Unit 6 - Physics 225 Relativity and Math Applications...

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Unformatted text preview: Physics 225 Relativity and Math Applications Spring 2010 Unit 6 Conservation, Conversion, and Nuclear Power N.C.R. Makins University of Illinois at Urbana-Champaign 2010 Physics 225 6.2 6.2 Physics 225 6.3 6.3 Unit 6: Conservation and Mass-Energy Conversion Last time, we discovered that Newtonian mechanics has to be modified when we are dealing with speeds close to the speed of light. We followed Einsteins thinking in the early years of the 20 th century, and proceeded to calculate all the essential relations of relativistic mechanics : ! F = d ! p dt ! p = m ! v W = ! F i d ! l = ! E " E = mc 2 = ( pc ) 2 + ( m c 2 ) 2 That is the minimal but complete set of relations, from which everything else can be derived. The first three of these formulas are exactly those of Newton, but some important changes have occurred: The energy E of a particle is now defined to be the particles total internal energy , including its kinetic energy (due to its motion) and its internal energy not related to its motion (due to its rest mass, and for complex particles, any other sources of internal energy like binding energy or heat). For a pointlike, structureless particle, the only source of internal energy is rest mass, and so we can obtain its kinetic energy as follows: KE = E ! m c 2 You may be wondering about potential energy where is it? If your particle is sitting near a planet, it has some potential energy due to the planets gravitational field. The potential energy of a particle in a force field of external origin is not included in E = mc 2 precisely because it involves an outside force: it is not internal to the particle, but instead reflects the particles environment. The inertial mass m of a particle the mass which appears in Newtons second law and in the definition of momentum is not a constant any more. It increases with a particles momentum, as given by the fourth equation above. We introduce the symbol m to indicate a particles rest mass , which is a constant. The rest mass is what you find when you look up particle properties in a reference book. The two are related by m = ! m . The following relations are very useful combinations of the above essential formulas: m = ! m ! p = ! m ! v E = ! m c 2 ! = pc E ! = E m c 2 where ! = v / c and ! = 1/ 1 " ( v / c ) 2 as always. Today we will add two crucial ingredients of relativistic mechanics : energy and momentum conservation . We will find that these lead to a profound and famous consequence of relativity: mass-to-energy conversion . Physics 225 6.4 6.4 Exercise 6.0: Warmup The formulas of relativistic mechanics take some getting used to (just practice, like anything else). Lets start off with a few warmup questions....
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This note was uploaded on 10/06/2011 for the course PHYS 225 taught by Professor Makins during the Spring '10 term at University of Illinois, Urbana Champaign.

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Unit 6 - Physics 225 Relativity and Math Applications...

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