Prozorov_34 - PHYSICS 222 Introduction to Classical Physics...

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Unformatted text preview: PHYSICS 222 Introduction to Classical Physics II Prof. Ruslan Prozorov Iowa State University Fall 2011 LECTURES 34 Relativity relativity theories • Special relativity - formulated by Albert Einstein in 1905 based on works of Henri Poincaré and Hendrik Lorentz o is the physical theory of measurement in an inertial frame of reference • General relativity - Einstein's theory of gravitation o the geometric theory of gravitation published by Albert Einstein in 1916. PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 2 The Twin Paradox One of the twin sisters travels 3 light years from Earth at a speed of 0.6c and returns back home to see her twin sister. On Earth, 1o years have passed, whereas the travelling twin calendar only showed 8 years. How can this be? PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 3 The Ladder Paradox let a long ladder fly lengthwise at great velocity past a normal garage. The proper length of the ladder is more than the length of the garage. Due to the length contraction of special relativity, the passing ladder can for an instant (or two) fit into the garage, as observed by the garage owner. However, for an observer riding on the ladder, it is the garage that moves at high speed and is length contracted. For the ladder observer there is no way that the ladder can fit into the garage, even for the briefest of moments. Hence, the whole idea seems paradoxical. http://en.wikipedia.org/wiki/Ladder_paradox PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 4 Albert Einstein’s solution • “The insight fundamental for the special theory of relativity is this: The assumptions of relativity and light speed invariance are compatible if relations of a new type ("Lorentz transformation") are postulated for the conversion of coordinates and times of events... “ • “The universal principle of the special theory of relativity is contained in the postulate: The laws of physics are invariant with respect to Lorentz transformations (for the transition from one inertial system to any other arbitrarily chosen inertial system). This is a restricting principle for natural laws...” A. Einstein, Autobiographical Notes, 1949. PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 5 Einstein’s First Postulate o The laws of physics apply in the same fashion everywhere. Newton’s Laws work no matter what your point of view. o A magnet moving in a coil of wire will induce a current, but … who’s to say if the coil is moving over the magnet or if the magnet is moving through the coil. PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 6 Einstein’s Second Postulate The speed of light is always the same. It matters not how fast you are going or in which direction you travel, the speed of light is always the same. PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 7 information – causality! information (signal) cannot be relayed between any two objects in any reference frame with a speed greater than the speed of light, c PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 8 information – causality! • signal from can reach • but NO information can be exchanged between and (in any direction) (what about and ?) PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 9 simultaneity—watching lightning Perception of one reference frame from another makes it hard to correctly interpret perceptions of physical events. PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 10 Galilean transformation t't x ' x ut y' y z' z PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University u 14 November 2011 11 The Lorentz transformation ux t ' t 2 c x ' x ut y' y z' z 1 u u2 1 2 c Lorentz factor PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 12 same point in two reference frames all reference frames are equal and indistinguishable PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 13 The - factor’s exponential impact PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 14 relativity of time intervals A “light clock” can be used to gain a graphic understanding of the gamma term that modifies the size of any quantity modified by relativistic effects. PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 15 Lengths parallel or perpendicular to motion o Lengths perpendicular to the relativistic velocities are not affected. o Lengths parallel are directly involved. PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 16 Contraction examples PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 17 transformation of EM fields PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 18 Appearance of objects in relativistic motion PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 19 the Doppler effect PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 20 The Doppler effect applied to EM waves The Doppler effect can cause enough change in the velocity of light to change the wavelength perceived by an observer at a distance. “RED SHIFT” is used to measure velocities of distant galaxies PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 21 Relativistic mechanics The energy and momentum of an object with invariant mass m0 (also called rest mass in the case of a single particle), moving with velocity v with respect to a given frame of reference. What if m0=0? -> massless particles MUST move with the speed of light! E m0 c mc 2 p m0 v 2 m is relativistic mass E pc m0 c 2 2 2 2 E pc 0 2 2 E pc m0 c m0vc 2 vc PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 22 relativistic momentum • As velocities near the speed of light, the gamma factor would cause momentum/mass/kinetic energy needed to cause the motion to approach infinity. It’s simply impossible. PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 23 most famous equation EVER: E = mc2 • when a single massive object is at rest (v = 0, p = 0), there is a non-zero mass remaining: mrest = E/c2. • The corresponding energy, which is also the total energy when a single particle is at rest, is referred to as "rest energy". • In systems of particles which are seen from a moving inertial frame, total energy increases and so does momentum. • However, for single particles the rest mass remains constant, and for systems of particles the invariant mass remain constant, because in both cases, the energy and momentum increases subtract from each other, and cancel. • Thus, the invariant mass of systems of particles is a calculated constant for all observers, as is the rest mass of single particles. PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 24 Relativistic examples PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 25 nuclear fusion 3.3445 x 10-27 kg 5.0083 x 10-27 kg m = 3.1375×10-29 kg 6.6465 x 10-27 kg 1.6749×10-27 kg 14.1 + 3.5 = 17.6 MeV = 2.8 X 10-12 J PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 26 the Sun & other stars PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 27 General Relativity Einstein’s tensor (space-time curvature) energy – momentum tensor 1 8 G R g R 4 T 2 c PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University equation of cosmos 14 November 2011 28 gravitation and acceleration produce the same effect PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 29 Newtonian mechanics and relativity PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 30 collapsing star PHYS222 - Lecture 34 - Prof. Ruslan Prozorov - Iowa State University 14 November 2011 31 ...
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This note was uploaded on 11/14/2011 for the course PHYS 5863005 taught by Professor Meyer during the Fall '09 term at Iowa State.

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