I realize we are straying from the original question, but I would like to
give my 2-cents worth for the general question.
In the typical physics textbook, Newton's 2nd law (N2L) and F_g = mg
typically occur around chapter 4. Newton's law of gravitation (NLG)
typically occurs around chapter 11. I think this separation is a
pedagogical mistake, and I introduce gravitation within a day or two of
introducing N2L. I do this both in majors-level physics and in
general-education physics. I believe I started getting a lot better student
understanding once I put these together.
When doing F=ma we can say that an object's mass is sort of a resistance to
acceleration, and we call this inertia, and m is called the inertial mass to
indicate it is the mass that appears in N2L.
Then I introduce NLG and say that exactly how gravity works is difficult to
explain, but it is not difficult to describe it as an action-at-a-distance
force that is directly proportional the product of both masses and inversely
proportional to the square of the separation. The constant of
proportionality of G gets F in newtons when the masses are in kilograms and
the separation is in meters.
I point out that gravity and acceleration are not identical concepts and it
is interesting that they both depend on mass. One wonders whether "mass" is
the same number for both. In case they are not the same, we can name the
mass in the gravity equation as the "gravitational mass" to distinguish it
from the inertial mass in N2L.
When we drop something, and we want to predict the acceleration as it falls,
we can use N2L in the form a = F/m. If we put NLG in for F, then we can
visualize the result as the product of two portions. One portion contains
the constant G times the gravitational mass of the earth and divided by the
square of the distance from the center of the earth. The other portion
contains the gravitational mass of the falling object divided by the
inertial mass of the falling object. If it turns out that these are really
the same number, then this term is just the number one (1). This means the
mass of the object no longer appears in the free-fall acceleration equation.
All objects falling in the same region will have the same acceleration of
Gm(earth)/r^2. If we evaluate this number for the mass of the earth and for
the average radius of the earth, we come up with 9.82 m/s^2. We abbreviate
this as g. We can also use this abbreviation in NLG so we have F_g =
gm(object) = m(object)g.
I then say that maybe they learned g = 9.81 m/s^2. I explain that the
actual value measured for g depends on your location on earth. When we
calculated 9.82 we assumed earth was a sphere, that we are all at the same
average altitude, the mass within the earth is homogeneous, and the earth is
not spinning. None of these are true, so the value of g varies from place
to place. The generally used number of 9.81 is in fact not correct for
Bluffton. The value here, rounded to three figures, is 9.80 m/s^2.
I have found that students generally find this presentation fascinating.
Every year several students come up after class and tell me that F = mg
never made sense to them until now. As mentioned, I do essentially the same
presentation for both majors physics and non-majors physics. The non-majors
don't seem to have problems, and they seem to understand F_g = mg better
than before. I have them see if they can find the value of g for Bluffton
to better than three decimal places. Prior to the Internet this required a
trip to the library. Now they can find it online.
I think once students have combined N2L and NGL in their minds, all the
correction terms for altitude, latitude, non-spherical earth, etc. make a
lot more sense. Prior to that, the whole idea of g is nebulous for them and
all the corrections just add to the nebulosity.
Michael D. Edmiston, Ph.D.
Professor of Chemistry and Physics
Bluffton University
1 University Drive
Bluffton, OH 45817
419.358.3270
edmiston@bluffton.edu