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Re: g, acceleration, field



At 08:03 PM 1/25/01 -0500, Herbert H 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?

I agree that "g" is a standard symbol denoting acceleration, especially in
the context F=mg.

On Thu, 25 Jan 2001 16:53:34 -0700 Jim Green <JMGreen@SISNA.COM> writes:

> carful explanation of the concept of the artificial mathematical
> invention of a field.

I agree that the concept of a field is something that needs to be
explained. But it takes only a few minutes to explain that a field is a
quantity (typically a number or a vector) that varies from place to place.

In elementary mathematics there is a distinction between
-- the sine function, is a function, which we can represent by a graph
-- the sine function evaluated at a point, e.g. sin(3), which is
a number, which we can represent by a numeral.

So it is with fields in general, and gravity in particular. The concept of
gravity might refer to
-- the gravitational field throughout space, or
-- the gravitational field evaluated at a point.
The latter is an acceleration.

> My position is quite simple: "g" should not be called "acceleration
> due to gravity" because it is not acceleration!

Maybe, maybe not. It is quite conventional to use "g" to denote the
gravitational field evaluated at the point of interest, in which case it
most certainly is an acceleration. It also quite conventional to use "g"
or "g sub n" to denote the _standard_ acceleration of gravity, which has
the assigned value
9.806 65 m s-2 (exactly)
and which is most definitely an acceleration. Reference:
http://physics.nist.gov/cgi-bin/cuu/Value?gn


Alas we do not have a widely-accepted good notation for denoting the whole
field; all we have is a number of mediocre (or worse) notations.

One good notation would be to write:
for_all x, g(x)
or equivalently, using the formal notation of lambda calculus:
(lambda x . g(x))
and both of those clearly denote functions. There are also set-theoretic
expressions (the set of ordered pairs such that ...). Various computer
languages have various good and not-so-good ways of talking about functions
and their values.

Other possible notations include
g()
and
g(...)
which are passable ways to indicate functions.

In contrast, g(y) by itself looks like the value of the function when
evaluated at the particular point y.

> If the student sits stationary in the
> lab, "g" is not zero but there is no acceleration -- at least no
> local acceleration.

In such a case it would be much better to say that g is an acceleration,
but there is no _net_ acceleration because there is a countervailing
acceleration provided by the seat.

Having no _net_ acceleration is very different from having no acceleration(s).