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Einstein’s Theory of Special Relativity VCE Physics Unit 3 2017 Jill Detez [email protected] An overview of how I will teach Einstein’s Relativity in Unit 3. This session is aimed at teachers who are new to teaching this topic. An outline of my approach and sharing of resources, activities and assessment ideas will be provided. Study Design References: Pages 39 & 40 Teaching Special Relativity in Unit 3 • Time allocation: 4 to 5 x 50 minute periods • Text reference: Heinemann Physics 12 4th Edition • Assessment: Informal topic test. • My SAC on Unit 3 Area of Study 3 will consist of annotations of practical activities (circular motion, strain potential energy, etc.) and may include a reference to the modelling γ and muon analyses. Frames of reference A reference frame allows us to refer to the position of a particle. Reference frames consist of an origin together with a set of axes. • In Physics, we generally use the Cartesian reference frame. (x, y, z and t) • Others can be used: – Sailors use latitude and longitude – Astronomers use angles to define the position of a star (altitude and azimuth, or right ascension Inertial reference frames We could roughly define an inertial reference frame as a frame that is at rest or in constant motion (i.e. not accelerating). But then the question arises: at rest with respect to what? Michelson-Morley experiment • Discuss the Michelson-Morley experiment at this point: determining the relative motion of the Earth with respect to the aether. http://scienceworld.wolfram.com/physics/MichelsonMorleyExperiment.html Laws of Physics and inertial reference frames We can more easily define an inertial reference frame as a frame in which an object obeys Newton’s first law. There are some good videos to introduce this concept. Try • Introduction to Special Relativity (Art Doing Physics, 4.5 min) • https://www.youtube.com/watch?v=PpJ-LPOuvPc • Also UNSW Physclips: http://www.animations.physics.unsw.edu.au/jw/Newt on.htm#frames Galilean relativity and Newton’s postulates Galileo’s principle of relativity states that all motion is relative to some particular frame of reference, but there can be no frame of reference that has an absolute zero velocity. Newton assumed that space and time are constant, uniform and straight (i.e. a metre ruler has the same length whether vertical or horizontal, at ground level or in space, and time flows at the same rate no matter the location of the observer) Einstein Michelson and Morley demonstrated that the speed of light (determined by Maxell’s equations) in vacuum is constant in all directions, the accepted value is c = 299 792 458 ms-1 (≈ 3.0 × 108 ms-1) In trying to explain this fixed upper limit for the speed of light, Einstein realised that Newton’s assumptions of space-time may not hold true for all motion, especially motion on scales involving huge distances and at speeds approaching the speed of light. If the speed of light is considered to be absolute, then Newtonian physics cannot be used to explain this phenomenon. The two postulates of special relativity Einstein’s two postulates: • The laws of Physics are the same in all inertial (non-accelerated) frames of reference • The speed of light has a constant value for all observers regardless of their motion or the motion of the source. http://www.pbslearningmedia.org/resource/phy03.sci.phys.energy.sprelativit y/einsteins-special-theory-of-relativity/ (7 min) Heinemann Physics 12 Questions • 6.1 Review page 211 • Questions 1 - 10 Some of the most dramatic differences between our everyday perceptions of space and time and the predictions made by special relativity concern time and length differences that arise between frames of reference moving relative to each other. Simultaneity Simultaneity is the relation between two events assumed to be happening at the same time in a frame of reference. According to Einstein's theory of relativity, simultaneity is not an absolute relation between events; what is simultaneous in one frame of reference will not necessarily be simultaneous in another. https://www.youtube.com/watch?v=fqVN2XmH_48 (2 min) Proper Time Einstein’s thought experiments regarding the concept of simultaneity lead him to question Newton’s concept of time being a constant in all reference frames. Watch this video: 60 Second adventure in Special Relativity (Nice for light clock) • www.youtube.com/watch?v=mU04-vJB6gc Evidence for time difference in different reference frames The difference in time for the events discussed by observers in a train would be so infinitesimally small as to be negligible. • However, aircraft flying at supersonic speeds recording time using atomic clocks can detect the difference. • To particles travelling at speeds approaching the speed of light, such as those in particle accelerators, the differences in time between the frames of reference become significant. Proper time • Proper time, Δto, is the term given for the time interval in the frame of reference that is ‘stationary’. • In the VCE Study Design, we will define proper time, to, as the the time interval between two events in a reference frame where the two events occur at the same point in space. • Alternatively, it is the shortest possible measured time interval between two events. Proper time (continued) • Consider the idea of the light clock. A beam of light is reflected between two mirrors, and ticks each time the light is reflected off the bottom mirror. • An observer sits in the room watching the light reflected between