Consider the gravitational field of a slightly idealized
planet. We assume uniform density, but just for fun we
allow it to be neither static nor spherical. If it's
rotating, if it's in mechanical equilibrium it will be
an ellipsoid.
Of course you could have the corresponding question
in electrostatics, for an ellipsoidal distribution of
material with a built-in uniform charge density.
Suppose we want to calculate the magnitude and direction
of the field at various places, including everywhere on
the surface.
It outlines the contributions of Newton, Maclaurin, Jacobi, Meyer,
Liouville, Dirichlet, Dedekind, Riemann, Poincaré, and Cartan.
It gives formulas you can use, if you like evaluating lots of
spherical harmonics.
Let me briefly mention some things that you *cannot* do.
The field is less at the equator, and the equator is farther
away from the center, but you cannot account for the field
by treating the planet as a point mass at the center and
simply applying the 1/r^2 factor. That gives the wrong
answer by a wide margin. It overestimates the effect. Note
that at the equator, there is more stuff under your feet,
and at the pole there is less, so this tends to compensate
for the 1/r^2 dependence.
The other thing you cannot do is ignore centrifugal effects.
The centrifugal field exists in the rotating frame (and not
otherwise). It has nothing to do with whether or not any
particular object is rotating; it depends entirely on
the rotation of the chosen reference frame. I suppose
you could analyze the planet as seen from a non-rotating
reference, but that would be weird, because the ordinary
lab frame *is* a rotating frame. It rotates along with
the earth, and the centrifugal contribution to g in that
frame is nontrivial. It is easily measurable using the
simplest instruments, especially if you measure it at
widely separated places and compare notes.
Most of the introductory physics books I've seen get this
wrong.
If you want to say that rotating frames are outside the
scope of the introductory course, that's fine ... but then
you MUST NOT claim to have explained the value of g in the
ordinary lab frame, or the place-to-place variation thereof.
One possibility: Split the difference: Say that the
centrifugal field exists in the rotating frame and not
otherwise. Say it multiple times if need be. Then
say that the centrifugal field is a smallish but
nontrivial and readily observable correction to the
g value in the ordinary terrestrial lab frame. Then
say that the details are beyond the scope of the
course. We will just use the operational empirical
value of g, and *not* pretend to derive it from
Newton's law of universal gravitation.
BTW this is intimately related to the weightlessness of
astronauts in the space station. The astronauts know
they are weightless, and any child who has seen the
video knows they are weightless. So when the physics
teacher says they are not "really" weightless -- in
defiance of overwhelming evidence -- it just makes
people think that physicists are idiots.
If you want to say that the equivalence principle is
beyond the scope of the course, that's fine ... but
you MUST NOT claim that the equivalence principle is
wrong. It's not wrong. Relative to the relevant
reference frame, those guys really are weightless.
Theory says so, overwhelming evidence says so, and
every child knows it.