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In gravitation, we have gravitational potential energy U and gravitational potential Φ where: $$Φ = \frac{U}{m}$$

In electrostatics, we have this instead: $$Φ = \frac{U}{q}$$

In a spring-mass system, we have elastic potential energy U defined as: $$U=\frac{1}{2}kx^2$$ where the point of reference is taken to be $x=0$.

Does a elastic potential similarly exist for spring-mass systems?

Unlike gravitation and electrostatics, a spring-mass system does not have a field associated with it, since the force exerted by the spring does not depend on the mass itself. And since potentials and related to fields by: $$\phi=-\int_{ref}^{\vec{r}} \vec{g}\cdot d\vec{r}$$, it would seem likely that the non-existence of an elastic field suggests the non-existence of an elastic potential.

But I know roughly that the concept of fields was not simply created to separate the problem solving of gravitation/electrostatics into 1. analysis of the effect of other masses/charges (aka finding the field), and 2. analysis of this net effect on the mass/charge in question (aka relating field to force).

Something about fields being more fundamental* than just $\frac{F}{m}$ or $\frac{F}{q}$, and thus the Maxwell equations are written in terms of electric fields, but I'm not too familiar with that, though this nevertheless makes me think that perhaps the idea of fields and potentials can still be applied to springs despite the elastic force being independent of the mass. Can they? Why or why not?

*I read this from Div, Grad, Curl and All That:

The second reason for introducing the electrostatic field is more basic. It turns out that all classical electromagnetic theory can be codified in terms of four equations, called Maxwell's equations, which relate fields (electric and magnetic) to each other and to the charges and currents which produce them. Thus, electromagnetism is a field theory and the electric field ultimately plays a role and assumes an importance which far transcends its simple elementary definition as "force per unit charge".

Note: This question had been asked before at Elastic potential, but both answers seemed to misinterpret what the question was asking.

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  • $\begingroup$ Do you mean to ask whether an elastic potential exists for electrostatic systems? $\endgroup$
    – J. Murray
    Aug 19, 2023 at 14:23
  • $\begingroup$ No, I mean to ask if an elastic potential exists for spring-mass systems. I raised the part about electrostatic fields to explain why I thought that perhaps even though elastic forces were independent of the mass, a field could still be defined meaningfully, and thus a potential could also be defined meaningfully. Anyway, could you enlighten me about how elastic potentials can exist for electrostatic systems? Surely charges and springs are 2 different things? $\endgroup$
    – bluesky
    Aug 19, 2023 at 14:27

2 Answers 2

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You were right in your assumption, since U does only depend on x (and k) and the force is not dependent of the body on the spring, potential is not meaningful.

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Interesting question. It would seem elastic potential has no meaning for a massless spring and I’ve found no reference to elastic potential in the literature. But I'm not so sure that is the case for a spring with mass or for a spring-mass system.

Usually, for a massless spring, we assign the elastic potential energy to some mass attached to the end of an extended or compressed spring. Then perhaps we could define the elastic potential of the spring-mass system as

$$\phi=\frac{1}{2}\frac{kx^2}{m}$$

However, I’d be interested in other’s thoughts on subject.

Hope this helps.

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