the mirrors from a distance, the time taken for the light to return to a mirror is t = 2L/c (diagram 1) • Now a second observer moves sideways at a constant velocity v relative to the mirrors. In the frame of this observer, the light appears to travel at an angle to the direction of motion, but of course at the same speed of light. • Hence the moving person sees the light travelling a greater distance at the same speed, so the measured time for this must be greater. Time dilation and the Lorentz Factor We can use the Pythagorean theorem to show that the relationship between the time measured in the reference frame, to, is related to the time measured in the moving frame, t, is: Note that the t = toγ derivation of the where time dilation relationship and the Lorentz Factor are not required for the VCE course. Time dilation and the Lorentz Factor (cont.) H. A. Lorentz (shared Nobel Prize in Physics 1902) first introduced the γ factor while attempting to explain the results of the Michelson-Morley experiment, hence it is known as the Lorentz factor. • v is the speed of the moving frame of reference • c is the speed of light in vacuum • t is the time observed in the stationary frame • to is the time observed in the moving frame Using time dilation - example • Suppose a news report stated that the starship Enterprise has just returned from a 5.0 year voyage (as measured by Earth time) while travelling at 0.89c. How much time has elapsed on the ship? Time dilation • Note that the observer on Earth measures the time for the same event as longer. In other words, the time is ‘dilated’ (expanded). Log-book Exercise: Using Excel to model the effect of speed on the value of γ • Investigate the effect of changing speed to obtain values for γ for a range of values of the ratio v/c. • Plot a graph of γ vs. v/c for values of 0 ≤ v/c ≤ 1 Observations: Describe the trend of your graph. At what speeds do relativistic effects become noticeable? Model the effect of speed on the value of γ Velocity (ms-1) 3.00E+02 3.00E+05 3.00E+06 3.00E+07 5.70E+07 8.40E+07 1.11E+08 1.38E+08 1.65E+08 1.92E+08 2.19E+08 2.46E+08 2.73E+08 2.83E+08 2.93E+08 2.96E+08 2.99E+08 3.00E+08 v/c γ 1.00E-06 1.00E-03 1.00E-02 1.00E-01 1.90E-01 2.80E-01 3.70E-01 4.60E-01 5.50E-01 6.40E-01 7.30E-01 8.20E-01 9.10E-01 9.43E-01 9.77E-01 9.87E-01 9.97E-01 1.00E+00 1.0000000000 1.0000005000 1.0000500038 1.0050378153 1.0185538856 1.0416666667 1.0763894729 1.1262289640 1.1973686802 1.3014480157 1.4631709377 1.7471413945 2.4119153510 3.0134397773 4.6563421829 6.1442394039 12.2576676967 #DIV/0! 10 9 8 7 6 γ 5 4 3 2 1 0 0.0 0.2 0.4 0.6 v/c 0.8 1.0 Useful videos Time Dilation with Stephen Hawking Ph.D: A video that describes what it might be like to travel on a spaceship at relativistic speeds. • https://www.youtube.com/watch?v=RSeomJx _f5E Explanation of time dilation.wmv (Mechanical Universe series explains time dilation) • https://www.youtube.com/watch?v=uIGegibP V-c Muons • Muons are high energy sub-atomic particles that are created in the Earth’s atmosphere when cosmic rays strike oxygen atoms at about 15 km above the Earth. Muons are unstable, with a mean lifetime of 2.196 μs. • These muons typically travel at 99.97% of the speed of light, and Newtonian physics predicts that a muon would travel about 659 m in one half life. • After 10 lifetimes there would be essentially no muons remaining, and so after 6.59 km ( a height of 8.42 km above the Earth) we would expect that no muons would be detected. http://hyperphysics.phyastr.gsu.edu/hbase/Relativ/muon.html Explaining high-altitude muons • Time dilation explains the observation that muons are, in fact, detected at the Earth’s surface. • If we apply the time dilation equation to the muons, we see that the mean life (in the muon’s reference frame) becomes 49.21 μs, which is more than 22 times longer than in the stationary frame. • This is plenty of time for the muons to cascade down to the Earth’s surface and hence be detected. Muon Investigation (Log Book) Cosmic radiation from distant parts of the universe interacts with particles in the atmosphere and produces muons. • Muons are unstable with a half life of approximately 1.5 µs (1 µs = 1 x 10 -6 s). Muons travel at approximately 0.995 c. • If the number of muons at time t = 0 is N0 and the muons enter the atmosphere at a height of 2000 m, calculate the percentage of muons which reach sea level. • The value you have calculated does not agree with empirical research, which detects 70 % of the original muons reaching the ground. Use the theory of relativity to find how many of the initial muons survive and calculate the half life of the muons in the Earth’s frame of reference. Heinemann Physics 12 Questions • 6.2 Review Page 219 • Questions 1 - 10 Proper Length In addition to time dilation, the length of an object also changes when observed in frames moving relative to the object. • We define proper length, Lo, as the length of the object in the frame which is at rest (the frame which contains both the measuring device and the object). Length Contraction The length of an object moving at relativistic speeds can be found by again applying the Lorentz factor. • The measured length in the moving frame becomes This equation shows that an object with a proper length Lo will have a shorter length, L, parallel to the motion of its moving frame of reference when measured by an observer in a stationary frame of reference. https://www.nobelprize.org/educational/physi cs/relativity/transformations-2.html http://www.staff.science.uu.nl/~hooft 