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Re: interpreting mass and energy



"... if the three particles are moved "very far" apart from each other and are at rest when "very far" apart, the interaction energies vanish and the
system's total energy is just the sum of the three particle energies."



Don't forget the interaction energy is not lost. Assuming it is
negative, [hope I've got the sign correct.] work is done to separate
them. This means the atoms will have an increased mass.

"NOTE: I do not like the term *rest energy*. It has been my experience that students associate *rest energy* with an object that is literally at rest."

I must be behind the times. I've heard of rest mass but not rest energy. Once one is convinced E = MC^2, there should be no confusion. Note, confusion may arise as a result of my first point.


bc


Joe Heafner wrote:

Good morning.

I have a question that I've been wrestling with since a recent
classroom discussion on mass and energy within the context of special
relativity. I hope I'm able to articulate the questions correctly
enough to get a useful answer.

Classically, an object's mass is apparently related to the number of
atoms that make up that object (e.g. a block of aluminum has a certain
number of aluminum atoms in it). The only way to change the object's
mass is to change the number of atoms in the object (I don't know that
the distinction between inertial mass and gravitational mass is
important here because the two have been shown experimentally to be
equivalent. I may be wrong though.). Furthermore, there's nothing at
all about an object's motion that can make that object have more or
fewer atoms in it and so the object's motion cannot affect the object's
*mass* as defined classically (e.g. a small block of aluminum tossed
across the room doesn't gain atoms by virtue of that motion).

Einstein showed that mass is but one form of energy. Consider a small
isolated multiparticle system made up of, say, three interacting
particles. This system's total energy is just the sum of the
individual particle energies (given by E = mc^2/sqrt(1-v^2/c^2) for
each particle) and the three mutual interaction energies (one
interaction energy for each unique pair of particles). Obviously, if
the three particles are moved "very far" apart from each other and are
at rest when "very far" apart, the interaction energies vanish and the
system's total energy is just the sum of the three particle energies.
If the particles are at rest, the expression for E above reduces to
simply mc^2 for each particle, and the system's total energy is the sum
of the three *rest energies* (see note below regarding this name). Now,
if the particles are at some finite distance from each other, the
system's total energy will be the sum of the particles' existence
energies and kinetic energies (both embodied in the expression for
particle energy, E) and the pairwise interaction energies. Obvsiously,
in this configuration the total energy is greater than that when the
particles are at rest "very far" apart. If we interpret the system's
total energy, E_sys, as being the product of total mass times c^2
(E_sys = M_tot c^2), then since the total energy is greater the total
mass must be greater than the sum of the existence energies. We can
calculate the difference between this new total mass and the sum of the
three existence energies, and it is tremendously small. We can't hope
to even measure it. If we devise an experiment of some kind to measure
the number of particles in our system when the system is in the higher
energy state, we would no doubt find just what we expect, three
particles. Yet, the system's total mass, we can argue, is greater than
the existence energy of just these three particles.

So, my question (at long last) is this. When we speak of the total mass
of a system such as the one described above, we're not permitted to
attribute any excess over and above the total existence energy to
simply an increase in number of particles despite the fact that if we
measure the system's total mass with, say, a balance we would get a
number that we could POSSIBLY, but SHOULDN'T, attribute to an increase
in number of particles. Is this correct? Our balance can't distinguish
between mass as particles and mass as energy. I'm basically looking for
correctness and clarity.



NOTE: I do not like the term *rest energy*. It has been my experience
that students associate *rest energy* with an object that is literally
at rest. They forget that a particle has mc^2 worth of energy even if
that particle is moving, despite the warning that the expression for E
(above) accounts for this. Therefore, in a bold attempt to prevent this
confusion, I now use the term *existence energy* in my classes as the
name for the quantity mc^2. It's energy that a particle has by virtue
of its mere existence; there's no concept of motion implied because a
proton is a proton whether it's at rest or in motion. Use of this name
has indeed seemed to help.

Cheers,
Joe Heafner

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New email address: heafnerj@ctc.net