1 / 66

LECTURE # 28

LECTURE # 28. RELATIVITY V MASS ENERGY EQUIVALENCE EXAMPLES. PHYS 270-SPRING 2010 Dennis Papadopoulos. APRIL 23, 2010. 1. Relativistic Energy. The total energy E of a particle is. This total energy consists of a rest energy. and a relativistic expression for the kinetic energy.

lorie
Download Presentation

LECTURE # 28

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. LECTURE # 28 RELATIVITY V MASS ENERGY EQUIVALENCE EXAMPLES PHYS 270-SPRING 2010 Dennis Papadopoulos APRIL 23, 2010 1

  2. Relativistic Energy The total energyE of a particle is This total energy consists of a rest energy and a relativistic expression for the kinetic energy This expression for the kinetic energy is very nearly ½mu2 when u << c.

  3. Eo=mc2 Invariant

  4. Energy conservation requires that M=2m+2K/c2 Mass is not conserved M M=2m+2K/c2 Pair creation DK=2mec2 e-(fast)+e-(at rest)e-+e-+e-+e+ Law of conservation of total energy

  5. Conservation of Energy The creation and annihilation of particles with mass, processes strictly forbidden in Newtonian mechanics, are vivid proof that neither mass nor the Newtonian definition of energy is conserved. Even so, the total energy—the kinetic energy and the energy equivalent of mass—remains a conserved quantity. Mass and energy are not the same thing, but they are equivalent in the sense that mass can be transformed into energy and energy can be transformed into mass as long as the total energy is conserved.

  6. Energy : The measure of a system’s capacity to do work • Units of Energy: Joule = Nt x m, eV= 1.6 x 10-19 J, Cal = 4.2 x 103 J • Examples : It takes 100 Joules to lift 10 kg by 1 meter against the • Earth’s gravity (g= 10 m/sec2); It takes .4 MJ to accelerate a 1000 kg • car to 30 m/sec (105 km/hr); It takes about 1010J to accelerate a • missile to 5 km/sec. [E=1/2 M(kg) v2 (m/sec) J] • Chemical Energy Storage : Chemical energy is stored in the • chemical bonds of molecules. As an order of magnititude a few eV • per bond. A 1 kg steak store approximately 1000 Cal or approximately 4 MJ. This is the amount (4-10 MJ/kg)stored in one kg of chemical explosives (TNT). Also a typical battery has few MJ of stored energy. • Energy Transformations: Energy has many forms, e.g. potential, • kinetic, chemical,acoustic, radiation, light etc. Each can be transformed • to the other, but overall energy is conserved.

  7. Explosion : In ordinary, e.g. TNT, explosion chemical energy is • transformed in kinetic energy of the fragments, acoustic energy of • the snow-plowed air (shock or blast wave) and some radiation from • the heated air. In an explosion we deliver the energy fast. Power: P= Energy/time. Units are Watt = J/sec. Hair dryer 1 kW, Light bulb 100 W. Power worldwide 1.3 Twatts, In a year multiply by 2x107 secs to get 3x1019 J or 104 MT (MT=4x1015 J)

  8. CHEMICAL BINDING ENERGY TWO OXYGEN ATOMS ATTRACT EACH OTHER TO FORM O2 WHILE RELEASING 5 eV OF ENERGY. THEREFORE 2 OXYGEN ATOMS ARE HEAVIER THAN AN OXYGEN MOLECULE BY Dm= 5 eV/c2 =9x10-36kg MASS OF OXYGEN MOLECULE IS 5x10-26kg. Dm/m=2x10-10 FORM 1 GRAM OF O2 AND GET 2x104 JOULES

  9. All matter is an Assembly of Atoms Atomic number vs. Mass number

  10. Radioactivity alpha decay Ra(226,88)->Rn(222,86)+He(4,2) U(238,92)->Th(234,90)+He(4,2) p n Beta decay C(14,6)->N(14,7)+e-+n

  11. E=mc2 1 kg has the potential to generate 9x1016 Joules Could provide electric power to city of 800000 for 3 years h=1 300 kg/year Efficiency=h mc2 Chemical reactions (0il, coal, etc) h= 10-9 Nuclear power – Fission h= 10-3 Nuclear power – Fusion h=10-2 1010 tons/year 300 tons/year 30 tons/year

  12. EXAMPLES OF CONVERTING MASS TO ENERGY • Nuclear fission (e.g., of Uranium) • Nuclear Fission – the splitting up of atomic nuclei • E.g., Uranium-235 nuclei split into fragments when smashed by a moving neutron. One possible nuclear reaction is • Mass of fragments slightly less than mass of initial nucleus + neutron • That mass has been converted into energy (gamma-rays and kinetic energy of fragments)

  13. FISSION One case of the fission of 236U. The net mass of the initial neutron plus the 235U nucleus is 219,883 MeV/c2. The net mass of the fission products (two neutrons, a 95Mo nucleus and a 139La nucleus) is 219,675 MeV/c2 - smaller because of the stronger binding of the Mo and La nuclei. The "missing mass'' of 208 MeV/c2 goes into the kinetic energy of the fragments (mainly the neutrons), which of course adds up to 208 MeV.

