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UW-Madison PHYSICS 107 - Homework - Exam

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1Phy107 Fall 2006 1Hour Exam 2: Wednesday, October 25th• In-class, covering waves, electromagnetism, and relativity• Twenty multiple-choice questions• Will cover: Chapters 8, 9 10 and 11Lecture material• You should bring– 1 page notes, written single sided– #2 Pencil and a Calculator– Review Monday October 23rd– Review test online on MondayHW#6:Chap 10 Conceptual: 36, 42 Problem 7, 9Chap 11 Conceptual: 5, 10Homework - ExamPhy107 Fall 2006 2From last time…• Einstein’s Relativity– All laws of physics identical in inertial ref. frames– Speed of light=c in all inertial ref. frames• Consequences– Simultaneity: events simultaneous in one framewill not be simultaneous in another.– Time dilation– Length contraction– Relativistic invariant: x2-c2t2 is ‘universal’ in thatit is measured to be the same for all observersPhy107 Fall 2006 3Review: Time Dilation and Length ContractionT =Tp=Tp1 v2c2Time in otherframeTime in object’s restframeL =Lp= Lp1v2c2Length in otherframeLength in object’srest frameTimes measured in otherframes are longer(time dilation)Distances measured in otherframes are shorter(length contraction)• Need to define the rest frameand the “other” frame which is moving withrespect to the rest framePhy107 Fall 2006 4Relativistic Addition of VelocitiesvdbFrame bFrame dvadObject avab=vad+ vdb1+vadvdbc2• As motorcycle velocityapproaches c,vab also gets closer andcloser to c• End result: nothingexceeds the speed oflightPhy107 Fall 2006 5Observing from a new frame• In relativity these eventswill look different inreference frame movingat some velocity• The new referenceframe can berepresented as sameevents along differentcoordinate axesCoordinates inoriginal frameCoordinates innew framect’x’xctNew frame movingrelative to originalPhy107 Fall 2006 6Time interval = 1.413 yrsTime interval = 4.526 yrsEvent separation = 0 LYEvent separation = 4.3 LYShip FrameEarth FrameA relativistic invariant quantity separation()2 c2time interval()2= 0  c 1.413yrs()()2= 2.0 LY2 separation()2 c2time interval()2= 4.3()2 c 4.526yrs()()2= 2.0 LY2• The quantity (separation)2-c2(time interval)2 isthe same for all observers• It mixes the space and time coordinates2Phy107 Fall 2006 7‘Separation’ between events•Views of the samecube from twodifferent angles.• Distance betweencorners (length of redline drawn on the flatpage) seems to bedifferent dependingon how we look at it.• But clearly this is just because we are not considering thefull three-dimensional distance between the points.• The 3D distance does not change with viewpoint.Phy107 Fall 2006 8Newton again• Fundamental relations of Newtonian physics– acceleration = (change in velocity)/(change in time)– acceleration = Force / mass– Work = Force x distance– Kinetic Energy = (1/2) (mass) x (velocity)2– Change in Kinetic Energy = net work done• Newton predicts that a constant force gives– Constant acceleration– Velocity proportional to time– Kinetic energy proportional to (velocity)2Phy107 Fall 2006 9Forces, Work, and Energy in RelativityWhat about Newton’s laws?• Relativity dramatically altered our perspective ofspace and time– But clearly objects still move,spaceships are accelerated by thrust,work is done,energy is converted.• How do these things work in relativity?Phy107 Fall 2006 10Applying a constant force• Particle initially at rest,then subject to a constant force starting at t=0, momentum =momentum = (Force) x (time)• Using momentum = (mass) x (velocity),Velocity increases without bound as time increasesRelativity says no. The effect of the force gets smaller and smaller as velocity approaches speed of lightPhy107 Fall 2006 1100.20.40.60.81012345SPEED / SPEED OF LIGHTTIMENewtonEinsteinvc=t /tot /to()2+ 1, to=FmocRelativistic speed of particlesubject to constant force• At small velocities(short times) themotion is describedby Newtonian physics• At higher velocities,big deviations!• The velocity neverexceeds the speed oflightPhy107 Fall 2006 12Momentum in Relativity• The relationship between momentum andforce is very simple and fundamental change in momentumchange in time= ForceMomentum is constant for zero forceandThis relationship is preserved in relativity3Phy107 Fall 2006 13Relativistic momentum• Relativity concludes that the Newtoniandefinition of momentum(pNewton=mv=mass x velocity)is accurate at low velocities,but not at high velocitiesThe relativistic momentum is: prelativistic=mv=11 (v /c)2massvelocityRelativistic gammaPhy107 Fall 2006 14Was Newton wrong?• Relativity requires a different concept ofmomentum• But not really so different!• For small velocities << light speed1, and so prelativistic  mv• This is Newton’s momentum• Differences only occur at velocities that are a substantialfraction of the speed of lightprelativistic=mv=11 (v /c)2Phy107 Fall 2006 15Relativistic Momentum• Momentum can be increasedarbitrarily, but velocity neverexceeds c• We still use• For constant force we still havemomentum = Force x time,but the velocity never exceeds c• Momentum has been redefinedprelativistic=mv =mv1 (v /c)200.20.40.60.81012345SPEED / SPEED OF LIGHTRELATIVISTIC MOMENTUMvc=p / pop / po()2+ 1, po= moc change in momentumchange in time= ForceNewton’smomentumRelativistic momentum fordifferent speeds.Phy107 Fall 2006 16How can we understand this?• accelerationmuch smaller at high speeds than at low speeds• Newton said force and acceleration related by mass.• We could say that mass increases as speed increases. =change in velocitychange in time     prelativistic=mv =m()v  mrelativisticv• Can write this— mo is the rest mass.— relativistic mass m depends on velocityprelativistic= mov = mo()v  mv =11 (v /c)2, m = moPhy107 Fall 2006 17Relativistic mass• The the particlebecomes extremelymassive as speedincreases ( m=mo )• The relativisticmomentum has newform ( p= mov )• Useful way of thinkingof things rememberingthe concept of inertia0123450 0.2 0.4 0.6 0.8 1RELATIVISTIC MASS / REST MASSSPEED / SPEED OF LIGHTPhy107 Fall 2006 18Example• An object moving at half the speed of lightrelative to a particular observer has a restmass of 1 kg. What is it’s mass measured bythe observer?=11 (v /c)2=11 (0.5c /c )2=11 0.25=10.75= 1.15So measured mass is 1.15kg4Phy107 Fall 2006

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