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Last Lecture Pendulums and Kinetic Energy of rotation Today Energy and Momentum of rotation Important Concepts Equations for angular motion are mostly identical to those for linear motion with the names of the variables changed. Kinetic energy of rotation adds a new term to the same energy equation, it does not add a new equation. Momentum of rotation gives an additional equation 8.01L IAP 2007 1/18/2006 Important Reminders Recitation problem solving in class tomorrow. Pset due tomorrow at 4:30 pm. Last MasteringPhysics due next Thursday at 10 pm (this is a change from the posted syllabus). Last Experiment is next Tuesday. Initial Final Exam info (sample problems and formula sheet) are now posted. Last MasteringPhysics assignment also includes many sample problems. 8.01L IAP 2007 1/18/2006 Torque Checklist Make a careful drawing showing where forces act Clearly indicate what axis you are using Clearly indicate whether CW or CCW is positive For each force: If force acts at axis or points to or away from axis, =0 Draw (imaginary) line from axis to point force acts. If distance and angle are clear from the geometry =Frsin() Draw (imaginary) line parallel to the force. If distance from axis measured perpendicular to this line (lever arm) is clear, then the torque is the force times this distance Don’t forget CW versus CCW, is the torque + or 8.01L IAP 2007 1/18/2006 Friction on a Circular Object (by request) For a circular object, friction is mostly identical to friction on a non-circular object: If slipping, then f N If rolling without slipping, then f N Friction acts exactly at the one point of contact and is tangential, i.e. perpendicular to the radius One peculiarity: Sliding along a surface, friction does negative work Rolling without slipping, friction does zero work 8.01L IAP 2007 1/18/2006 Kinetic Energy with Rotation Adds a new term not a new equation! Rotation around any fixed pivot: KE 12 I pivot 2 2 Moving and rotating: KE ICM MTot vCM 1 2 8.01L IAP 2007 2 1 2 1/18/2006 Everything you need to know for Linear & Rotational Dynamics r r F Ma r r I This is true for any fixed axis and for an axis through the center of mass, even if the object moves or accelerates. Rolling without slipping: v R a R f N Friction does NOT do work! Rolling with slipping: v R a R f N Friction does work, usually negative. Rarely solvable without using force and torque equations! 8.01L IAP 2007 1/18/2006 Kinematics Variables Position x Angle Velocity v Angular velocity Acceleration a Angular acceleration Force F Torque Mass M Moment of Inertia I Momentum p Angular Momentum L d dt 8.01L IAP 2007 d d 2 2 dt dt 1/18/2006 Angular Momentum Conserved when external torques are zero or when you look over a very short period of time. True for any fixed axis and for the center of mass r Formula we will use is simple: L I Vector nature (CW or CCW) is still important r r Point particle: L r p Conservation of angular momentum is a separate equation from conservation r of linear momentum dL Angular impulse: dt 8.01L IAP 2007 r r L dt 1/18/2006