We opened Chapter 9 by introducing the energy principle—basically a statement of energy accounting—but noted that we would need to develop many new ideas to make sense of energy. We’ve now explored kinetic energy, potential energy, work, conservative and nonconservative forces, and much more. It’s time to return to the basic energy model and start pulling together the many ideas introduced in Chapters 9 and 10.

FIGURE 10.28 shows several objects that are acted on by forces. The objects might be particles, but they could also be macroscopic objects that have thermal energy. Their kinetic and thermal energy $K+E_{\text{th}}$ changes as the forces do work on the objects. Suppose we divide the work $W$ into the work $W_{\text{c}}$ done by conservative forces and the work $W_{\text{nc}}$ done by nonconservative forces. With this distinction, the energy principle for the change in the kinetic and thermal energy is

FIGURE 10.28 A system with both internal interactions and external forces.

Let’s define the system so that all the conservative interactions are inside the system. This is the system shown in Figure 10.28. As we’ve learned, the work done by these conservative forces becomes potential energy: $W_{\text{c}}=-\text{Δ}U$. The remaining forces—the nonconservative forces—are now external forces from the environment.

To illustrate what we mean by a nonconservative external force, suppose you pick up a book that is at rest on the floor and place it at rest on a table. The book $\text{+}$ earth system gains gravitational potential energy, but $\text{Δ}K=0$ and $\text{Δ}{E}_{\text{th}}=0$. Where did the energy come from? Or consider pulling the book across the table with a string. The book gains kinetic and thermal energy, but not by transforming potential energy. The force of your hand and the tension in the string are nonconservative forces—they’re not associated with a potential energy—that “reach in” from the environment to change the system. Because these are external forces, the work they do—transferring energy to or from the system—could be labeled $W_{\text{ext}}$ rather than $W_{\text{nc}}$. We will define $W_{\text{ext}}=W_{\text{nc}}$ to be consistent with the notation introduced in Chapter 9. With these changes, Equation 10.28 becomes

Grouping all the energy terms together, we can write Equation 10.29 as

where $E_{\text{sys}}=K+U+E_{\text{th}}=E_{\text{mech}}+E_{\text{th}}$ is the energy of the system. Equation 10.30, the energy principle but with all the terms now defined, is our most general statement about how the energy of a mechanical system changes.

In Section 10.4 we defined an isolated system as a system that does not exchange energy with its environment. That is, an isolated system is one on which no work is done by external forces: $W_{\text{ext}}=0$. Thus an immediate conclusion from Equation 10.30 is that the total energy $E{}_{\text{sys}}$ of an isolated system is conserved. If, in addition, the system is nondissipative (i.e., no friction forces), then $\text{Δ}{E}_{\text{th}}=0$. In that case, the mechanical energy $E_{\text{mech}}$ is conserved. You’ll recognize this as the law of conservation of energy from Section 10.4. The law of conservation of energy is one of the most powerful statements in physics.

FIGURE 10.29 reproduces the basic energy model of Chapter 9. Now you can see that this is a pictorial representation of Equation 10.30. $E_{\text{sys}}$, the total energy of the system, changes only if external forces transfer energy into or out of the system by doing work on the system. The kinetic, potential, and thermal energies within the system can be transformed into each other by forces within the system. And $E_{\text{sys}}$ is conserved in the absence of interactions with the environment.

FIGURE 10.29 The basic energy model.