perpendicular to its cover? DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Manipulating the Moments of Inertia of the Human Body Flexing the knee and hip of the recovery leg reduces the moment of inertia of the leg about the sprinter’s hip joint. DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Moments of Inertia & Linear Velocity Golf club designers longer length of a club gives larger MoI. Longer drivers are lighter and less massive then shorter ones. The designers have to consider the Balance of length and mass To avoid twisting the club during swing designers have distributed more of the mass towards the heel and toe of the club head DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Angular Momentum • Linear momentum quantifies the linear motion of an object • Angular momentum quantifies the angular motion of an object. DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Angular Momentum of a Rigid Body Formula (7.4) • Angular Momentum (H) is a vector quantity size & direction • Angular momentum is the angular version of linear momentum, so it is the product of the angular version of mass (moment of inertia) times the angular version of linear velocity (angular velocity). • Units are kilogram meters squared per second DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Angular Interpretation of Newton’s First Law of Motion 1. Law of inertia An object will remain at rest or continue moving at constant speed in a straight line, unless a force is exerted on it. DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Angular Interpretation of Newton’s First Law of Motion Formula (7.7) This is the equation for Angular Momentum of the human body. It is constant unless external torques act on it. The body’s MoI is variable and can be changed by alterling limb positions, the body’s Angular Velocity also changes to accomodate the changes in the MoI. DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Angular Interpretation of Newton’s First Law of Motion • Conservation of Angular Momentum can be described by a figure skater doing a spin • Large moment of inertia about the longitudinal axis due to arms and one leg held away from body • The figure skater brings them closer to the body reducing the moment of inertia which increases the angual velocity (see formula) DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Angular Interpretation of Newton’s Second Law of Motion 2. Law of acceleration Any time an object starts, stops, speeds up, slows down, or changes direction a net external force is acting to cause the acceleration. DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Angular Interpretation of Newton’s Second Law of Motion • The change in angular momentum of an object is proportional to the net external torque exerted on it, and this change is in the direction of the net external torque. • Rigid object with constant moment of inertia • From the Linear equation (F=m*a) we substitute Torque for Force, Formula (7.9) Angular acceleration for acceleration and Moment of Inertia for mass. DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Angular Interpretation of Newton’s Second Law of Motion • Nonrigid object with a variable MoI • Note the line over T, this = average • As with the linear version of Newton’s second law, Formula (7.10) the angular version indicates only what happens at an instant in time when a net torque acts. DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Angular Impulse & Angular Momentum • Angular impulse is the change in angular momentum • Angular Impulse is the product of Torque and the time that Torque is applied • In activities in which the goal is to spin very fast, the athlete should start with a large moment of inertia during the torque production period so that the duration of torque application is maximized Formula (7.11) Angular Impulse Change in Angular Momentum DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Angular Interpretation of Newton’s Third Law of Motion 3. Law of action-reaction If an object exerts a force on another object, the other object exerts the same force on the first object but in the opposite direction. DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Angular Interpretation of Newton’s Third Law of Motion • How does a long pole help him keep his balance? • One way is that it can lower the center of gravity of the acrobat and pole system as well as increase its moment of inertia. • Think of Newton’s 3rd Law • What happens if he falls clockwise to his left? • If he exerted a clockwise torque on the pole and moved it clockwise in the same direction of his fall, the pole would exert an equal but oppositely directed torque on him. DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Summary • The basics of angular kinetics, are explained by angular interpretations of Newton’s laws of motion • Moment of Inertia (I) The property of an object to resist changes in its angular motion • Radius of gyration (k) length dimension representing, on average, how far an object’s mass is located from an axis of rotation • Angular Momentum (H) is a measure of an object’s motion. It is the product of MoI and AV • Angular Impulse change in Angular Momentum DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Summary • Newtons 1st Law objects do not change their angular momentum unless a net external torque acts on them • Newtons 2nd Law A rigid object will accelerate angularly in the direction of the net external torque, and its angular acceleration will be inversely related to its moment of inertia • Newtons 3rd Law For every torque, there is an equal torque acting on another object but in the opposite direction DEPARTMENT OF FOOD AND NUTRITION, AND SPORT SCIENCE Preparation for Examination Linear variable Angular variable Linear velocity v Angular velocity ω Linear acceleration a Angular acceleration α Mass m Moment of Inertia I Center of mass Radius of gyration k Force F Torque τ M Linear momentum ρ Angular momentum H.