I've proposed exercises that I thought might reveal some conceptual
difficulties previously in this thread. Nobody took me up on them, so
maybe I shouldn't bother trying to do it again. Nevertheless, undaunted
...
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.
b) What would a properly functioning scale read if it were used to
weigh the object?
c) If the object were allowed to fall, what would be its acceleration
relative to the nearby planet?