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Re: Mass



fred bucheit wrote:

Perhaps it is better to define mass as: what a beam balance measures
while
at rest in a gravitational field.

Mass measured with a beam balance in a gravitational field is
gravitational mass. This method of measurement provides an operational
definition of gravitational mass. As I recall, this was the definition
given in high school texts such as _PSSC Physics_ and _Foundations of
Physics_ by Robert Lehrman
and Clifford Swartz. Both texts later introduced inertial mass. In _PSSC
Physics_ students measured inertial mass with an inertial balance
constructed with parallel hacksaw blades which supported a metal
platform between them. The period of vibration of the balance which
vibrates in a horizontal plane depends only on the inertial mass for a
given spring system -- a consequence of Newton's second law in which the
"m" is inertial mass.

PSSC used the mass of a small C-clamp as the unit of mass. The period
was determined for 1, 2, 3 ... identical C-clamps. Then a calibration
curve of mass (in C-clamp units) vs. period was plotted. The inertial
mass of an arbitrary object that could be clamped to the platform
could be determined with the help of the calibration curve. The
gravitational mass of 1, 2, 3, ... C-clamps was determined with an
equal arm beam balance. This showed that, to within the precision of the
experiment, that gravitational mass and inertial mass were proporional.
The PSSC platform had a circular hole just big enough for a metal
cylinder to fit through it. Rather than being clamped to the platform,
it was suspended by a thread fastened to the ceiling of the room. The
idea was that the weight of the cylinder, balanced by the tensional
force exerted by the string on the cylinder (not an action-reaction
pair), had nothing to do with the inertial mass as determined by the
period of the inertial balance.(I did not consider the suspended string
part of the experiment very convincing. I believe it was left out of the
most recent edition. The C-clamp was in vertical equilibrium in the
non-suspended case.)

_Foundations of Physics_ by Lehrman and Swartz appealed to a thought
experiment on the leaning tower of Pisa such as described, if not
performed, by Galileo. If two stones, one with twice the gravitational
mass of the other (as measured with a beam balance), were released
together, they would
strike the ground at the same time, allowing for minor discrepancies.
This would imply equal acceleration. The (gravitational) force F in
Newton's second law for the larger stone would be twice as as great as
for the smaller one. The inertial mass m for the larger stone would also
have to be twice as great as for the smaller stone in order to keep the
acceleration the same, the acceleration being the ratio
of force to mass. L&S did this more formally and in greater detail, as I
wrote on one of these forums several months ago, but the idea is that
inertial mass and gravitational mass are proportional -- equal if the
same object is used as the standard for both.

L&S equate inertial mass with inertia, even to the point of giving it
the symbol "I" until the
proportionality of inertial and gravitational mass is arrived at. They
emphasize operational definitions throughout their text.

The oft-repeated description of inertia, and hence inertial mass, as a
_resistance_ to change in state of motion, is not an operational
definition, but I suspect it has some value in developing intuition.

Hugh Logan