CHAPTER 2 Special Theory of Relativity Part 2

Transcription

1 CHAPTER 2 Special Theory of Relativity Part The Apparent Need for Ether 2.2 The Michelson-Morley Experiment 2.3 Einstein s Postulates 2.4 The Lorentz Transformation 2.5 Time Dilation and Length Contraction 2.6 Addition of Velocities 2.7 Experimental Verification 2.8 Twin Paradox 2.9 Space-time 2.10 Doppler Effect 2.11 Relativistic Momentum 2.12 Relativistic Energy 2.13 Computations in Modern Physics 2.14 Electromagnetism and Relativity

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11 2.11: Relativistic Momentum p mumu 0 p x y 0 p x 0 p mumu 0 y u u u u

12 u u u Now look at the same collision in moving frame (one moving with particle on the right). u p p x y 2mu 0 Using the relativistic velocity addition of 2u In this case, moving frame velocity vu; ux u u' x 1 u / c 2mu So, px before is mu' x 1 u / c p after is mu' 2mu x x Momentum is not conserved in moving frame

13 Relativistic Momentum Linear momentum is not conserved if we use the conventions for momentum from classical physics even if we use the velocity transformation equations from the special theory of relativity. There is no problem with the y direction, but there is a problem with the x direction along the direction the ball is moving in each system. Question How do we modify the definition of momentum so that linear momentum is conserved in all frames? Answer This accomplished by redefining momentum to be: mu p mu u / c However keep in mind that u in refers to the particle velocity, not the frame (v).

14 mu mu 1 u / c Relativistic Momentum p 2 2 Frank (fixed or stationary system) is at rest in system K holding a ball of mass m. Frank throws his ball along his +y axis at speed u 0. Mary (moving system) holds a similar ball in system K that is moving in the x direction with velocity v with respect to system K. Mary throws her ball along her y axis at the same speed u 0. The two balls collide and each of them catches their own balls as it rebounds.

15 In system K according to Frank u 0 Frank s ball u 0 If we use the classical definition of momentum, the momentum of the ball thrown by Frank is entirely in the y direction pfy mu 0 The change of momentum after collision as observed by Frank is p 2mu Fy 0

16 In system K according to Frank Mary measures the initial velocity of her own ball to be u' 0, u' u Mx My 0 u 0 In order to determine the velocity of Mary s ball as measured by Frank we use the velocity transformation equations: u Mx u' Mx 1 u' v/ c Mx v 2 v u ' u 1 / My umy u 2 0 v c 1 u' Mx v/ c Mary s ball u Mx u My

17 In system K according to Frank Before the collision, the momentum of Mary s ball as measured by Frank (the Fixed frame) becomes Before Before Mary s ball For a perfectly elastic collision, the momentum after the collision is After After The change in momentum of Mary s ball according to Frank is

18 In system K according to Frank, The total change in momentum of the collision, p F + p M, does not zero! p F = p Fy = 2mu 0 p p 2 mu (1/ 1) 0 F M 0 Similarly, in system K according to Mary, p' p' 0 F M Linear momentum is not conserved if we use the classical momentum even if we use the velocity transformation equations from special relativity.

19 Relativistic Momentum Modification of the definition of linear momentum is required for preserving both linear momentum and Newton s second law. p mu mu p mu u / c keep in mind that u in refers to the particle velocity, not the frame (v). p mu p Classical expression is accurate to within 1% as long as u < 0.14 c. mu

20 mu 1 u / c mu Relativistic Momentum p 2 2 [Example 2.9] Show that relativistic momentum is conserved in the above case. For the Frank s ball in system K according to Frank, Fy 0 2 p mu mu mu mu 1 u / c 2 2 o For the Mary s ball in system K according to Frank, umx v u umy u0 1 v / c u u u v u (1 v / c ) M Mx My o for the relativistic momentum: 1 1 u / c 2 2 M 2mu 1 v / c 2mu pmy mumy mumy 2muMy 1 / 1 / um c u0 c Wow! Same form! pp p 0 Same result for the system K according to Mary. Fy Fy

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22 2.12: Relativistic Energy

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26 Relativistic and Classical Kinetic Energies K mc 2 ( 1) K mc mc 2 2 For speeds u << c, K 1 2 mu 2

27 Total Energy and Rest Energy Rewriting in the form K mc mc 2 2 The term mc 2 is called the rest energy and is denoted by E 0. This leaves the sum of the kinetic energy and rest energy to be interpreted as the total energy of the particle. The total energy is denoted by E and is given by

28 The Equivalence of Mass and Energy

29 Example: energy stored in a stationary golf ball Example: two blocks of wood that collide and stick together

30 Relationship of Energy and Momentum We square this result, multiply by c 2, and rearrange the result. E E p c Or, E m c p c

31 Massless Particles must have a speed equal to the speed of light c 1 1 u / c u u 2 2 Photons!

32 2.13: Computations in Modern Physics

33 Electron Volt (ev)

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35 Example: carbon-12 Mass ( 12 C atom) Mass ( 12 C atom)

36 Binding Energy

37 Binding Energy

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40 Electromagnetism and Relativity Einstein was convinced that magnetic fields appeared as electric fields observed in another inertial frame. That conclusion is the key to electromagnetism and relativity. Einstein s belief that Maxwell s equations describe electromagnetism in any inertial frame was the key that led Einstein to the Lorentz transformations. Maxwell s assertion that all electromagnetic waves travel at the speed of light and Einstein s postulate that light speed is invariant in all inertial frames seem intimately connected.

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43 A moving charge on conducting Wire According to wire frame, the force is magnetic. F B According to moving charge (q 0 ) frame, the force is electric. F E