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Mass and Energy Conservation

Mass and Energy Conservation - variable mass it’s force =...

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Mass and Energy Conservation: Kinetic Energy and Mass for Very Fast Particles As everyone has heard, in special relativity mass and energy are not separately conserved, in certain situations mass m can be converted to energy E = mc 2 . This equivalence is closely related to the mass increase with speed, as we shall see. Suppose a constant force F accelerates a particle of rest mass m 0 in a straight line. The work done by the force in accelerating the particle as it travels a distance d is Fd , and this work has given the particle kinetic energy. As a warm up, recall the elementary derivation of the kinetic energy ½ mv ² of an ordinary non- relativistic (i.e. slow moving) object of mass m . Suppose it starts from rest. Then after time t , it has traveled distance d = ½ at 2 , and v = at . From Newton’s second law, F = ma , the work done by the force Fd = mad = ½ ma 2 t 2 = ½ mv 2 . This won’t work if the mass is varying, because Newton’s Second Law isn’t always F = ma , for
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Unformatted text preview: variable mass it’s force = rate of change of momentum, and if the mass changes the momentum changes, even at constant velocity. An instructive extreme case is the kinetic energy of a particle traveling close to the speed of light, as particles do in accelerators. In this regime, the change of speed with increasing momentum is negligible! Instead, where as usual c is the speed of light. This is what happens in a particle accelerator for a charged particle in a constant electric field, with F = qE . Since the particle is moving at a speed very close to c , in time dt it will move cdt and the force will do work Fcdt . The equation above can be rewritten So the energy dE expended by the accelerating force in the time dt yields an increase in mass, and Provided the speed is close to c , this can of course be integrated to an excellent approximation, to relate a finite particle mass change to the energy expended in accelerating it....
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