Hooke's law from first principles

[Acut]

Following a discussion in this forum, I have a question: Is it possible to derive Hooke's law from first principles?

I think a purely electrostatic model is not adequate: Earnshaw theorem would imply there's no "relaxed" position. Also, electrostatic forces get weaker with increasing distance, the opposite trend of spring forces.

Is my reasoning correct? Have I overlooked something? Is it possible to create a purely electrostatic model of elasticity?

 

[Mapes]

Absolutely. In solids, atoms sit at an energy minimum (specifically, electrostatic attraction balanced by Pauli repulsion) that governs interatomic spacing. The energy values $E(x_0+\Delta x)$ around any minimum at $x_0$ can be expanded as a Taylor series,

$$E(x_0+\Delta x)=E(x_0)+\frac{\partial E(x_0)}{\partial x}\Delta x+\frac{1}{2}\frac{\partial^2 E(x_0)}{\partial x^2}(\Delta x)^2+\dots\approx \frac{1}{2}\frac{\partial E(x_0)}{\partial x}(\Delta x)^2$$

which is Hooke's Law where $k=\partial^2 E(x_0)/\partial^2 x$ (taking $E(x_0)$ as our energy reference and noting that $\partial E(x_0)/\partial x$ is zero because we're at an energy minimum). Does this answer your question?

 

[Acut]

You Taylor series reminded me we may model spring forces with nearly any kind of forces - since Hooke's law is linear, many forces will fit it for suficiently small displacements.

So there's something that avoids the consequences of Earnshaw's theorem- Pauli Repulsion.
Is Pauli Repulsion explained with quantum mechanics? Or is there a way of modelling this repulsion using only classical mechanics?

 

[Mapes]

Pauli repulsion is indeed explained with QM. And on the classical side, nuclear electrostatic repulsion may play a part too in controlling equilibrium atomic spacing, though I don't know offhand to what extent.

These effects are often modeled empirically with a repulsion term (e.g., in the http://en.wikipedia.org/wiki/Lennard-Jones_potential" ).

 

[PJC01]

I would like to clarify that Hooke's law is a fundamental law of physics that describes the relationship between the force applied to an elastic material and the resulting deformation. It states that the force applied is directly proportional to the displacement produced, as long as the material remains within its elastic limit. This law was discovered by the English scientist Robert Hooke in the 17th century, and it has been extensively studied and validated through experiments and observations.

Now, to answer the question at hand, it is indeed possible to derive Hooke's law from first principles. In fact, this is exactly what Hooke did when he first proposed the law. He observed the behavior of springs and deduced the relationship between the force applied and the resulting deformation. This was later confirmed by experiments and further developed by other scientists.

However, it is important to note that Hooke's law is a macroscopic description of the behavior of elastic materials. It does not take into account the microscopic interactions between atoms and molecules that give rise to the macroscopic behavior. Therefore, while a purely electrostatic model may not fully explain the phenomenon of elasticity, it can still be used as a simplified model to illustrate the concept.

Furthermore, the Earnshaw theorem states that a system of static charges cannot be in a stable equilibrium. However, this does not apply to elastic materials as they are not made up of static charges but rather moving atoms and molecules. Therefore, the theorem does not contradict Hooke's law.

In conclusion, while a purely electrostatic model may not fully explain the phenomenon of elasticity, it is still possible to derive Hooke's law from first principles. The law has been extensively studied and validated through experiments and observations, and it remains a fundamental principle in understanding the behavior of elastic materials.