Non-oblateness: the deviation of the earth’s shape from a true oblate spheroid; e.g., the north pole is farther from the center of the earth than is the south pole. There are also additional bulges and flattened spots on the earth which cause some people to say the earth is pear-shaped [1]. I believe these non-oblateness features are on the order of 20 meters, tiny
compared to the 21 km oblateness, which is tiny compared to the 6000 km radius of the earth.
mountains and basements: a mountain is a natural formation of extra mass at larger radius. Basements are artificial formations of removed mass at smaller radius. My “mountains and basements” phrase is short for “mountains and valleys and buildings and basements” by which I mean the natural and artificial bumpiness of the earth’s surface. I was thinking of an anecdote from a “Fifth Force” seminar I heard years ago [2].
Dex: I think the objections to my original answer are better stated in the other thread: the same “forces” that fling the earth into oblateness also fling the plumb bob, such that the plumb bob is flung into line with the normal. This may be true for an elastic liquid perfectly oblate spheroid earth, but I am not going to answer that question (we particle physicists hate non-inertial reference frames.) I prefer to evaluate the assumptions by rephrasing the question: are the “forces” that fling the plumb bob the same ones that cause the shape of the earth? The answer is an unequivocal NO. If the rotational and self-gravitational forces are the only effects determining the shape of the earth, then the earth WOULD be a perfect oblate spheroid. We KNOW the earth is not a perfect oblate spheroid; people have measured the difference between the north pole radius and the south pole radius; people have measured the bulges and flattened spots; I myself have seen a mountain or two. Thus, we are forced to conclude that there are additional effects at
work, presumably the rigidity of the earth’s crust, plate tectonics, tidal effects from the sun and the moon, and maybe the Chandler wobble for all I know. (There is also considerable earth-shaping caused by those bulldozers that wake me every morning with their bleepity bleep beeping.) The shape of
the earth is in fact a hot topic of current research; I’m sure the GPS folks tear their hair out in frustration everyday trying to model it.
Therefore the earth’s shape is much more complex than a simple oblate spheroid, and as such we cannot expect that a plumb bob would ever line up with the normal. I will admit that the plumb bob is probably flung a bit by the earth’s rotation, as well as the local mass distribution. Nonetheless,
the plumb bob measures what you typically want to know – the direction
stuff will fall, aka “down”.
If you insist that the normal of an oblate spheroid goes through its center, then I can’t help you.
Footnotes:
[1] I believe it is more correct to say that the average radius at a specific latitude (i.e., integrated over longitude) resembles the cross section of half a pear.
[2] As you know, gravitational and electromagnetic forces are very long range, while the weak and strong forces are very short range. This irks some people and about 15 years ago, scientists were all hot to find a medium range “fifth” force. To measure this hypothetical force, you have to
account for very subtle effects of gravity, and the scientists therefore had to know the distribution of mass in their location, so they went down the street knocking on the doors of serious corporations, in truly Cecilian fashion, asking the dimensions of their basements. As you can imagine, the
corporations were suspicious of such inquiries, and the scientists had to do a lot of explaining before the corporations would give them their basement blueprints.