https://www.physik.hu-berlin.de/qom/pdfs/Schmidt2011.pdf
I think we would have the technology. This research team measured 10^-10 times Earth's surface gravity.
No, you can't really figure it out.
You get stuck when you narrow it down to a 1D problem (you can find this axis by lots of spinning and help from the Intermediate Axis Theorem). You can then measure the total mass and perpendicular moment of inertia. This leaves 2 equations and 4 unknowns...
That's incorrect, but I thought the same thing. It's one of the logical traps built into the problem.
As a start, think of the tension in the string if you submerge a hanging ball. The tension will reduce from the work done by displacing water. Try pushing an object into a glass of water (I...