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ROCHESTER PHY 121 - Lecture 12 Notes - Magnetism

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1Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterPhysics 121.Thursday, February 28, 2008.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterPhysics 121.Thursday, February 28, 2008.• Course Information• Quiz?• Topics to be discussed today:• The center of mass• Conservation of linear momentum• Systems of variable massFrank L. H. Wolfs Department of Physics and Astronomy, University of RochesterPhysics 121.Thursday, February 28, 2008.• Homework set # 5 is now available on the WEB and will bedue next week on Saturday morning, March 8, at 8.30 am.• This homework set contains WeBWorK problems and avideo analysis.• The most effective way to work on the assignment is totackle 1 or 2 problems a day.• If you run into problems, please contact me and I will try tohelp you solve your problems (Physics 121 problems thatis).2Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterHomework Set # 5.WeBWorK problems.Collision force2D collisionInelastic collisionsand conservationof energyConservation ofenergy (including“friction” losses)Finding U(x)The escape velocityRocket motionFrank L. H. Wolfs Department of Physics and Astronomy, University of RochesterHomework Set # 5.WeBWorK problems.Explosion in 1DElastic collisionsand conservationof energy2D collisionFrank L. H. Wolfs Department of Physics and Astronomy, University of RochesterPhysics 121Thursday, February 28, 2008.• We will grade Exam # 1 this weekend, and the grades willbe distributed via email on Monday.• The exam will be returned to you during workshops nextweek. Please carefully look at the exam and if you madeany mistakes, try to understand why what you did was notcorrect. If you disagree with the grade you received, youneed to come and talk to me. Your TAs can not changeyour exam grade.• The next exam will take place on March 25. Please do notwait until the day before the exam to start studying!3Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterPhysics 121.Quiz lecture 12.• The quiz today will have 3 questions and provide me withfeedback on Exam # 1.• Note: each answer you submit is correct!Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterThe Center of Mass.• Up to now we have ignored theshape of the objects were arestudying.• Objects that are not point-likeappear to carry out morecomplicated motions than point-like objects (e.g. the object maybe rotating during its motion).• We will find that we can usewhatever we have learned aboutmotion of point-like objects if weconsider the motion of the center-of-mass of the extended object.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterBut what is the center of massand where is it located?• We will start with considering one-dimensional objects. For an objectconsisting out of two point masses, the center of mass is defined asdXcmm1m2 xcm=m1x1+ m2x2m1+ m2=1Mmixii!4Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterBut what is the center of massand where is it located?• Let’s look at this particularsystem.• Since we are free to choose ourcoordinate system in a wayconvenient to us, we choose itsuch that the origin coincideswith the location of mass m1.• The center of mass is located atNote: the center of mass does notneed to be located at a positionwhere there is mass!dXcmm1m2 xcm=m2dm1+ m2Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterBut what is the center of massand where is it located?• In two or three dimensions thecalculation of the center of massis very similar, except that weneed to use vectors.• If we are not dealing withdiscreet point masses we need toreplace the sum with an integral. !rcm=1M!rdmV!Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterBut what is the center of massand where is it located?• We can also calculate theposition of the center of mass of atwo or three dimensional objectby calculating its componentsseparately:• Note: the center of mass may belocated outside the object. xcm=1Mmixii!ycm=1Mmiyii!zcm=1Mmizii!5Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterCalculating the position of the center of mass. Example problem.• Consider a circular metal plate ofradius 2R from which a disk ifradius R has been removed. Letus call it object X. Locate thecenter of mass of object X.• Since this object is complicatedwe can simplify our life by usingthe principle of superposition:• If we add a disk of radius R,located at (-R/2, 0) we obtain asolid disk of radius R.• The center of mass of this disk islocated at (0, 0).x-axisy-axis(b)x-axisy-axis(a)Mass M (solid disk)Mass M/4 Mass 3M/4 Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterCalculating the position of the center of mass.Example problem.• The center of mass of the soliddisk can be expressed in terms ofthe disk X and the disk D weused to fill the hole in disk X:• This equation can be rewritten as• Where we have used the fact thatthe center of mass of disk D islocated at (-R/2, 0).• Thus, xcm,X = R/3.x-axisy-axis(b)x-axisy-axis(a)Mass M (solid disk)Mass M/4 Mass 3M/4 xcm,C=xcm,XmX+ xcm,DmDmX+ mD= 0 xcm,X= !xcm,DmDmX=RmDmXFrank L. H. Wolfs Department of Physics and Astronomy, University of RochesterMotion of the center of mass.• To examine the motion of the center of mass we start withits position and then determine its velocity and acceleration: M!rcm= mi!rii!M!vcm= mi!vii!M!acm= mi!aii!6Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterMotion of the center of mass.• The expression for Macm can be rewritten in terms of theforces on the individual components:• We conclude that the motion of the center of mass is onlydetermined by the external forces. Forces exerted by onepart of the system on other parts of the system are calledinternal forces. According to Newton’s third law, the sumof all internal forces cancel out (for each interaction thereare two forces


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