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Homework - Exam From last time… HW#7: M Chap 11: Questions A, C, Exercises 4, 8 M Chap 12: Exercises 4, 6 • Einstein’s Relativity – All laws of physics identical in inertial ref. frames – Speed of light=c in all inertial ref. frames Hour Exam 2: Wednesday, March 8 • Consequences • In-class, covering waves, electromagnetism, and relativity • Twenty multiple-choice questions • Will cover: March Chap 6-12 Griffith Chap 12,15,16 All 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” link Wed. Mar. 1, 2006 Phy107, Lect 18 – Simultaneity: events simultaneous in one frame will not be simultaneous in another. – Time dilation – Length contraction – Relativistic invariant: x2-c2t2 is ‘universal’ in that it is measured to be the same for all observers Wed. Mar. 1, 2006 1 – tproper measured in frame where events occur at same spatial location • L=Lproper / γ ! – Lproper measured in frame where events are simultaneous (or object is at rest) Wed. Mar. 1, 2006 "= A shipship parked at a street corner is 12 meters long. It then cruises around the block and moves at 0.8c past someone standing on that street corner. The street corner observer measures the ship to have a length of: 1 1# (v /c) 2 γ always bigger than 1 γ increases as v increases A. 20.0 m B. 5.7 m C. 0.6 m D. 7.2 m γ would be infinite for v=c Suggests some limitation on velocity as we approach speed of light Phy107, Lect 18 Wed. Mar. 1, 2006 3 ‘Separation’ between events Phy107, Lect 18 4 The real ‘distance’ between events • Need a quantity that is the same for all observers • A quantity all observers agree on is • Views of the same cube from two different angles. • Distance between corners (length of red line drawn on the flat page) seems to be different depending on how we look at it. 2 x 2 " c 2 t 2 # (separation) " c 2 (time interval) • But clearly this is just because we are not considering the full three-dimensional distance between the points. Phy107, Lect 18 ! 2 • Need to look at separation both in space and time to get the full ‘distance’ between events. • In 4D: 3 space + 1 time x 2 + y 2 + z 2 " c 2t 2 • The same or ‘invariant’ in any inertial frame • The 3D distance does not change with viewpoint. Wed. Mar. 1, 2006 2 Question Time dilation, length contraction • t= γ tproper Phy107, Lect 18 5 Wed. Mar. 1, 2006 Phy107, Lect 18 6 ! 1 Space travel example, earth observer Astronaut observer • Event #1: leave earth Ship stationary, Earth and star move 0.95c d=4.3 light-years d=4.3 light-years (LY) • Event #2: arrive star "t earth = Wed. Mar. 1, 2006 0.95c Astronaut measures proper time Earth observer time is dilated, longer by factor gamma 0.95c Event 1: ship and earth together Event 2: ship and star together Spatial separation in astronaut’s frame is zero! d 4.3 light - years = 4.526 yrs = v 0.95Phy107, c Lect 18 Wed. Mar. 1, 2006 7 Phy107, Lect 18 8 ! ! A relativistic invariant quantity Earth Frame Event separation = 4.3 LY Event separation = 0 LY Time interval = 4.526 yrs Time interval = 1.413 yrs (separation) " c 2 (time interval) 2 (separation) 2 Suppose space-time distance between events is zero. " c 2 (time interval) 2 What does this mean? 2 = ( 4.3) " (c ( 4.526yrs)) = "2.0 LY 2 = 0 " (c (1.413yrs)) = "2.0 LY (space separation)2 = c2(time interval)2 • The quantity (separation)2-c2(time interval)2 is ! the same for all observers • It mixes the space and time coordinates This 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. 2 ! 