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Re: Mass



fred bucheit wrote:

I think there is a great advantage in defining mass as what a beam
balance measures. It teaches that we can measure mass even though we
can not
actually define it in words. I think it is important that students
realize
this irony. We spend too much time trying to do the impossible.

Fred Bucheit

~ Measurement is the backbone of experiment which is the backbone of
science~

I agree with the spirit of what Fred is saying. He is giving an
operational definition of gravitational
mass. However, I think the same can be done for inertial mass, as I
wrote previously. A beam balance
is usually more convenient than an inertial balance, but once the
equivalence of inertial and gravitational
mass is demonstrated, one may use a beam balance to determine mass.

I think the more sophisticated high school definitions of gravitational
and inertial mass serve well for grownup physicists. Although mass has
been described as "quantity of matter," this is vague and not easily
susceptible of measurement. What is the operational definition of
"quantity of matter?"

As for E=mc^2, I believe the interpretation of m as relativistic energy
(including the kinetic energy associated with the CM motion of a body)
has fallen out of favor among physicists. In the more modern
formulations of special relativity, mc^2 is the rest energy of a body of
rest mass m. Most texts don't even call m "rest mass" (or denote it
"m_0"), because mass is invariant. However, the mass of a body
depends on its internal energy. A stretched spring has more mass than an
unstretched spring. If we had sensitive enough beam balances or inertial
balances, the difference could be detected. There are no
balances that sensitive, but the energy released in nuclear fission or
fusion provides evidence that the idea is correct. Incidentally, the
masses of charged particles are determined using the laws of mechanics
(special relativistic) and electromagnetism. In classical mechanics, the
"m" in Newton's second law is inertial mass, and Newton's law can be
used as the basis for an operational definition of inertial mass
once force is known. (The vibrating inertial balance that I described
provides an instrument for the determination of inertial mass, but
inertial mass could be detemined directly from Newton's second law with
straight line motion experiments once force is defined operationally.
The force could even be held constant, so that mass would be the only
variable.) I think that the "m" in relativistic mechanics can be
interpreted similarly as inertial mass. (It is still equivalent to
gravitational mass as in classical mechanics.)

General relativists might see mass as something that warps spacetime,
the warped spacetime determining the paths that free objects follow --
along geodesics.

When one asks what one means when he says that a body "has mass," it is
something like asking what one means when he says someone "has money."
Twins might look alike, but one might turn out to be rich and the other
poor. Or two very different people could have the same wealth. People
can be characterized by the the value of their wealth, whether it be
value of real estate or money in the bank, etc. Quoting from _Principles
of Mechanics, 2nd ed._ by Synge and Griffith (1949), p. 9 at the
beginning of their section on "Mass," "Primitive trade was a matter of
barter; later, money was introduced as a standard scale for comparision
of values, and equivalence in value is now expressed by equality of
price. This exemplifies a process of deep importance in science, namely,
a concentration on some characteristic (value) of a thing and its
expression by a number (price). A barrel of apples is very different
from a pair of shoes, but they may be equivalent if value is the only
characteristic in which we are interested. That the price is the same
expresses complete equivalance as far as our purse is concerned." They
discuss how objects might be different in many characteristics, but can
be equivalent in certain experiments, such as when they placed on a
spring balance or used as a projectile in a ballistic pendulum under the
same conditions. They state, "As we assign a price to each article in
trade, so we may assign a number to each piece of matter, equality of
these numbers implying mechanical equivalence. This number is called
_mass_ and is usually denoted by 'm.'..." They end up saying that
comparison of the mass of objects is usually made by weighing them,
using a balance rather than a spring scale for reasons of accuracy -- in
agreement with Fred and John D. Presumably, the location is the same for
the masses being compared by weighing. The results are essentially
gravitational mass. S&G say very little about inertial mass in their
book, other than the ballistic experiment in which two objects could be
equivalent. [No one knew more about inertial and gravitational mass than
John L. Synge, who was a significant person in the field of relativity.
Around the time that _Principles of Mechanics, 2nd ed._ was written,
John L. Synge wrote an important paper treating mathematically what
happens inside the event horizon of a black hole, based on the
Schwarzschild metric (1950).
<http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Synge.html>. I
am struck with the way his emphasis on mathematical models and his use
of analogies are similar to those used by present-day physics educators.]