Physics 207, Lecture 16, Oct. 29Chap. 13: Rotational DynamicsSlide 3Rotational Dynamics: A child’s toy, a physics playground or a student’s nightmareRotational VariablesRotational Variables...Summary (with comparison to 1-D kinematics)Slide 9Example: Wheel And RopeSlide 11System of Particles (Distributed Mass):Slide 13System of Particles: Center of MassSlide 15Sample calculation:Slide 17Rotational Dynamics: What makes it spin?Lecture 16, Exercise 1 TorqueSlide 20Slide 21TOT = m r2 a and inertiaCalculating Moment of InertiaCalculating Moment of Inertia...Lecture 16, Home Exercise Moment of InertiaLecture 16, Home Exercise Moment of InertiaSlide 27Moments of InertiaMoments of Inertia...Slide 30Rotation & Kinetic EnergySlide 32Lecture 16, Exercise 2 Rotational Kinetic EnergySlide 34Rotation & Kinetic Energy...Moment of Inertia and Rotational EnergyWork (in rotational motion)Work & Kinetic Energy:Lecture 16, Home exercise Work & EnergyLecture 16, Home exercise Work & EnergyExample: Rotating RodSlide 42Slide 43Slide 44Connection with CM motionConnection with CM motion...Rolling MotionExample : Rolling MotionSlide 49Slide 50MotionAngular Momentum:Example: Two DisksSlide 54Lecture 16, Oct. 29Example: Bullet hitting stickExample: Throwing ball from stoolSlide 58An example: Neutron Star rotationAngular Momentum as a Fundamental QuantityFundamental Angular MomentumIntrinsic Angular MomentumAngular Momentum of a MoleculeAngular Momentum of a Molecule (It heats the water in a microwave over)Physics 207: Lecture 16, Pg 1Physics 207, Physics 207, Lecture 16, Oct. 29Lecture 16, Oct. 29Agenda: Chapter 13Agenda: Chapter 13Center of Mass Center of Mass TorqueTorqueMoment of InertiaMoment of InertiaRotational EnergyRotational EnergyRotational MomentumRotational MomentumAssignment: Assignment: Wednesday is an exam review session, Exam will be Wednesday is an exam review session, Exam will be held in rooms B102 &held in rooms B102 & B130 in Van Vleck at 7:15 PMMP Homework 7, Ch. 11, 5 problems, MP Homework 7, Ch. 11, 5 problems, NOTE: Due Wednesday at 4 PMNOTE: Due Wednesday at 4 PMMP Homework 7A, Ch. 13, 5 problems, available soonMP Homework 7A, Ch. 13, 5 problems, available soonPhysics 207: Lecture 16, Pg 2Chap. 13: Rotational DynamicsChap. 13: Rotational DynamicsUp until now rotation has been only in terms of circular motion with ac = v2 / R and | aT | = d| v | / dtRotation is common in the world around us.Many ideas developed for translational motion are transferable.Physics 207: Lecture 16, Pg 3Conservation of angular momentum has consequencesConservation of angular momentum has consequencesHow does one describe rotation (magnitude and direction)?Physics 207: Lecture 16, Pg 4Rotational Dynamics: A child’s toy, a physics Rotational Dynamics: A child’s toy, a physics playground or a student’s nightmareplayground or a student’s nightmare A merry-go-round is spinning and we run and jump on it. What does it do?We are standing on the rim and our “friends” spin it faster. What happens to us?We are standing on the rim a walk towards the center. Does anything change?Physics 207: Lecture 16, Pg 5Rotational VariablesRotational VariablesRotation about a fixed axis: Consider a disk rotating aboutan axis through its center:]How do we describe the motion:(Analogous to the linear case ) R (rad/s) 2TangentialvTdtddtdxvPhysics 207: Lecture 16, Pg 6Rotational Variables...Rotational Variables...Recall: At a point a distance R away from the axis of rotation, the tangential motion: x = R v = R a = R Rv = Rx rad)in position (angular 21rad/s)in elocity (angular v )rad/sin accelation(angular constant 20002tttPhysics 207: Lecture 16, Pg 7Summary Summary (with comparison to 1-D kinematics)(with comparison to 1-D kinematics) Angular Linearconstant0t 0 0212t tconstantaat0vv20021v attxx And for a point at a distance R from the rotation axis:x = R v = Ra = RPhysics 207: Lecture 16, Pg 9Lecture 15, Lecture 15, Exercise 5Exercise 5Rotational DefinitionsRotational DefinitionsA goofy friend sees a disk spinning and says “Ooh, look! There’s a wheel with a negative and with antiparallel and !” Which of the following is a true statement about the wheel?(A)(A) The wheel is spinning counter-clockwise and slowing down.(B) (B) The wheel is spinning counter-clockwise and speeding up.(C) (C) The wheel is spinning clockwise and slowing down.(D) The wheel is spinning clockwise and speeding up Physics 207: Lecture 16, Pg 10Example: Wheel And RopeExample: Wheel And RopeA wheel with radius r = 0.4 m rotates freely about a fixed axle. There is a rope wound around the wheel. Starting from rest at t = 0, the rope is pulled such that it has a constant acceleration a = 4m/s2. How many revolutions has the wheel made after 10 seconds? (One revolution = 2 radians)aarPhysics 207: Lecture 16, Pg 11Example: Wheel And RopeExample: Wheel And RopeA wheel with radius r = 0.4 m rotates freely about a fixed axle. There is a rope wound around the wheel. Starting from rest at t = 0, the rope is pulled such that it has a constant acceleration a = 4 m/s2. How many revolutions has the wheel made after 10 seconds? (One revolution = 2 radians)Revolutions = R = ( and a = rt + ½tR = ( + ½a/rtR = (0.5 x 10 x 100) / 6.28aarPhysics 207: Lecture 16, Pg 12System of Particles (Distributed Mass):System of Particles (Distributed Mass):Until now, we have considered the behavior of very simple systems (one or two masses).But real objects have distributed mass !For example, consider a simple rotating disk and 2 equal mass m plugs at distances r and 2r.Compare the velocities and kinetic energies at these two points.12Physics 207: Lecture 16, Pg 13System of Particles (Distributed Mass):System of Particles (Distributed Mass):An extended solid object (like a disk) can be thought of as a collection of parts. The motion of each
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