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

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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
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Physics 225 6.2 6.2
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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 Einstein’s 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 0 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 particle’s 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 0 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 planet’s 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 particle’s environment. The inertial mass m of a particle – the mass which appears in Newton’s second law and in the definition of momentum – is not a constant any more. It increases with a particle’s momentum, as given by the fourth equation above. We introduce the symbol m 0 to indicate a particle’s 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 0 . The following relations are very useful combinations of the above essential formulas: m = ! m 0 ! p = ! m 0 ! v E = ! m 0 c 2 ! = pc E ! = E m 0 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 .
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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). Let’s start off with a few warmup questions. (a) Newton’s most famous formula is F = ma … yet it is conspicuously absent from our summary of relativistic mechanics. Hmm. Most of Newton’s other formulas are preserved, like p = mv . Honestly, it looks like all the relations underlying F = ma are still there, yet our familiar friend is absent. What has happened to F = ma ? Maybe it’s fine, maybe it’s so easy to derive from the other relations that we just didn’t bother to write it down.
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