If you press a ball against a spring and release it, the ball shoots forward. It certainly seems like the spring had a supply of stored energy that was transferred to the ball. Or imagine tossing the ball straight up. Where does its kinetic energy go as it slows? And from where does it acquire kinetic energy as it falls? There’s again a sense that the energy is stored somewhere as the ball rises, then released as the ball falls. But is energy really stored? And if so, where? And how? Answering these questions is key to expanding our understanding of the basic energy model.

The key idea of Chapter 9 was that energy is a property of a system and that forces from the environment—external forces—can change the system’s energy by doing work on the system. Chapter 9 focused on systems with one or more particles in which all forces are external forces that originate in the environment. But that’s not the only way to define the system. An alternative is to define the system with the forces inside. How does that affect the system’s energy?

For example, suppose objects A and B are connected by a compressed spring. If we release them, they’ll fly apart and gain kinetic energy. FIGURE 10.1a is a before-and-after representation and Figure 10.1b is an analysis, based on the discussion of Chapter 9, in which we’ve defined system 1 to consist of only the two objects. The spring forces $\vec{F}$_{Sp on A} and $\vec{F}$_{Sp on B} do work $W_{\text{A}}$ on A and $W_{\text{B}}$ on B. For system 1, the energy principle

tells us that the work done by the spring changes the system’s kinetic energy.

FIGURE 10.1 Two ways to think about a spring pushing objects apart.

But recall that forces are interactions. With that in mind, we can consider a different choice of system, system 2, in which the spring—the interaction—is inside the system. FIGURE 10.1c is an interaction diagram, like those in Chapter 7, showing that A and B are interacting with each other via the spring. It’s important to recognize that a system is an analysis tool, not a physical thing. We can define the system however we want; our choice does not change the behavior of physical objects. Objects A and B are oblivious to our choice of system, so $\text{Δ}{K}_{\text{tot}}$ for system 2 is exactly the same as for system 1.

But system 2 differs from system 1 in one important way. Unlike system 1, system 2 has no external forces that transfer energy to or from the environment; hence $W=0$. Consequently, the energy principle for system 2 is

But we know that system 2 has a changing kinetic energy, so how can $\text{Δ}{E}_{\text{sys}2}=0?$

Because system 2 has an interaction inside the system that system 1 lacks, let’s postulate that system 2 has an additional form of energy associated with the interaction. That is, system 1 has $E_{\text{sys}1}=K_{\text{tot}}$, because particles have only kinetic energy, but system 2 has $E_{\text{sys}2}=K_{\text{tot}}+U$, where $U$, called potential energy, is the energy of the interaction.

If this is true, we can combine $\text{Δ}{E}_{\text{sys}2}=0$, from Equation 10.2, with our knowledge of $\text{Δ}{K}_{\text{tot}}$ from Equation 10.1 to write

That is, system 2 can have $\text{Δ}{E}_{\text{sys}}=0$ if it has a potential energy that changes by just the right amount to offset the change in kinetic energy:

where $W_{\text{int}}$ is the total work done inside the system by the interaction forces.

Equation 10.3 tells us that the system’s kinetic energy can increase $(\text{Δ}K>0)$ if its potential energy decreases $(\text{Δ}U<0)$ by the same amount. In effect, the interaction stores energy inside the system with the potential to be converted to kinetic energy (or, in other situations, to thermal energy)—hence the name potential energy. This idea will become more concrete as we start looking at specific examples. And, since we postulated the existence of an energy associated with interactions, we’ll need to investigate the types of interactions for which this is true.