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Slide 1AnnouncementsRotational KinematicsAngular Displacement, Velocity, and AccelerationTorque and Vector ProductMore Properties of Vector ProductMoment of InertiaTotal Kinetic Energy of a Rolling BodyAngular Momentum of a ParticleAngular Momentum of a Rotating Rigid BodyConservation of Angular MomentumSimilarity Between Linear and Rotational MotionsConditions for EquilibriumHow do we solve equilibrium problems?Fluid and PressurePascal’s Principle and HydraulicsAbsolute and Relative PressureBuoyant Forces and Archimedes’ PrincipleMore Archimedes’ PrincipleSlide 20Bernoulli’s Equation cont’dSimple Harmonic MotionEquation of Simple Harmonic MotionVibration or Oscillation PropertiesSimple Block-Spring SystemEnergy of the Simple Harmonic OscillatorCongratulations!!!!Wednesday, Dec. 1, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu11. Review2. Problem solving sessionPHYS 1443 – Section 003Lecture #25Wednesday, Dec. 1, 2004Dr. Jaehoon YuFinal Exam, Monday, Dec. 6!!Homework #12 is due midnight, Friday, Dec. 3, 2004!!Wednesday, Dec. 1, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu2Announcements•Final Exam–Date: Monday, Dec. 6–Time: 11:00am – 12:30pm–Location: SH103–Covers: CH 10 – CH 14Wednesday, Dec. 1, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu3Rotational KinematicsThe first type of motion we have learned in linear kinematics was under a constant acceleration. We will learn about the rotational motion under constant angular acceleration, because these are the simplest motions in both cases.fJust like the case in linear motion, one can obtainAngular Speed under constant angular acceleration:Angular displacement under constant angular acceleration:fOne can also obtain 2fti221ttii ifi 22Wednesday, Dec. 1, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu4Using what we have learned in the previous slide, how would you define the angular displacement?Angular Displacement, Velocity, and AccelerationHow about the average angular speed?And the instantaneous angular speed?By the same token, the average angular accelerationAnd the instantaneous angular acceleration?When rotating about a fixed axis, every particle on a rigid object rotates through the same angle and has the same angular speed and angular acceleration.ififififtttttlim0dtdififtttttlim0dtdUnit? rad/sUnit? rad/sUnit? rad/s2Unit? rad/s2Wednesday, Dec. 1, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu5xyzOTorque and Vector ProductThe magnitude of torque given to the disk by the force F isLet’s consider a disk fixed onto the origin O and the force F exerts on the point p. What happens?sinFrBAC The disk will start rotating counter clockwise about the Z axisThe above quantity is called Vector product or Cross productFrxFrpBut torque is a vector quantity, what is the direction? How is torque expressed mathematically? Fr What is the direction? The direction of the torque follows the right-hand rule!!What is the result of a vector product?Another vectorWhat is another vector operation we’ve learned?Scalar productcosBABAC Result? A scalarsinBABAC Wednesday, Dec. 1, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu6More Properties of Vector ProductThe relationship between unit vectors, kji and ,kkjjii BAVector product of two vectors can be expressed in the following determinant form 0ji ij kkj jk iik ki jzyxzyxBBBAAAkjizyzyBBAAizxzxBBAAjyxyxBBAAk iBABAyzzy  jBABAxzzx kBABAxyyxWednesday, Dec. 1, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu7Moment of Inertia Rotational Inertia:What are the dimension and unit of Moment of Inertia?iiirmI22mkg  2MLMeasure of resistance of an object to changes in its rotational motion. Equivalent to mass in linear motion.Determining Moment of Inertia is extremely important for computing equilibrium of a rigid body, such as a building.dmrI2For a group of particlesFor a rigid bodyWednesday, Dec. 1, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu8Total Kinetic Energy of a Rolling BodyWhere, IP, is the moment of inertia about the point P.Since it is a rotational motion about the point P, we can write the total kinetic energySince vCM=R, the above relationship can be rewritten as221PIK What do you think the total kinetic energy of the rolling cylinder is?PP’CMvCM2vCMUsing the parallel axis theorem, we can rewriteK222121CMCMMvIK What does this equation mean?Rotational kinetic energy about the CMTranslational Kinetic energy of the CMTotal kinetic energy of a rolling motion is the sum of the rotational kinetic energy about the CMAnd the translational kinetic of the CM221PI 2221MRICM2222121MRICMWednesday, Dec. 1, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu9Angular Momentum of a ParticleIf you grab onto a pole while running, your body will rotate about the pole, gaining angular momentum. We’ve used linear momentum to solve physical problems with linear motions, angular momentum will do the same for rotational motions.sinmvrL Let’s consider a point-like object ( particle) with mass m located at the vector location r and moving with linear velocity vprL The angular momentum L of this particle relative to the origin O is What do you learn from this?If the direction of linear velocity points to the origin of rotation, the particle does not have any angular momentum.What is the unit and dimension of angular momentum? 22/ smkg Note that L depends on origin O. Why? Because r changesThe direction of L is +zWhat else do you learn? Since p is mv, the magnitude of L becomesIf the linear velocity is perpendicular to position vector, the particle moves exactly the same way as a point on a rim.][22 TMLWednesday, Dec. 1, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu10Angular Momentum of a Rotating Rigid BodyLet’s consider a rigid body rotating about a fixed axisiiiivrmL Each particle of the object rotates in the xy plane about the z-axis at the same angular speed, IdtdLzextThus the torque-angular momentum relationship becomesWhat do you see?Since I is constant for a rigid bodyMagnitude of the angular momentum of a particle of mass


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