This is a follow-up to a previous question How can you weigh your own head in an accurate way?.
My purpose is not to restart the flurry of more or less humorous jokes (which are not such a bad thing when infrequent) but to try to draw some conclusions, as I found the event instructive, and hopefully to get a real answer.
Beyond the jokes, I think the question was also interesting from a more social point of view: how fallacies operate, how people vote, what is the effect how the fast answer inducement by site rules.
It is clear now, at least to me, that the real physical problem was to determine the mass density distribution of a (partly ?) solid object without using destructive means. Of course the concept of destructiveness may depend on the concerned body, and that was one source of jokes.
Still, it is a relevant topic to review the means, if any, for achieving such a purpose, as it seems that there are not that many. This is actually the question I am asking.
But I would like also, at the same time, to review some of the answers given to the previous qiestion.
One that attracted my attention was the micro-satellite solution. Does it really work ? So we could precisely ask: given a precise knowledge of the gravity field created by a solid object, can we deduce from it the mass density distribution in the solid. The answer seems to be no. Take the trivial case of a sphere with a uniform radial density distribution. The gravity outside the sphere depends only on the total mass, so that it tells us nothing about the internal density distribution. Is that an exceptional degenerate case, or is it a strong hint that the analysis of the gravity field is not enough ?
The slicing solution I proposed was an interesting mistake, certainly a silly one. But it is ever so tempting to believe the solution is near when you have a set of equations, and easy ones to boot. A cute trap. It becomes more obvious when you try to do it in continuous rather than discrete form.
It should have been obvious, as we know that the torque produced by weighing a massive object depend only on total mass and distance of the center of mass to the axis.
What was less obvious, at least to me (but I have not done much
physics since college), was the use of the moment of inertia. One
could also think of measuring moments of inertia after virtual slicing
of the object as previously with torque. Unfortunately the moment of
inertia depends only on 3 values, the previous two and a reference
moment of inertia (for a given direction of the rotation axis). It is
a useful remark for computing moments of inertia, but shows that there
is no hope there for solving our problem.
This was remarked by @Ben Crowell in another question (see below).
More generally, it seems there is no hope from any discretized measurement of a quantity that depends polynomially on the distance. One cannot get more unknowns than the degree of the polynomial.
I will not comment on Compton scattering, and other uses of indirect physical phenomena, if only for lack of competence. There is also the fact that I would like to know of a solution involving only mechanics.
One technique I heard of is measuring wave propagation across the object. I hinted to that in a comment on the Compton scattering solution. I have no expertise on that, but I think that is how the structure of the planet is analyzed by geophysicists (earthquake waves). I also heard of underground analyses by similar means, using explosives. But I am not sure about the type of information that can be obtained in this way. Does one get density distribution. Does it always work ?
The other technique I thought of is ultrasound based medical imagery. Could it measure densities.
Now since we are considering waves, could we get something out of the mesurement of gravity wave propagation, assuming it is possible to do the necessary measurements. But I have no idea what this actually entails or means.
Interestingly, a simpler version of the problem has already been discussed to some extent two months ago in How can I determine whether the mass of an object is evenly distributed? Some ideas were repeated for the recent question, but new ideas emerged too (gravity field analysis) even if they have weaknesses, and are somewhat hard to use in most situations.
Then, what techniques have been used to know the density distribution of Earth, the Sun or possibly other bodies. Is there anything systematic that we overlooked.
So, are there means of solving the problem purely with mechanics and gravity?
A related question is whether it is easier, or as hard, to determine whether the mass density distribution is uniform.
More generally, I am wondering whether there is a way of characterizing the properties of phenomena that can help determine the density distribution inside a solid object.
Alternatively, could one prove it is not achievable by purely mechanical means.