  14. Fusion • Nuclear fusion (e.g. hydrogen) • Fusion – the sticking together of atomic nuclei • Much more important for Astrophysics than fission • e.g. power source for stars such as the Sun. • Explosive mechanism for particular kind of supernova • Important example – hydrogen fusion. • Ram together 4 hydrogen nuclei to form helium nucleus • Spits out couple of “positrons” and “neutrinos” in process

  15. Fusion • Mass of final helium nucleus plus positrons and neutrinos is less than original 4 hydrogen nuclei • Mass has been converted into energy (gamma-rays and kinetic energy of final particles) • This (and other very similar) nuclear reaction is the energy source for… • Hydrogen Bombs (about 1kg of mass converted into energy gives 20 Megaton bomb) • The Sun (about 4109 kg converted into energy per second)

  16. Annihilation • Anti-matter • For every kind of particle, there is an antiparticle… • Electron  anti-electron (also called positron) • Proton  anti-proton • Neutron anti-neutron • Anti-particles have opposite properties than the corresponding particles (e.g., opposite charge)… but exactly same mass. • When a particle and its antiparticle meet, they can completely annihilate each other… all of their mass is turned into energy (gamma-rays)!

  17. EXAMPLES OF CONVERTING ENERGY TO MASS • Particle/anti-particle production • Opposite process to that just discussed! • Energy (e.g., gamma-rays) can produce particle/anti-particle pairs • Very fundamental process in Nature… shall see later that this process, operating in early universe, is responsible for all of the mass that we see today!

  18. PAIR PRODUCTION Electron-positron PAIR PRODUCTION by gamma rays (above) and by electrons (below). The positron (e+) is the ANTIPARTICLE of the electron (e-). The gamma ray (g ) must have an energy of at least 1.022 MeV [twice the rest mass energy of an electron] and the pair production must take place near a heavy nucleus (Z) which absorbs the momentum of the g .

  19. Particle production in a particle accelerator • Can reproduce conditions similar to early universe in modern particle accelerators…

  20. A real particle creation event

  21. Space-time Diagrams

  22. Moving objects Stationary object light ct x World lines of events Space-time diagrams • Because space and time are “mixed up” in relativity, it is often useful to make a diagram of events that includes both their space and time coordinates. • This is simplest to do for events that take place along a line in space (one-dimensional space) • Plot as a 2D graph • use two coordinates: xand ct • Can be generalized to events taking place in a plane (two-dimensional space) using a 3D graph (volume rendered image):x, yandct • Can also be generalized to events taking place in 3D space using a 4D graph, but this is difficult to visualize

  23. Time, the fourth dimension? “Spacetime” In x,y space the two space dimensions are interchangeable if they have the same units. A similar relationship can be used to express the relationship between space and time in relativity. Light propagating in one dimension in a spacetime coordinate system as viewed from a frame S. The distance traveled is equal to the speed of light times the time elapsed. v<c ct v=c t in years distance in lightyears t in secs distance in lighteseconds q v>c 45° x b=v/c=tanq

  24. SIMULTANEITY

  25. Spacetime diagrams in different frames • Changing from one reference frame to another… • Affects time coordinate (time-dilation) • Affects space coordinate (length contraction) • Leads to a distortion of the space-time diagram as shown in figure. • Events that are simultaneous in one frame are not simultaneous in another frame ct q q x

  26. A

  27. ctJ ctM xM -1 xJ

  28. Light cone for event “A” “Light Cone”

  29. “light like” “time like” “Space like” Different kinds of space-time intervals Time-like:s2>0 Light-like: s2=0 Space-like: s2<0 “Light Cone”

  30. Past, future and “elsewhere”. “Future of A” (causally-connected) “Elsewhere” (causally- disconnected) “Past of A” (causally-connected)

