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Hooke's law from first principles

  1. May 10, 2010 #1
    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?
     
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  3. May 11, 2010 #2

    Mapes

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

    [tex]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[/tex]

    which is Hooke's Law where [itex]k=\partial^2 E(x_0)/\partial^2 x[/itex] (taking [itex]E(x_0)[/itex] as our energy reference and noting that [itex]\partial E(x_0)/\partial x[/itex] is zero because we're at an energy minimum). Does this answer your question?
     
  4. May 11, 2010 #3
    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?
     
  5. May 11, 2010 #4

    Mapes

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    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" [Broken]).
     
    Last edited by a moderator: May 4, 2017
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