I understand $E=mc^2$ calculates the amount of energy inherent in a given mass. Mass meaning "an object's resistance to acceleration (a change in its state of motion) when a net force is applied" (1) and Energy meaning "the property that must be transferred to an object in order to perform work on, or to heat, the object." (2) This Mass-energy equivalence makes more intuitive sense to me when thinking of those explicit definitions and how they are inherently related to one another and tied together through fundamental forces.
As I consider my question, I may just be unclear on the nature of mass and how it is independent of other properties. But I can't help to wonder - does a gram of sugar, a gram of soil, a gram of water, and a gram of lead all contain the same inherent amount of energy, because they are all of equal mass (or, said in a different way, because they are all of equal resistance to acceleration)? It could be that the only difference between equal masses of those materials are in other properties than their inherent Energy (for example, a gram of water and a gram of lead have different volume and conductivity).
Answer
To answer your actual question:
Does a gram of sugar, a gram of soil, a gram of water, and a gram of lead all contain the same inherent amount of energy, because they are all of equal mass?
Yes, you are essentially correct. There are several types of what you call "inherent energy". In the ordinary matter, about 99% of it comes from the energy of gluons, the force particles (or more precisely the strong field) holding protons and neutrons together in the nucleus of atoms. The bulk of the remaining 1% comes from quarks (of which the protons and neutrons consist) interacting with the Higgs field. Then electrons (whose mass comes from other sources) add roughly about 0.05%. The total number of different types of energy contributing to mass is quite large (perhaps a few dozen depending on the situation) and may include the heat, chemical energy of bonds, and many other types, but again, the bulk of it is what is mentioned above. So the exact composition of different types of energy in lead and sugar may be slightly different, but overall only in a tiny fraction of a percent with the rest being essentially the same.
Please note that I am describing the normal everyday matter moving at ordinary speeds. There are things in the universe with mass from very different sources, such as black holes, neutron stars, relativistic objects (if you count the kinetic energy), etc.
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