  31. Causality Could an event at O cause A? Yes, because a “messenger” at O would not have to travel at a speed greater than the speed of light to get there. where light that is here now may go in the future x=-ct ct x=ct A B Could an event at O cause B? C A light signal sent from O could reach B. x here,now O Could an event at O cause C? No, the spacetime distance between O and C is greater than could be covered by light. It would require time travel. where light that is here now may have been in the past

  32. The twin paradox • Suppose Andy (A) and Betty (B) are twins. • Andy stays on Earth, while Betty leaves Earth, travels (at a large fraction of the speed of light) to visit her aunt on a planet orbiting Alpha Centauri, and returns • When Betty gets home, she finds Andy is greatly aged compared her herself. • Andy attributes this to the time dilation he observes for Betty’s clock during her journey • Is this correct? • What about reciprocity?Doesn’t Betty observe Andy’s clock as dilated, from her point of view? Wouldn’t that mean she would find him much older, when she returns? • Who’s really older??What’s going on???

  33. B (return) A ct B (outbound) x Andy’s point of view • Andy’s world line, in his own frame, is a straight line • Betty’s journey has world line with two segments, one for outbound (towards larger x) and one for return (towards smaller x) • Both of Betty’s segments are at angles 45 to vertical, because she travels at vc • If Andy is older by t years when Betty returns,he expects that due to time dilation she will have aged by t/ years • Since 1/ = (1-v2/c2)1/2 1, Betty will be younger than Andy, and the faster Betty travels, the more difference there will be L

  34. B (return) B (return) A A ct ct B (outbound) B (outbound) x x Betty’s point of view • Consider frame moving with Betty’s outbound velocity • Andy on Earth will have straight world line moving towards smaller x • Betty’s return journey world line is not the same as her outbound world line, instead pointing toward smaller x • Both Andy’s world line and Betty’s return world line are at angles  45  to vertical (inside of the light cone) • Betty’s return world line is closer to light cone than Andy’s world line • For frame moving with Betty’s return velocity, situation is similar

  35. B (return) A ct B (outbound) x Solution of the paradox • From any perspective, Andy’s world line has a single segment • From any perspective, Betty’s world line has two different segments • There is no single inertial frame for Betty’s trip, so reciprocity of time dilation with Andy cannot apply for whole journey • Betty’s proper time is truly shorter -- she is younger than Andy when she returns

  36. B (return) B (return) A A ct ct B (outbound) B (outbound) x x Different kinds of world lines • Regardless of frame, Betty’s world line does not connect start and end points with a straight line, while Andy’s does • This is because Betty’s journey involves accelerations, while Andy’s does not

  37. Questions

  38. Which of these is an inertial reference frames (or a very good approximation)? • A rocket being launched • A car rolling down a steep hill • A sky diver falling at terminal speed • A roller coaster going over the top of a hill • None of the above

  39. Which of these is an inertial reference frames (or a very good approximation)? • A rocket being launched • A car rolling down a steep hill • A sky diver falling at terminal speed • A roller coaster going over the top of a hill • None of the above

  40. Ocean waves are approaching the beach at 10 m/s. A boat heading out to sea travels at 6 m/s. How fast are the waves moving in the boat’s reference frame? • 4 m/s • 6 m/s • 16 m/s • 10 m/s

  41. Ocean waves are approaching the beach at 10 m/s. A boat heading out to sea travels at 6 m/s. How fast are the waves moving in the boat’s reference frame? • 4 m/s • 6 m/s • 16 m/s • 10 m/s

  42. A carpenter is working on a house two blocks away. You notice a slight delay between seeing the carpenter’s hammer hit the nail and hearing the blow. At what time does the event “hammer hits nail” occur? • Very slightly after you see the hammer hit. • Very slightly after you hear the hammer hit. • Very slightly before you see the hammer hit. • At the instant you hear the blow. • At the instant you see the hammer hit.

  43. A carpenter is working on a house two blocks away. You notice a slight delay between seeing the carpenter’s hammer hit the nail and hearing the blow. At what time does the event “hammer hits nail” occur? • Very slightly after you see the hammer hit. • Very slightly after you hear the hammer hit. • Very slightly before you see the hammer hit. • At the instant you hear the blow. • At the instant you see the hammer hit.

  44. A tree and a pole are 3000 m apart. Each is suddenly hit by a bolt of lightning. Mark, who is standing at rest midway between the two, sees the two lightning bolts at the same instant of time. Nancy is at rest under the tree. Define event 1 to be “lightning strikes tree” and event 2 to be “lightning strikes pole.” For Nancy, does event 1 occur before, after or at the same time as event 2? • before event 2 • after event 2 • at the same time as event 2

More Related