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Re: "acceleration due to gravity"



Herb Gottlieb wrote:

Kindly help me understand the difference between
F=ma and F=mg.
In the first equation "a" stands for any
acceleration regardless of the force or mass
that may be involved? In the second equation
"g" stands for the acceleration due to a
gravitational force resulting from a position
of a mass in a gravitational field?

Let me have a go at that, Herb. Basically, F=ma, is a law whereby we
can determine the dynamics of an object of mass m if we know the
(net) force F applied to it. F=mg is a rule for calculating the force
on an object of mass m due to the gravitational interaction of the
earth and the object, after correcting for the rotation of the earth,
and all the other corrections Leigh talked about in his post on the
subject.

An object can have an acceleration numerically equal to *g* under
certain circumstances, but *g* is not, in and of itself, an
acceleration. It only has common units of acceleration because we
know empirically that the "inertial" mass. m-sub-I, and the
"gravitational" mass, m-sub-G, are, to the degree that we can
determine, the same thing, and so we drop the subscripts and cancel
the m from the equation mg=ma, and find that, in the absence of
friction or air resistance, or other impeding factors, the measured
acceleration turns out to be numerically equal to *g.* But when we
are interested in the force balancing the force of gravity in an
equilibrium situation, we still call the force of gravity on the
object mg, even though it isn't accelerating at all.

In other words, F=ma talks about inertial mass, and F=mg talks about
gravitational mass. So the units of *g* are properly N/kg and not
m/s^2. Since we don't have any a priori reason to think they are the
same, we don't have the right to "cancel the masses" out of the
equations until such time as we are satisfied that they are the same.

Let me offer an example where they are not the same. Consider the
electric force. Everybody knows that it is found by multiplying the
Electric Field strength by the "electrical mass." Right? But we don't
even try to cancel the inertial and electrical masses in our
equation, do we, because we know they are not the same. How do we
know? It's easy--we give them different symbols and different names.
We call the electrical mass "charge" and use the symbol *q.* So the
students never even think of cancelling them, as they should not. But
the mathematical formalism between the electrical case and the
gravitational case is identical. And furthermore, we don't know any
more about what "charge" is than we do about what "gravitational
mass" is, except that they are the properties of matter that interact
with the respective fields. And by some accident, exploited by
Einstein in his general theory of relativity, it turns out that
inertial mass and gravitational mass are equivalent.

So, since we have no problem using the concept of electrical field
strength (E) in units of N/C, we should have no problem using the
gravitational field strength (g) in units of N/kg. And we can be
pleasantly surprised when, every so often the acceleration turns out
to be numerically equal to *g.*

Here the particle physicists on the list can correct me, but it is my
understanding that the "mass" that he Higgs field is supposed to
endow matter with is the inertial mass and *not* the gravitational
mass. It still remains a mystery why the two are the same.

A little long-winded, but that's my take on it.

Hugh
--

Hugh Haskell
<mailto://haskell@ncssm.edu>
<mailto://hhaskell@mindspring.com>

(919) 467-7610

Let's face it. People use a Mac because they want to, Windows because they
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