Imagine two tops made up of exactly one thousand atoms. One is kept at 4 degrees Kelvin, the other at room temperature.
1. Would they weigh the same given an arbitrarily precise scale in the Earth's gravitational field?
2. If both were spun at exactly 100 rpm, would they have the same moment of inertia?
Here is my own initial idea. Gravitational forces bind to energy density. As such, the hotter top would weigh more on a scale because heat contributes to the energy density of the top. The rest mass of a thousand atoms is the same no matter what its temperature. As such, the cold and hot top would have the same moment of inertia.