When solving a problem, you get to define the system. But your choice has consequences! is the energy of the system, so a different system will have a different energy. Similarly, is the work done on the system by forces originating in the environment, and that will depend on the boundary between the system and the environment.
In Figure 10.1, system 1 is a restricted system of just the particles, so system 1 has only kinetic energy. All the interaction forces are external forces that do work. Thus system 1 obeys
System 2 includes the interaction, so system 2 has both kinetic and potential energy. But the choice of the system boundary means that no work is done by external forces. So for system 2,
Both mathematical statements are true because they refer to different systems. Notice that, for system 2, kinetic energy can be transformed into potential energy, or vice versa, but the total energy of the system does not change. This is our first glimpse of the idea of conservation of energy.
The point to remember is that any choice of system is acceptable, but you can’t mix and match. You can define the system so that you have to calculate work, or you can define the system where you use potential energy, but using both work and potential energy is incorrect because it double counts the contribution of the interaction. Thus the most critical step in an energy analysis is to clearly define the system you’re working with.