Why is inertial mass always equal to gravitational mass?
This was the great mystery of Newtonian physics that was "resolved"
in General Relativity in terms of the Principle of Equivalence (PoE).
The PoE says that there is no way to distinguish the effects of a
local gravitational field from
the effects of being in an absolutely accelerating reference frame.
In simpler words, the PoE says that the gravitational force on an
object is nothing more than an apparent effect due to observing it
from an absolutely accelerated reference frame. It arises for
exactly the same reason that all "inertial forces"--like the
centrifugal and coriolis forces--arise. Inertial forces are always
equal to the product of the inertial mass of the object and the
opposite of the local absolute vector acceleration of the reference
frame from which the object is being observed.
So now the questions (and answers) are,
Q1 How do we determine that we are absolutely accelerating?
A1 Drop a ball. If it "falls", then there appears to be a
gravitational force on *it* and, therefore, *you* are absolutely
accelerating.
Q2 With respect to what am I absolutely accelerating?
A2 You are absolutely accelerating relative to frames that aren't
absolutely accelerating. That is not a trivial statement since, by
definition, frames that aren't absolutely accelerating will be those
in which objects do not "fall." A moment's thought will reveal that
those are the "freely falling frames." (The argument goes as
follows: If I fall with the ball and then release another ball, it
will not "fall" relative to me. Therefore, I observe no
gravitational force on the second ball. Therefore, I am not
absolutely accelerating.)
Q3 How does one determine one's absolute acceleration?
A3 The answer should now be plainly evident. Drop a ball. Measure
its acceleration relative to you. Your absolute acceleration is the
opposite of its acceleration relative to you.
In the case of us standing on the earth, we have an absolute
acceleration of 9.8 m/s^2 upward. As a result, local objects
experience an inertial force equal to their inertial mass times 9.8
m/s^2 downward. We call this the gravitational force.
I don't think I would recommend talking about all of this to an
entire high school class. I'd save it for after class discussions
with those who are unusually curious and capable. These arguments
are subtle and deceptively simple. They will be lost on (and worse,
confusing to) any student who is not exceptionally comfortable
thinking about appearances from different reference frames. It's
hard enough for most novices to think about appearances in frames
that have a constant velocity with respect to them. It is almost
impossible for them to imagine appearances in frames that accelerate
relative to them.
--
A. John Mallinckrodt http://www.csupomona.edu/~ajm
Professor of Physics mailto:ajm@csupomona.edu
Physics Department voice:909-869-4054
Cal Poly Pomona fax:909-869-5090
Pomona, CA 91768-4031 office:Building 8, Room 223