A system’s internal energy is defined as its total energy minus the CM’s energy. It is the part of the energy left over once you move into the centre-of-mass frame and “subtract” the overall motion. But you must be careful about what actually belongs to it.
Warning — Gravitational energy is not internal
The total coincides with : it is therefore fully summarised in the CM. Consequently, when calculating internal energy (total energy minus CM energy), gravitational potential energy does not appear in the internal energy, because it has already been entirely subtracted as part of the CM’s energy.
So what does internal energy contain? Three types of contribution, all tied to what happens inside the system, regardless of where it is or how it moves as a whole:
- internal kinetic energy (motion of the parts relative to the CM);
- internal potential energies (springs between the parts, electrostatic forces between the system’s components);
- thermal energy.
The guiding idea is the separation between the system’s “collective motion” (captured by the CM) and its “internal life” (captured by the internal energy). Gravity acts on the system as a whole — through its centre of gravity — and for this reason belongs to the first group, not the second.
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Topics: Centre of mass Concepts: Centre of mass · Internal energy · Gravitational potential energy
Related exercises: Challenge: chain on a table · Worked exercise — The cat on the trolley · Rocket exploding mid-trajectory