A system in which particles interact gravitationally or elastically or, as we’ll discover later, electrically has a potential energy. But do all forces have potential energies? Is there a “tension potential energy” or a “friction potential energy”? If not, what’s special about the gravitational and elastic forces? What conditions must an interaction satisfy to have an associated potential energy?

FIGURE 10.27 shows a particle that can move from point A to point B along two possible paths while a force $\vec{F}$ acts on it. Recall that a force does work and changes the particle’s kinetic energy. In general, the force experienced along path 1 is not the same as the force experienced along path 2. Consequently, the work done and the particle’s kinetic energy when it arrives at B depend on which path was followed.

FIGURE 10.27 A particle can move from A to B along either of two paths.

Let’s assume that there is a potential energy $U$ associated with force $\vec{F}$ just as the gravitational potential energy $U_{\text{G}}=mgy$ is associated with the gravitational force $\vec{F}{}_{\text{G}}=-mgĵ$. What restrictions does this assumption place on $\vec{F}?$ There are three steps in the logic:

1. Potential energy is an energy of position. $U$ depends only on where the particle is, not on how it got there. The system has one value of potential energy when the particle is at A, a different value when the particle is at B. Thus the change in potential energy, $\text{Δ}U=U_{\text{B}}-U_{\text{A}}$, is the same whether the particle moves along path 1 or path 2.

2. Potential energy is transformed into kinetic energy, with $\text{Δ}K=-\text{Δ}U$. If $\text{Δ}U$ is independent of the path followed, then $\text{Δ}K$ is also independent of the path. The particle has the same kinetic energy at B no matter which path it follows.

3. According to the energy principle, the change in a particle’s kinetic energy is equal to the work done on the particle by force $\vec{F}$. That is, $\text{Δ}K=W$. Because $\text{Δ}K$ is independent of the path followed, it must be the case that the work done by force $\stackrel{\mathit{⃗}}{F}$ as the particle moves from A to B is independent of the path followed.

In other words, if there is a potential energy, then the work done on the particle has to be the same along every path from A to B.

A force for which the work done on a particle as it moves from an initial to a final position is independent of the path followed is called a conservative force. The importance of conservative forces is that a potential energy can be associated with any conservative force. Specifically, the potential-energy difference between an initial position i and a final position f is

where the notation $W_{\text{c}}(\text{i}\to \text{f})$ is the work done by a conservative force as the particle moves along any path from i to f. Equation 10.27 is a general definition of the potential energy associated with a conservative force.

A force for which we can define a potential energy is called conservative because the mechanical energy $K+U$ is conserved for a system in which this is the only interaction. We’ve already shown that the gravitational force is a conservative force by showing that $\text{Δ}{U}_{\text{G}}$ depends only on the vertical displacement, not on the path followed; hence mechanical energy is conserved when two masses interact gravitationally. Similarly, mechanical energy is conserved for a mass on a spring—an elastic interaction—if there are no other forces. Conservative forces do not contribute to any loss of mechanical energy.