Newton s Laws of Motion Angular Analogues Newton s First Law A rotating body will maintain a state of constant rotational motion unless acted on by an external torque Resistance to Acceleration Resistance to linear acceleration Mass Resistance to angular acceleration Moment of Inertia Moment of inertia is determined by mass and mass distribution relative to axis of rotation Determining Moment of Inertia Determining Moment of Inertia Moment of inertia is the sum of the products of all the mass elements of an object and the square of the distances of the mass elements from the axis of rotation A more practical approach Iaxis miri2 where is an experimentally determined length known as the radius of gyration that applies to the whole object r1 m1 axis Iaxis mbody 2 m2 r2 Iaxis m1r12 m2r22 mnrn2 is not the same as the distance to segment CM magnitude of is different for different axes of rotation Unit for moment of inertia consists of unit of mass multiplied by unit of length squared kg m2 1 Human Body Moment of Inertia Can be computed for Applications Chocking up on a baseball softball bat individual body segments forearm thigh etc the whole body Tuck vs layout position of a diver or gymnast Typically expressed for an axis through the center of mass or through the proximal or distal end layout tuck Position of a runner s leg during the swing phase moment of inertia of the whole body about different axes of rotation hip 13 5 kg m2 11 5 kg m2 4 5 kg m2 1 5 kg m2 2 5 kg m2 Angular Momentum Angular Momentum Momentum A 57 g tennis ball is struck by a racket giving it an angular momentum of 6 8 10 4 kg m2 s If the radius of gyration of the ball is 2 4 cm how fast will it be spinning For linear motion For angular motion Or M mv H I topspin forehand shot H m 2 Factors that affect angular momentum mass of the object m distribution of mass relative to axis of rotation angular velocity of the object Units for angular momentum kg m2 s H I m 2 0 00068 kg m2 s 0 057 kg 0 024 m 2 0 00068 kg m2 s 0 000033 kg m2 20 6 rad s 1180 deg s 3 28 rev s 2 Conservation of Angular Momentum The total angular momentum of a given system remains constant in the absence of external torques H is constant Conservation of Angular Momentum A 60 kg diver is in a layout position with radius of gyration of 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 reducing his radius of gyration to 0 25 m However I and can change so long as their product remains constant recall that H I Conservation of Angular Momentum First find H when diver leaves the board H m 2 H 60 kg 0 5 m 2 4 rad s 60 kg m2 s H is constant so now find when is reduced to 0 25 m 60 kg m2 s 60 kg 0 25 m 2 0 5 m 0 25 m Transfer of Angular Momentum Total body angular momentum is constant while the body is airborne no external torques However even with total angular momentum constant it is possible to Transfer angular velocity from one part of the body to another Change the body axis of rotation 16 rad s 3 Transfer of Angular Momentum The Falling Cat Phenomenon Early in the dive H is concentrated in upper body while later in the dive H is concentrated in the lower body H is constant during the fall but the cat always rights itself How does it do this HTOTAL constant HLOWER changes HUPPER changes Newton s Laws of Motion Angular Analogues Newton s Second Law A net torque produces angular acceleration of a body that is directly proportional to the magnitude of the torque in the same direction as the torque and inversely proportional to the body s moment of inertia T I compare with F m a and more properly TP IP Torque Angular Acceleration How much torque does a board reaction force of 850 N create about the CM of a diver r 0 67 m What is the angular acceleration if the diver s moment of inertia about his CM is 10 5 kg m2 r F T F r 850 N 0 67 m 569 5 N m Tcm Icm T I 569 5 N m 10 5 kg m2 54 2 rad s2 3000 deg s2 4 Angular Impulse Momentum Angular Impulse Momentum Depends on the magnitude and direction of the applied torque and the duration time of torque application A person is swinging their leg forward at 200 s 3 5 rad s If the leg has a moment of inertia about the hip of 0 7 kg m2 how much torque must be generated by the hip extensor muscles to stop the leg in 0 4 s Linear Impulse force time Ft Angular Impulse torque time Tt Impulse momentum relationship Linear Angular Ft M Tt H Ft mv f mv i Tt I f I i T t I f I i T T t 0 7 kg m2 0 0 7 kg m2 3 5 rad s T t 2 45 kg m2 s T 2 45 kg m2 s 0 4 s 6 13 N m Angular Work Power Energy Angular Work Power Energy When a torque T acts over an angular displacement mechanical work is done W T must be in rad While doing dumbbell curls a person generated an elbow joint flexion moment of 25 6 N m while flexing their elbow over a 120 range how much mechanical work was done Mechanical power is the rate at which a torque does work on a system It took the person 0 78 s to lift the weight from A to B in the diagram what was the mechanical power P W t or W T P T W 25 6 N m 2 09 rad 53 5 J must be in rad s Note that the units for angular work and power are J N m and W J s respectively the same as for linear work and power P W t P 53 5 J 0 78 s 68 6 W 5 Angular Work Power Energy Total mechanical energy TE is the sum of kinetic energy KE and potential PE energy TE KE PE Kinetic energy can be due to linear TKE and or angular RKE motion TKE m v2 RKE I 2 Newton s Laws of Motion Angular Analogues Newton s Third Law For every torque exerted by one body on another there is an equal and opposite torque exerted by the second body on the first Therefore the total mechanical energy of a system is given by TE m v2 I 2 m g h Action Reaction Torques In running the arms swing in …