1Phy107, Lect 18 1Wed. Mar. 1, 2006Hour Exam 2: Wednesday, March 8• In-class, covering waves, electromagnetism, and relativity• Twenty multiple-choice questions• Will cover: March Chap 6-12Griffith Chap 12,15,16All lecture material• You should bring– 1 page notes, written double sided– #2 Pencil and a Calculator– Review Monday March 6– Review questions online under “Review Quizzes” linkHW#7:M Chap 11: Questions A, C, Exercises 4, 8M Chap 12: Exercises 4, 6 !Homework - ExamPhy107, Lect 18 2Wed. Mar. 1, 2006From 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, Lect 18 3Wed. Mar. 1, 2006Time dilation, length contraction• t= γ tproper– tproper measured inframe where eventsoccur at same spatiallocation• L=Lproper / γ– Lproper measured inframe where eventsare simultaneous (orobject is at rest)! "=11# (v /c)2γ always bigger than 1γ increases as v increasesγ would be infinite for v=cSuggests some limitationon velocity as weapproach speed of lightPhy107, Lect 18 4Wed. Mar. 1, 2006QuestionA shipship parked at a street corner is 12 meters long.It then cruises around the block and moves at 0.8c pastsomeone standing on that street corner.The street corner observer measures the ship to have alength of:A. 20.0 mB. 5.7 mC. 0.6 mD. 7.2 mPhy107, Lect 18 5Wed. Mar. 1, 2006‘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, Lect 18 6Wed. Mar. 1, 2006The real ‘distance’ between events• Need a quantity that is the same for all observers• A quantity all observers agree on is• Need to look at separation both in space and timeto get the full ‘distance’ between events.• In 4D: 3 space + 1 time• The same or ‘invariant’ in any inertial frame! x2" c2t2# separation( )2" c2time interval( )2! x2+ y2+ z2" c2t22Phy107, Lect 18 7Wed. Mar. 1, 2006Space travel example, earth observer• Event #1: leave earth• Event #2: arrive star0.95c0.95cd=4.3 light-years (LY)! "tearth=dvc 95.0years-light 3.4= ! = 4.526 yrsPhy107, Lect 18 8Wed. Mar. 1, 2006Astronaut observer0.95cd=4.3 light-yearsShip stationary, Earth and star moveAstronaut measures proper timeEarth observer time is dilated, longer by factor gammaEvent 1: ship and earth togetherEvent 2: ship and star togetherSpatial separation in astronaut’s frame is zero!Phy107, Lect 18 9Wed. Mar. 1, 2006Time 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 coordinatesPhy107, Lect 18 10Wed. Mar. 1, 2006(Space-time ‘distance’)2 = (space separation)2-c2(time interval)2Suppose space-time distance between events is zero.What does this mean?(space separation)2 = c2(time interval)2This is a light beam propagating. Event 1 is shooting out the light beam. Event 2 is receiving the light signal.Time interval can only be zero if events are at same spatial location.Phy107, Lect 18 11Wed. Mar. 1, 2006QuestionWhich events have a space-time separation of zero?A. A & BB. A & CC. C & DD. None of themxctWorldline oflight beamABCDPhy107, Lect 18 12Wed. Mar. 1, 2006Universal space-time distance(space separation)2-c2(time interval)2 = (space-time distance)2Think of all the different observers measuring different spatialseparations, different time intervals.Suppose the universal (space-time distance)2 < 0Minimum time interval:space separation is zero -> events occur at same spatial location.This was our condition for measuring the proper time.In another frame, spatial separation not zeromeasured time interval must be longerTime dilation!3Phy107, Lect 18 13Wed. Mar. 1, 2006Newton 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, Lect 18 14Wed. Mar. 1, 2006Forces, 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, Lect 18 15Wed. Mar. 1, 2006Applying a constant force• Particle initially at rest,then subject to a constant force starting at t=0, Δmomentum =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, Lect 18 16Wed. Mar. 1, 200600.20.40.60.810 1 2 3 4 5SPEED / SPEED OF LIGHTTIMENewtonEinstein! vc=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, Lect 18 17Wed. Mar. 1, 2006Momentum 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 relativityPhy107, Lect 18 18Wed. Mar. 1, 2006Relativistic 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: !
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