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2. Antimatter |
Energy is Relative |
In classical physics energy is defined as the capacity to do work, where work is defined as force acting through a distance, and force is defined as the time rate of change of momentum. Therefore, the kinetic energy of an object is equal to the work done by the force accelerating the object from zero to its final speed. Both Serway pp. 34-35 and Tipler p. 74-apply these definitions to derive the equation:
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K = kinetic energy of an object m = g mo = relative mass of the object mo = proper mass of the
object |
K
= mc2 – moc2 |
According
to this equation, kinetic energy is the difference between two quantities. We
call the first of these quantities the total energy of the particle and the
second the rest energy. In terms of these quantities we obtain the following
energy relationships:
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K = kinetic energy of an object E = g Eo = total energy of the
object Eo = rest energy of the
object |
K
= E – Eo E
= mc2 = g Eo = K
+ Eo Eo
= moc2 |
According
to these equations, mass and energy are equivalent (as long as one uses the
conversion factor c2). Every mass has its equivalent energy,
and every energy its equivalent mass. Relativistic mass is simply the total
energy of an object, and proper mass is its rest energy.
Because
mass and energy are the same thing, we now have two words representing the same
physical quantity. This is why many scientists today prefer to restrict the
usage of the words. By using the word “mass” only in reference to energy at
rest and “energy” only in reference to mass in motion, communication becomes
more efficient. There is no longer a need to continually use the qualifiers
“rest” and “relativistic” in front of mass and energy. But the true
justification for discouraging the use of such terms as “relativistic mass,”
“rest mass,” “relativistic energy,” and “rest energy,” is not that they are
invalid or meaningless, but because they are redundant and wordy.
Because
mass and energy are the same thing, it is possible to convert mass into energy
and energy into mass. In fact, any time energy is added to a system, the mass
of that system increases. And when energy is removed, the mass decreases. This
includes energy of every conceivable form: kinetic energy, potential energy,
chemical energy, excitation energy, binding energy, mass, and even antimatter.
The
center-of-mass frame of reference for a system of particles is defined to be that
frame in which the total momentum is zero.
The
proper mass of a system of particles is defined in its center-of-mass frame to
be the mass equivalent of the total energy contained in that system. Therefore,
a spinning flywheel has more mass than one at rest, a compressed spring more
than before compression, a hot pot more than a cold one, an excited atom more
than an unexcited one, and an atomic nucleus less than the total mass of its
components.
The
law of conservation of energy is just as valid in relativistic physics as in
classical physics. The total energy of an isolated system must remain constant
regardless of what happens within the system. Of course, the masses and
energies of the components may vary because they are not isolated but interact
with one another.
Although
mass and energy are the same thing, momentum is not. Nevertheless, mass,
energy, and momentum are not completely unrelated quantities because the mass
and velocity of a particle determine both its energy and its momentum. One
relationship between energy and momentum is expressed through the Lorentz
Transformation of E and p. (See Tipler p.76-80.) Another relationship is
expressed through the following equation:
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E = energy of an object in
the lab frame p = magnitude of the momentum in lab frame mo = proper mass of the
object |
E2
= p2 + (moc2)
2 |
These
relationships are a consequence of the fact that mass, energy, and momentum are
three different aspects of a four-dimensional vector in spacetime. Mass is the
length of the vector, energy the time component, and momentum the space
component. But these concepts are difficult to understand from a TV3D
perspective. They are much more obvious when seen from the invariant point
of view.
Problems:
P1. Kinetic Energy
Show
(a) that the kinetic energy equation can be written as K = (g – 1)moc2 , and (b) that this reduces
to the classical equation K = (1/2)mov2
in the classical limit.
P2. If an electron and a positron
annihilate one another, creating two photons in the process, what is the energy
of each photon? me = 0.5110 MeV/c2.
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ModPhy1/Unit1/SpecialRelativity/RelativeView/