2 (Space-time ‘distance’)2 = (space separation)2-c2(time interval)2 Ship Frame 2 2 Wed. Mar. 1, 2006 Phy107, Lect 18 9 Question ct Phy107, Lect 18 10 Universal space-time distance Which events have a space-time separation of zero? A. A & B B. A & C C. C & D D. None of them Wed. Mar. 1, 2006 D Worldline of light beam (space separation)2-c2(time interval)2 = (space-time distance)2 Think of all the different observers measuring different spatial separations, different time intervals. Suppose the universal (space-time distance)2 < 0 Minimum time interval: C space separation is zero -> events occur at same spatial location. This was our condition for measuring the proper time. A B x Wed. Mar. 1, 2006 Phy107, Lect 18 11 In another frame, spatial separation not zero measured time interval must be longer Time dilation! Wed. Mar. 1, 2006 Phy107, Lect 18 12 2 Newton again Forces, Work, and Energy in Relativity What about Newton’s laws? • 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)2 Wed. Mar. 1, 2006 Phy107, Lect 18 13 • 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 increases Relativity says no. The effect of the force gets smaller and smaller as velocity approaches speed of light Phy107, Lect 18 – But clearly objects still move, spaceships are accelerated by thrust, work is done, energy is converted. • How do these things work in relativity? Applying a constant force Wed. Mar. 1, 2006 • Relativity dramatically altered our perspective of space and time 15 Wed. Mar. 1, 2006 Phy107, Lect 18 • At small velocities (short times) the motion is described by Newtonian physics • At higher velocities, big deviations! • The velocity never exceeds the speed of light Wed. Mar. 1, 2006 Momentum in Relativity Einstein 0.8 0.6 0.4 0.2 v = c t /t o (t /t o ) 2 +1 , to = F m oc 0 0 1 2 3 TIME ! Phy107, Lect 18 4 5 16 Relativistic momentum Momentum is constant for zero force and Relativistic gamma change in momentum = Force change in time The relativistic momentum is: prelativistic = "mv 1 "= 1# (v /c) 2 This relationship is preserved in relativity ! Phy107, Lect 18 Newton 1 • Relativity concludes that the Newtonian definition of momentum (p Newton=mv=mass x velocity) is accurate at low velocities, but not at high velocities • The relationship between momentum and force is very simple and fundamental Wed. Mar. 1, 2006 14 Relativistic speed of particle subject to constant force SPEED / SPEED OF LIGHT – – – – – 17 Wed. Mar. 1, 2006 Phy107, Lect 18 mass velocity 18 ! 3 Relativistic Momentum • Momentum can be increased arbitrarily, but velocity never exceeds c • We still use • Relativity requires a different concept of momentum prelativistic = "mv 1 "= 1# (v /c) 2 • But not really so different! change in momentum = Force change in time ! • For small velocities << light speed γ≈1, and so prelativistic ≈ mv • • For constant force we still have momentum = Force x time, but the velocity never exceeds c • Momentum has been redefined 1 Relativistic momentum 0.8 0.6 0.4 0.2 v = c 0 ! This is Newton’s momentum prelativistic = "mv = • Differences only occur at velocities that are a substantial fraction of the speed of light Wed. Mar. 1, 2006 Newton’s momentum SPEED / SPEED OF LIGHT Was Newton wrong? Phy107, Lect 18 1# (v /c) Wed. Mar. 1, 2006 19 0 mv 2 p / po ( p / po ) 2 +1 , po = moc 1 2 3 4 5 ! RELATIVISTIC MOMENTUM Relativistic momentum for different speeds. Phy107, Lect 18 20 ! • The the particle becomes extremely massive as speed increases ( m=γmo ) " change in velocity % • acceleration $#= change in time '& much smaller at high speeds than at low speeds • Newton said force and acceleration related by mass. ! • We could say that mass increases as speed increases. prelativistic = "mv = ("m)v # mrelativisticv • Can write this prelativistic = "mov = ("mo )v # mv ! "= — mo is the rest mass. 1 1$ (v /c) 2 • The relativistic momentum has new form ( p= γmov ) • Useful way of thinking of things remembering the concept of inertia , m = "m o — relativistic mass m depends on velocity RELATIVISTIC MASS / REST MASS Relativistic mass How can we understand this? 5 4 3 2 1 0 0 0.2 0.4 0.6 0.8 1 SPEED / SPEED OF LIGHT ! Wed. Mar. 1, 2006 Phy107, Lect 