KIN 3309 1nd Edition Lecture 19 Outline of Last Lecture I Outline II Linear vs Angular Motion III Angular Kinetics IV Torque V Moment Arm VI Manipulating Torque VII Levers VIII General Lever Mechanics IX Levers Class 1 X Levers Class 2 XI Levers Class 3 XII Rotation and Leverage XIII Eccentric off axis Forces Create Torques XIV Force Couples XV Linear vs Angular Terms XVI Rotational Analog of Newton s First Law XVII Moment of Inertia XVIII Radius of Gyration These notes represent a detailed interpretation of the professor s lecture GradeBuddy is best used as a supplement to your own notes not as a substitute XIX Parallel Axis of Theorem XX Angular Momentum XXI Conservation of Angular Momentum XXII Rotational Analog of Newton s Second Law XXIII Rotational Analog of Newton s Third Law XXIV Newton s Laws of Motion XXV Example 1 XXVI Example 2 XXVII Example 3 XXVIII Quiz Outline of Current Lecture I Torque II Levers III Rotational Analog of Newton s First Law IV Moment of Inertia V Angular Momentum VI Rotational Analog of Newton s Second Law VII Rotational Analog of Newton s Third Law VIII Conservation of Angular Momentum IX Outline X Centers of Mass Gravity XI Total Body Center of Mass XII Linear vs Angular Kinetic Analysis XIII Dynamic Analysis XIV Angular Impulse XV Angular Work XVI Angular Power XVII Angular Energy XVIII Work Energy Theorem XIX Quiz Current Lecture I Torque a Torque is often referred to as rotary force and is the angular equivalent of linear force b Torque is the product of force F and the perpendicular distance d from the force s line of action to the axis of rotation i Vector ii The units of torque are newton meters Nm II Levers a Three parts i Fulcrum provides the point of rotation ii Resistance the load you are trying to move iii Effort the force you are applying III Rotational Analog of Newton s First Law a A rotating body will continue in a state of uniform angular motion unless acted upon by an external torque b A body s resistance to a change in angular motion is called its moment of inertia IV Moment of Inertia a The inertial property for rotating bodies b Represents resistance to angular acceleration c Based on both mass and the distance the mass is distributed from the axis of rotation V Angular Momentum a The quality of angular motion that a body possess b H I x omega i Angular momentum is represented by H ii Angular momentum has units of kg x m2 s iii H is the rotational equivalent of linear momentum p mv VI Rotational Analog of Newton s Second Law a The rate of change of angular momentum of a body is proportional to the torque causing it and the change takes place in the direction in which the torque acts VII Rotational Analog of Newton s Third Law a For every torque that is exerted by one body on another there is an equal and opposite torque exerted by the second body on the first VIII Conservation of Angular Momentum a When gravity is the only external force acting on an object the angular momentum remains constant i This is because the gravitational force acts through the center of mass of an object point of rotation and therefore does not produce a torque b This allows divers and gymnasts to manipulate their angular velocities by changing their moments of inertia c When angular momentum is conserved there is a tradeoff between moment of inertia and angular velocity i Tuck position small I large omega ii Extended position large I small omega IX Outline a Torques i Static and dynamic analysis b Angular impulse c Angular work d Angular power e Angular energy f Work Energy Theorem Centers of Mass Gravity a The center of mass CM is the theoretical point about which the body mass is evenly distributed b CM is used interchangeably with more restrictive center of gravity CG which is the theoretical point which the sum of all torques due to gravity equal zero X XI Total Body Center of Mass a You can move your center of mass by redistributing your weight i e changing your body posture XII XIII Linear vs Angular Kinetic Analysis a Static Analysis i Linear 1 Systems at rest or constant velocity ii Angular 1 Applies to systems at rest or at constant b Dynamic Analysis i Linear 1 Systems in motion accelerating of decelerating ii Angular 1 Applies to systems in motion undergoing angular acceleration Dynamic Analysis a If angular acceleration is significant then a dynamic analysis is required b Biomechanists usually evaluate each segment from a distal end and work systematically up each segment link i This is known as an inverse dynamic analysis XIV Angular Impulse a Torque applied over a period of time i Impulse is vector quantity the direction is the same as the direction of the force ii Symbolized as L iii Units kgm2 s b Impulse change in momentum XV Angular Work a Torque applied over an angular displacement i Work is a scalar quantity b Positive work concentric muscle actions c Negative work eccentric muscle actions XVI Angular Power a The rate of change in angular work i Angular power is the rate of change in angular work ii Units of power are watts J s iii Power is a scalar quantity XVII Angular Energy a The capacity to do work due to rotational motion XVIII XIX Work Energy Theorem a Work angular change in rotational energy b This again assumes that no work is done to deform the object i e no strain energy is stored Quiz a The moment arm of the biceps brachii muscle about the elbow joint is largest when the angle at the elbow joint is approximately i 180 degrees ii 150 degrees iii 120 degrees iv 90 degrees b For a body to be in static equilibrium i All forces acting on the body must sum to zero ii All torques acting on the body must sum to zero iii The body must be stationary iv All of the above c Angular impulse is the product of i Force and the time that force is applied this is linear impulse ii Torque and the time that torque is applied T x delta T iii Angular inertia and angular momentum iv Mass and angular velocity v Force and torque