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Suppose the ball M is pushed suddenly toward the ball m (along
the line connecting the two centers of mass).
... It's called the /tide/.
That doesn't sound like any definition of "tide" I have heard. For
example, wikipedia's 'Tidal force' page opens with: "The tidal force
is a secondary effect of the force of gravity and is responsible for
the tides. It arises because the gravitational force exerted by one
body on another is not constant across it; the nearest side is
attracted more strongly than the farthest side."
The original example would work perfectly well with point masses.
Hence it is not a tidal force. It is simply the GMm2/r^2 force
between the centers of the two objects, with 'r' changing as one
ball is repositioned.