101/animations.html Length contraction: Example (Giancoli p 809) • A rectangular painting measures 1.00 m tall and 1.50 m wide. It is hung on the side wall of a spaceship which is moving past the Earth at a speed of 0.90c. (a) What are the dimensions of the painting as seen by the captain of the spaceship? (b) What are the dimensions as seen by an observer on the Earth? Using the Lorentz transformation Sometimes it is useful to make v the subject in the Lorentz factor. This produces: • Students are encouraged to include this on their notes sheet for use during the November examination. Heinemann Physics 12 Questions • 6.3 Review page 224 • Questions 1 – 10 • Also Chapter Review Question pages 225 - 226 Relativistic mass increase According to Newton’s Second Law, an object of mass m subjected to a force F will experience an acceleration a, as given by F = m×a. The Special Theory predicts that: • As the speed of the body increases, then to an observer at rest relative to the body the mass of the body will increase. If mo is the mass of the body when it is at rest relative to the observer, then this is the rest mass. When it is moving at speed v relative to the observer, then its mass m is given by m = mo Newton’s Second Law does not put any upper limit on the velocity an object can attain. • If, for example, a force of 1000 N were applied to a mass of 1.0 g for 1000 s, then it would reach a velocity………. This value is much greater than the speed of light! However, the Lorentz transformation equations show us that if the relative velocity between two objects is c, then the time dilation is infinite and the length contraction is zero. In other words, time stays still for objects travelling at the speed of light, and the object will have no length in the direction of travel. The mass equation shows that the mass would be infinite. Hence the Special Theory tells us that the maximum velocity an object can attain is c, and also that this velocity is unattainable. It would seem that the only thing that can travel at the speed of light is light itself! Mass-Energy As a consequence of the Special Theory, the mass of a moving object increases. A net force is needed to accelerate any object and this force does work on the object. Newtonian mechanics indicates that the conservation of energy laws imply that the work done in accelerating an object equals the increase in kinetic energy of the object. In other words: Work done = Force x displacement = gain in kinetic energy So F x s = (EK) = ½ m (v2) This equation does not apply in Special Relativity. Since the Laws of Physics hold in all reference frames, we need to find a new relationship between work done and energy transferred. In exploring these ideas, Einstein came to the idea of mass and energy being interchangeable, such that the gain in mass of an accelerated body could be equal to a gain in energy. Hence he established his famous equation: E = mc2 where E is the total energy of the object and m is its relativistic mass. If the object is at rest then it has rest mass energy given by Eo = moc2 These two equations can be combined in terms of the work done in accelerating and object from rest to a velocity v, then the object will have a total mass-energy mc2. Its mass energy will have changed by an amount EK, where EK = E - moc2 = mc2 – moc2 • This is the gain in kinetic energy, and is equal to the work done on the object. The key to understanding these equations is Energy and mass are entirely equal! Mass Energy Equivalence in the VCE Physics Study Design Etot = Ek + E0 = γmc2 where E0 = mc2 and where kinetic energy can be calculated by: Ek = (γ – 1)mc2 Example: Example: Calculate the mass of uranium that is converted into energy in one year (3.15 x 107 s ) in a nuclear power station that has an annual power output of 100 MW. Conservation of Mass-Energy We can no longer consider the laws of conservation of energy and mass as two separate laws of Physics, but rather we need to combine them into one single law. Consider the following example: An electron is accelerated through a potential difference of 2 x 106 volts. Calculate the final velocity of the electron applying: • classical Newtonian mechanics • relativistic mechanics Nuclear Fusion This is the process of joining together two smaller nuclei to form a larger, more stable nucleus. Extremely high temperatures are needed in order for this to occur, for example, in the Sun or other stars. A typical reaction in smaller stars (like our Sun) is: http://hyperphysics.phyastr.gsu.edu/hbase/Astro/ procyc.html Mass defect: The binding energy of the nucleus appears as a loss in mass, Δm, which can be calculated using E = mc2 http://slideplayer.com /slide/8701359/ Heinemann Physics 12 Questions • 7.5 Review page 261 • Questions 4 – 10 (questions 1 -3 relate to relativistic momentum, which is not mentioned in the new Study Design) Data Sheet References Sources of Sample Questions • Text book (Heinemann Physics 12 4th Edition Chapters 6 & 7 have a good selection) • Heinemann Physics 12 3rd Edition has a good selection of multiple choice questions in Chapter 6 (from the old Relativity Option) • Sample and Trial exam papers (2005 -2016 Option, 2017 core) • Other textbooks (Such as Giancoli) • Exam papers from alternative courses (IB, GCSE, etc.) Other aspects to discuss but not specified for the VCE course • Twin paradox: (Minute Physics) https://www.youtube.com/watch?v=Bg9MVRQY mBQ • Electricity and magnetism: (Veratisium) https://www.youtube.com/watch?v=1TKSfAkW WN0&t=14s Contact me…. Email: [email protected]