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It is a well known exercise on classical Newton's gravity that assuming that the Earth has an ideal round shape to compute its gravity force at the given distance from its center. (Answer: it is equal to the gravity produced by the centered ball in the Earth whose radius equals to that distance).

I am aware of the fact that the Earth is not an ideal ball, but nevertheless were there experimental attepmts to measure the gravity force of the Earth in holes of various depth and comparing them with the above exercise?

Qmechanic
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MKO
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    Related, possible duplicate: https://physics.stackexchange.com/q/18446/123208 although that question doesn't go into the details of the actual density measurements. – PM 2Ring Apr 25 '20 at 12:58
  • My question is different: I am asking about experiments, not a theory like that question. – MKO Apr 25 '20 at 13:07
  • Also see https://physics.stackexchange.com/q/132935/123208 David's answer there has some links to experimental work related to underground gravity. – PM 2Ring Apr 25 '20 at 14:12

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In the calculation that you refer to having the celestial body be a perfect sphere is not the only simplification. Addtionally the calculation is for the idealized case that the celestial body has uniform density. I haven't looked it up, but I imagine the Earth's iron core has several times the density of the mantle.(Inner core plus outer core has a radius of about 3000 km)

That non-unformity of density doesn't give a very large deviation from the idealized case, but I guestimate that the deviation is large enough to make gravity measurements at various depths in holes of very little use. At best you can go a couple of kilometers deep, that doesn't make a dent (pun intended).

Other than that:
While it is the case that the gravity that you are experiencing at any point on Earth is from the entire Earth, for the most part it is due to the mass in close proximity to you, as gravity falls off quadraticly with distance.

For example, in the Indian ocean basin there is an area that is referred as the 'low geoid anomaly'. At the lowest point the ocean level is a whopping 106 meters below the reference height.

That particular anomaly is extremely wide and deep, but in general there are anomalies all over the place. Precise measurement of gravitational acceleration at the Earth's surface gives information about the geology underneath.

Interpretation of those gravimetric raedings have to take everything into account, including the non-uniform density of the Earth as a whole. Small scale anomalies can only be identified after correcting for all known larger scale anomalies.


Overall:
Gravimetric measurements are done a lot, they play an important part in geology, but I don't expect any added value in taking gravimetric measurements down deep holes. So I don't expect anyone to have done that.

Cleonis
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Gravity measurements have been done in coal mines. Of course the deviations are small, as the depths are limited. The deepest drill into the Earth's crust was 12,262 meters. It should be very hard to measure gravity under the pressure and temperature conditions at such depths. Also the result is predicted to be very similar to the surface value by models, which indicate that gravity is more or less constant up to depths of several kilometers.

Community
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my2cts
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