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The Theory of Special Relativity

The Theory of Special Relativity. Ch 26. Two Theories of Relativity. Special Relativity (1905) Inertial Reference frames only Time dilation Length Contraction Momentum and mass (E=mc 2 ) General Relativity Noninertial reference frames (accelerating frames too)

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The Theory of Special Relativity

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  1. The Theory of Special Relativity Ch 26

  2. Two Theories of Relativity Special Relativity (1905) • Inertial Reference frames only • Time dilation • Length Contraction • Momentum and mass (E=mc2) General Relativity • Noninertial reference frames (accelerating frames too) • Explains gravity and the curvature of space time

  3. Classical and Modern Physics Classical Physics – Larger, slow moving • Newtonian Mechanics • EM and Waves • Thermodynamics Modern Physics • Relativity – Fast moving objects • Quantum Mechanics – very small

  4. 10% c Speed Atomic/molecular size Size

  5. Correspondence Principle • Below 10% c, classical mechanics holds (relativistic effects are minimal) • Above 10%, relativistic mechanics holds (more general theory)

  6. Inertial Reference Frames • Reference frames in which the law of inertia holds • Constant velocity situations • Standing Still • Moving at constant velocity (earth is mostly inertial, though it does rotate)

  7. Basic laws of physics are the same in all inertial reference frames • All inertial reference frames are equally valid

  8. Speed of Light Problem • According to Maxwell’s Equations, c did not vary • Light has no medium • Some postulated “ether” that light moved through • No experimental confirmation of ether (Michelson-Morley experiment)

  9. Two Postulates of Special Relativity Einstein (1905) • The laws of physics are the same in all inertial reference frames • Light travels through empty space at c, independent of speed of source or observer There is no absolute reference frame of time and space

  10. Simultaneity • Time always moves forward • Time measured between things can vary Lightning strikes point A and B at the same time O will see both at the same time and call them simultaneous

  11. Moving Observers 1 On train O2 - train O1 moves to the right On train O1 – train O2 moves to the left

  12. Moving Observers 2 • Lightning strikes A and B at same time as both trains are opposite one another

  13. Train O2 will observe the strikes as simultaneous • Train O1 will observe strike B first (not simultaneous Neither reference frame is “correct.” Time is NOT absolute

  14. Time Dilation • Consider light beam reflected and observed on a moving spaceship and from the ground

  15. Distance is shorter from the ship • Distance is longer from the ground • c = D/t • Since D is longer from the ground, so t must be too.

  16. On Spaceship: c = 2D/Dto Dto = 2D/c On Earth: c = 2 D2 + L2 Dt v = 2L/Dt L = vDt 2

  17. c = 2 D2 + v2 (Dt)2/4 Dt c2 = 4D2 + v2 Dt2 Dt = 2D c 1 –v2/c2 Dt = Dto 1 - v2/c2

  18. Dt = Dto √ 1 - v2/c2 Dto • Proper time • time interval when the 2 events are at the same point in space • In this example, on the spaceship

  19. Is this real? Experimental Proof • Jet planes (clocks accurate to nanoseconds) • Elementary Particles – muon • Lifetime is 2.2 ms at rest • Much longer lifetime when travelling at high speeds

  20. Time Dilation: Ex 1 What is the lifetime of a muon travelling at 0.60 c (1.8 X 108 m/s) if its rest lifetime is 2.2 ms? Dt = Dto √ 1 - v2/c2 Dt = (2.2 X 10-6 s) = 2.8 X 10-6 s 1- (0.60c)2 1/2 c2

  21. Time Dilation: Ex 2 If our apatosaurus aged 10 years, calculate how many years will have passed for his twin brother if he travels at: • ¼ light speed • ½ light speed • ¾ light speed

  22. Time Dilation: Ex 2 • 10.3 y • 11.5 y • 10.5 y

  23. Time Dilation: Ex 3 How long will a 100 year trip (as observed from earth) seem to the astronaut who is travelling at 0.99 c? Dt = Dto 1 - v2/c2 Dto = Dt 1 - v2/c2 Dto = 4.5 y

  24. Time Dilation: Ex 3 If our apatosaurus aged 10 years, and his brother aged 70 years, calculate the apatosaurus’ average speed for his trip. (Express your answer in terms of c). ANS: 0.99 c

  25. Length Contraction • Observers from earth would see a spaceship shorten in the length of travel

  26. Only shortens in direction of travel • The length of an object is measured to be shorter when it is moving relative to an observer than when it is at rest.

  27. Dto = Dt √ 1 - v2/c2 v = L Dto = L/v (L is from spacecraft) Dto Dt = Lo/v Lo = L v v √ 1 - v2/c2 L = Lo √ 1 - v2/c2

  28. L = Lo √ 1 - v2/c2 Lo = Proper Length (at rest) L = Length in motion (from stationary observer)

  29. Length Contraction: Ex 1 A painting is 1.00 m tall and 1.50 m wide. What are its dimensions inside a spaceship moving at 0.90 c?

  30. Length Contraction: Ex 2 What are its dimensions to a stationary observer? Still 1.00 m tall L = Lo √ 1 - v2/c2 L = (1.50 m)(√ 1 - (0.90 c)2/c2) L = 0.65 m

  31. Length Contraction: Ex 3 The apatosaurus had a length of about 25 m. Calculate the dinosaur’s length if it was running at: • ½ lightspeed • ¾ lightspeed • 95% lightspeed

  32. 21.7 m • 15.5 n • 7.8 m

  33. Four-Dimensional Space-Time Consider a meal on a train (stationary observer) • Meal seems to take longer to observer • Meal plate is more narrow to observer

  34. Move faster – Time is longer but length is shorter • Move slower – Time is shorter but length is longer • Time is the fourth dimension

  35. Momentum and the Mass Increase p = mov 1 - v2/c2 Mass increases with speed mo = proper (rest) mass m = mo 1 - v2/c2

  36. Mass Increase: Ex 1 Calculate the mass of an electron moving at 4.00 X 107 m/s in the CRT of a television tube. m = mo 1 - v2/c2 m = 9.11 X 10-31 kg = 9.19 X 10-31 kg 1 - (4.00 X 107 m/s)2/c2

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