I suppose I owe those few folks who have any remaining interest in this
topic a resolution to (and a reason for asking) my question from yesterday
morning. I asked:
Consider two identical, spherically symmetric, nonrotating planets of
radius R and mass M orbiting their common center of mass in a circle of
radius 2R (and don't worry about Roche limits and all that!)
a) What is the Newtonian "force of gravity" on an object of mass m near
the surface of one planet when the other is directly overhead.
Answer: Newtonian force of gravity = (8/9) GM/R^2
b) What would a properly functioning scale read if it were used to
weigh the object?
Answer: Reading of properly functioning scale = (137/144) GM/R^2
c) If the object were allowed to fall, what would be its acceleration
relative to the nearby planet?
Answer: Acceleration relative to surface of planet = (137/144) GM/R
Frankly, in retrospect, I'm not completely sure why I thought this
question might promote some useful cognitive dissonance beyond the mildly
interesting facts that
1) there is a difference despite the fact that the earth does
*not* rotate on its axis,
2) in this case the scale reads *more* than the Newtonian force
of gravity rather than less as in the case of the earth,
3) and, of course as *always*, the fact that the scale reading
agrees with m*g only if you interpret g as dv/dt of a freely
falling object.