Yes, hit post by mistake, went off a thinking, gone until now; couldn’t delete. Quite a stupid post, even for me, that.
Well wouldn’t you know it, Googling the topic sentence I found in this sub-board a thread six years after this OP on the same column. That one is https://boards.straightdope.com/t/tube-through-the-earth/508787/43
It is shorter and goes more directly to the physics (“elementary” in college according to more than one poster). But this is thread is the fons et origo of discussion, so I post this note here.
Cecil gave up on the math, and that is addressed in the later thread, even with a rough-ish estimate of the period.
Upthread someone says (sorry for no cite) “you can’t and never will make an experiment,” but one not only has been done, in some essential form, but it’ non-experimental use has changed our world and is still ubiquitous: inertial guidance systems, in all stuff that flies “by itself” or otherwise.
The OP subject was aced by M[aximillian] Schuller in 1923, in “Die Störung von Pendel und Kreiselapparaten durch die Beschleunigung des Fahrzeuges,” Physikalische Zeitschrifts XXIV:16. The article is translated into English here, having been translated at RAND as a (once-) Secret document: “The Perturbation of Pendulum and Gyroscope Instruments by Acceleration of the Vehicle” https://apps.dtic.mil/dtic/tr/fulltext/u2/b806120.pdf
A Schuler Pendulum has the (ideal) orbital period of a satellite at sea level. The physical practice of this (“the experiment”) simply flips the situation: a gyroscopically stable axes-platform is torqued in feedback by some system so that it remains "“pointing straight down,” done in respect to “Schuler Tuning.” From https://en.wikipedia.org/wiki/Schuler_tuning
An inertial navigation system is tuned by letting it sit motionless for one Schuler period. If its coordinates deviate too much during the period or it does not return to its original coordinates at its end it must be tuned to the correct coordinates.To implement Schuler tuning, the feedback loop is modified to tilt the platform as the vehicle moves in the north–south and east–west directions, to keep the platform facing “down.” To do this, the torquers that rotate the platform are fed a signal proportional to the vehicle’s north–south and east–west velocity. The turning rate of the torquers is equal to the velocity divided by the radius of Earth […]
It works out to a effortless back and forth through the earth of in Schuler period of 84.4 sec (the figure is posted in the '09 thread).
Surprising to me is that mechanical inertial systems (torque applied via servomotors in loop) are still out there in most airplanes, even with the technology more recently advanced to to stability measured by interference of laser beams moving along the axes, either in electrically charged gas (ring laser guidance) or fiber-optic laser guidance. Even post GPS, inertial guidance systems are still built in as complementary systems.
Although dated, from IEEE is the best general article introducing Schuler and inertial guidance systems is King’s 1998 “Inertial Guidance–40 years of Evolution,” GEC Review, XIII:3. https://www.imar-navigation.de/downloads/papers/inertial_navigation_introduction.pdf
A more comprehensive thing with refs is Wiki’s https://en.wikipedia.org/wiki/Inertial_navigation_system
ps: plus, @Riemann, my abandoned post obviously wasn’t (isn’t) “word salad.” It was just abandoned suddely and incomprehensibly, given the clarity of what was posted. Look up the term (my snarl is because the phrase is a term of art, and was created by practitoners, in types of mental illness. Do you say “quantum leap” incorrectly often?