(Path independence / Zero potential energy / Potential energy curves)
In this section, Feynman discusses conservation forces from the perspectives of path independence, zero potential energy, and potential energy curves.
1. Path independence:
“…in general the work depends upon the curve, but in special cases, it does not (Feynman et al., 1963, section 14–3 Conservative forces).”
In general, the work done by a force may be dependent on the path of an object. Specifically, the work done by a conservative force is independent on the path of the object, for example, whether it is curved or straight. Feynman mentions that the word “conservative” does not involve political ideas and it is misleading. In essence, the force is conservative (or “mechanical energy is conserved” during the work done) if the integral of the force times the distance in going from position 1 to position 2 is the same for any curve that connects this pair of points. Interestingly, Feynman explains in the next section that energy is still conserved even for non-conservative forces. However, some may criticize Feynman’s definition of conservative force.
Note: The terms non-dissipative force and dissipative force could be used instead of conservative force and non-conservative force respectively.
We should clarify that path independence is only a feature of a conservative force. A more comprehensive definition of conservative force may include the following: (1) The work done by a conservative force can be related to a difference between the initial and final value of a potential energy function: DW = -DU. (2) The work done is reversible and it is independent of the path of the object; the total work done is zero if the starting point and ending point are the same, DW = 0. (3) Conservative force is constant or time-independent. (4) The curl of a conservative force is zero, for example, the curl of an inverse square law force is zero. (5) The conservative force may be a spring force (F = -kr) or a central force (F = GMm/r2 or F = Qq/4pe0r2).
2. Zero potential energy:
“The constant has been chosen here so that the potential is zero at infinity. Of course, the same formula applies to electrical charges because it is the same law (Feynman et al., 1963, section 14–3 Conservative forces).”
According to Feynman, we can assume an object located at position P has zero potential energy, U(P) = 0. If we use any other point, say Q, instead of P, it would turn out that the potential energy of the object (at any point) to be changed only by the addition of a constant, C. Thus, the difference in potential energy between two points (1) and (2): [U(1) + C] – [U(2) + C] = U(1) − U(2) and this shows that it does not matter if we add a constant to the potential energy. Physics teachers may add that we are unable to determine the absolute potential energy (or absolute kinetic energy) by using any experiment. Essentially, physicists measure the change in potential energy (or its effect) and relate it to a conservative force.
We can choose a “standard” point P and set the potential energy at this point as zero. Although Feynman says that the point P can be arbitrarily chosen, he has demonstrated the usefulness of setting zero potential energy at infinity. Using a rocket problem, he assumes the kinetic energy of the rocket to be zero at a point somewhere far away and the potential energy is GmM divided by infinity, which is zero. In Tips on Physics, he writes that “K.E. + P.E. at a = K.E. + P.E. at ¥ (Feynman et al., 2006, p. 73).” Note that we can idealize an object that is significantly far from the Earth as located at infinity. Alternatively, physics teachers could explain that the potential energy –GMm/r is approximately zero where r >> R (R: radius of the Earth).
3. Potential energy curves:
“Therefore we find curves of potential energy in quantum mechanics books, but very rarely do we ever see a curve for the force between two molecules… (Feynman et al., 1963, section 14–3 Conservative forces).”
In a curve of the potential energy U(r) as shown in Fig. 14–3, the minimum of potential energy at r = d means that the work done (or the change in potential energy) is nearly zero if we move a short distance. There is only a little change in potential energy at the bottom of the curve, or equivalently, the force is approximately zero at the equilibrium point. One of Feynman’s reasons for discussing the potential energy curves is to bring out the idea that force (as compared to energy) is not particularly suitable for quantum mechanics. Physicists prefer to use the term interaction, but they still adopt phrases such as “strong force” and “weak force.” More important, we are unable to measure force by determining the acceleration of sub-microscopic particles; in a sense, the word force is used as a metaphor in quantum theory.
Feynman mentions that if several conservative forces are acting on an object at the same time, then the total potential energy of the object is the sum of the potential energies from each of the conservative forces. In addition, molecular forces are a complicated function of the positions of the atoms instead of simply a sum of terms from pairs of atoms. In Volume II, Feynman incorrectly reasons that “[f]or any radial force the work done is independent of the path, and there exists a potential. If you think about it, the entire argument we made above to show that the work integral was independent of the path depended only on the fact that the force from a single charge was radial and spherically symmetric (Feynman et al., 1964, section 4–4).” However, it can be shown that the curl of an inverse square force (1/r2) is zero, but the curl of a force that is inversely proportional to r is not definitely zero (Kleppner & Kolenkow, 2014).
Questions for discussion:
1. How would you criticize Feynman’s definition of conservative force?
2. Why do physicists define potential energy of an object to be zero when it is located at infinity?
3. Would you use the term conservative force in the context of quantum mechanics?
The moral of the lesson: in general, the work done by a force depends upon the path of an object, but we may say that the force is conservative if it does not depend on the path in special cases.
References:
1. Feynman, R. P., Gottlieb, & M. A., Leighton, R. (2006). Feynman’s tips on physics: reflections, advice, insights, practice: a problem-solving supplement to the Feynman lectures on physics. San Francisco: Pearson Addison-Wesley.
2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.
3. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.
4. Kleppner, D., & Kolenkow, R. (2014). An Introduction to Mechanics (2nd ed.). Cambridge: Cambridge University Press.
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