18 21 Wed. Mar. 1, 2006 Example Phy107, Lect 18 22 Question • An object moving at half the speed of light relative to a particular observer has a rest mass of 1 kg. What is it’s mass measured by the observer? A object of rest mass of 1 kg is moving at 99.5% of the speed of light. What is it’s measured mass? 1 1 1 = = 1# 0.25 1# (v /c) 2 1# (0.5c /c) 2 1 = = 1.15 0.75 "= A. 10 kg B. 1.5 kg C. 0.1 kg So measured mass is 1.15kg ! Wed. Mar. 1, 2006 Phy107, Lect 18 23 Wed. Mar. 1, 2006 Phy107, Lect 18 24 4 Work and Energy Relativistic Kinetic Energy • Might expect this to change in relativity. • Newton says that Work = Force x Distance, and that net work done on an an object changes the kinetic energy • Can do the same analysis as we did with Newtonian motion to find • Used this to find the classical kinetic energy KE relativistic = (" #1) moc 2 • Doesn’t seem to resemble Newton’s result at all 1 KE Newton = mv 2 2 • However for small velocities, it does reduce to the Newtonian form ! 1 KE relativistic " mov 2 for v << c 2 ! Wed. Mar. 1, 2006 Phy107, Lect 18 25 Wed. Mar. 1, 2006 Phy107, Lect 18 26 ! Relativistic Kinetic Energy • Kinetic energy gets arbitrarily large as speed approaches speed of light • Is the same as Newtonian kinetic energy for small speeds. Wed. Mar. 1, 2006 • The relativistic kinetic energy is 4 2 KE relativistic = (" #1) moc 2 o (KINETIC ENERGY) / m c • Can see this graphically as with the other relativistic quantities Total Relativistic Energy 3 = "moc 2 # moc 2 Relativistic 2 Depends on velocity 1 Newton ! this as • Write "moc 2 = KE relativistic + moc 2 0 0 0.2 0.4 0.6 0.8 1 SPEED / SPEED OF LIGHT Phy107, Lect 18 27 Constant, independent of velocity Total energy Wed. Mar. 1, 2006 Kinetic energy Rest energy Phy107, Lect 18 28 ! Mass-energy equivalence Example • In a frame where the particle is at rest, its total energy is E = moc2 • This results in Einstein’s famous relation 2 E = "moc , or E = mc 2 • Just as we can convert electrical energy to mechanical energy, it is possible to tap mass energy • This says that the total energy of a particle is related to its mass. ! • A 1 kg mass has (1kg)(3x108m/s)2=9x1016 J of energy • Even when the particle is not moving it has energy. • We could also say that mass is another form of energy – Just as we talk of chemical energy, gravitational energy, etc, we can talk of mass energy Wed. Mar. 1, 2006 Phy107, Lect 18 29 – We could power 30 million 100 W light bulbs for one year! (~30 million sec in 1 yr) Wed. Mar. 1, 2006 Phy107, Lect 18 30 5 Nuclear Power • Doesn’t convert whole protons or neutrons to energy • • Since γ depends on velocity, the energy is measured to be different by different observers • Extracts some of the binding energy of the nucleus – Energy and momentum Relativistic energy is E = "moc 2 • Momentum also different for different observers 90Rb and 143Cs + 3n have less rest mass than 235U +1n: E = mc2 – Can think of these as analogous to space and time, which individually are measured to be different by different observers ! • But there is something that is the same for all observers: E 2 " c 2 p 2 = ( m oc 2 ) 2 = Square of rest energy • Compare this to our space-time invariant Wed. Mar. 1, 2006 Phy107, Lect 18 31 ! Wed. Mar. 1, 2006 Phy107, Lect 18 2 x " c 2t 2 32 ! A relativistic perspective • The concepts of space, time, momentum, energy that were useful to us at low speeds for Newtonian dynamics are a little confusing near light speed • Relativity needs new conceptual quantities, such as space-time and energy-momentum • Trying to make sense of relativity using space and time separately leads to effects such as time dilation and length contraction • In the mathematical treatment of relativity, space-time and energy-momentum objects are always considered together Wed. Mar. 1, 2006 Phy107, Lect 18 33 6