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I am studying basic mechanics and have reached the chapter on potential energy. However I am a bit confused about the difference between potential energy and the formula for the potential energy due to work done by a conservative force. I am not sure which of the following interpretations is correct:

Possibility 1.

One possibility is that potential energy is a general idea which doesn't have anything to do with conservative forces. The formula for potential energy however would only be for conservative forces, because the textbook says that the work done by nonconservative forces depends on the path and not just the end points, hence a formula for them doesn't exist.

So in this possibility, it would seem as though potential energy is one thing, and the formula for it is another (more specific) thing which only exists for conservative forces. But then why are both referred to as "potential energy", and why does the textbook say that potential energy is defined by the formula (meaning that potential energy as a general concept is specifically only for conservative forces?).

Possibility 2.

Another possibility is that potential energy is only defined for conservative forces to begin with. In this case, there would be no such thing as potential energy for nonconservative forces.

But this confuses me a bit because the book defines potential energy without reference to conservative forces, as simply being the energy a system possesses due to its configuration. However, the reason why I think possibility 2 is correct is because all of the textbooks I have read say that potential energy is defined by the formula for potential energy due to a conservative force. This would mean that potential energy indeed only exists for conservative forces.

Conclusion

I am unable to decide which of the two possibilities is correct. There seems to be a contradiction here which I am not seeing how to avoid.

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Possibility 2 is correct.

Your textbook says that potential energy is the energy possessed by a system due to its configuration. This is the right way to think about potential energy, but it's not very precise - the more precise definition, which is not quite as physically intuitive, is given in terms of work done by conservative forces.

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  • $\begingroup$ So is it the case that "only conservative forces cause a system to possess energy due to its configuration"? (This would have to be true if possibility 2 is correct.) $\endgroup$
    – Raghib
    Commented May 31, 2019 at 2:26
  • $\begingroup$ Someone has answered this on another forum - thanks. Here is their answer: "If the force governing a system is non-conservative then there is not a well defined energy for a given configuration. The path taken to arrive at the configuration would also matter, not just the configuration." $\endgroup$
    – Raghib
    Commented May 31, 2019 at 2:29
  • $\begingroup$ @Raghib Yes. Conservative forces have the property that if you know the initial and final configurations of the system, then you know how much work the force has done. If a box starts on the floor and ends on my shelf, the amount of work gravity does in the interim is fixed (say, 10 J) whether I lifted the box straight up or mailed it to Japan and back first. That's why it makes sense to talk about the energy of a configuration - because it doesn't matter how the system got that way. For nonconservative forces, this is not true, so a configuration does not correspond to a unique energy. $\endgroup$
    – J. Murray
    Commented May 31, 2019 at 2:30

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