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According to the perspective of a high school student,this question has been asked so pardon me for any lack of intuition or knowledge about this topic.

As far as I know as per Coulomb's law or Newton's law, attraction force or repulsion force should decrease as the distance increases between two particles(charged). When a spring is expanded, shouldn't its attraction force decrease as interparticular distance is increasing gradually? Then why does restoring force increase? I know this is an universal truth but I want to know why does this happen

Qmechanic
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MSKB
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  • It is not a universal truth, and comes from the nature of Taylor expansions. See "derivation" of Hooke's Law: https://physics.stackexchange.com/a/408992/307237. – gmz Oct 04 '21 at 02:41
  • I knew Hooke's law is empirical and did not have any theoretical derivation atleast during Hooke's time – MSKB Oct 04 '21 at 05:05

3 Answers3

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As far as I know as per Coulomb's law or Newton's law, attraction force or repulsion force should decrease as the distance increases between two particles(charged).

This is true; for example, because of electron distribution fluctuations, even neutral atoms are attracted to each other as $\sim r^{-6}$, where $r$ is the separation distance. However, they're also repelled more strongly (as $\sim r^{-12}$, say) at very close distances. The sum gives us a pair potential:

The sum of the two terms produces an energy minimum that corresponds to the equilibrium separation distance. And because every smooth minimum looks like a parabola up close, for small strain $\varepsilon$, the energy scales as $E\varepsilon^2/2$, where $E$ is a constant stiffness.

Thus, it's the balance between attraction and repulsion that causes the restoring force of springs to increase as we stretch them: although the attraction forces between molecules are lessening, the repulsive force is lessening more. Does this make sense?

Chemomechanics
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  • I understood that repulsion decreases and increases the effectiveness of attrative force. But isn't attractive force decreasing too? Or is it that within elastic limit this decrement of attractive force is too low to be considered? – MSKB Oct 04 '21 at 04:56
  • @MSKB The graph above is for that of potential, not force. Are you aware that force is the derivative of potential wrt. displacement? – gmz Oct 04 '21 at 05:16
  • Yes I know that – MSKB Oct 04 '21 at 05:31
  • My textbook has the similar graph which involves force as well as potential – MSKB Oct 04 '21 at 05:31
  • @MSKB I wouldn't say "repulsion decreases and increases the effectiveness of attrative force". There is an attractive force (say, $\sim r^{-6}$) and a repulsive force (say, $\sim r^{-12}$) that sum to give a net force. When you stretch a crystal, the attractive force decreases, as you note, but the repulsive force decreases faster, which results in the net force increasing to restore the original spacing. Try plotting the individual forces or plugging in sample numbers to convince yourself. – Chemomechanics Oct 04 '21 at 05:54
  • Yes now I understood. After a certain distance, the effective restoring force will start to decline which is beyond the elastic limit ,right? – MSKB Oct 04 '21 at 05:56
  • If you mean the elastic limit that defines the start of irrecoverable deformation, typically involving the creation and movement of defects, then yes, this generally occurs before the pair potential curve changes its shape. – Chemomechanics Oct 04 '21 at 07:08
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Newton's and Coulomb's laws apply to point particles interacting by gravitational and electrostatic forces only. That's far from the case in a solid, or even a diatomic molecule for which there are many interacting particles that are subject to Coulomb's law (Newton is negligible) and the laws of quantum mechanics. The situation of a diatomic molecule is complex; that of a solid considerably more so. When you pull the atoms of a diatomic molecule further from each other the complicated combination of Coulomb's law and quantum mechanics wants to bring them back together. There are some hand-waving classical pictures that demonstrate how a rearrangement of charges can "explain" the attraction, but they are very poor arguments. For starters (and finishers) they assume that atoms exist, but in classical physics atoms do not exist. The argument starts with a contradiction. The arguments also ignore the fact that the so-called exchange symmetry (roughly speaking, that's the Pauli exclusion principle) works to keep electrons apart ... in addition to the Coulomb repulsion. As I said earlier, the situation is complicated.

garyp
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The interatomic displacements are very small, since an elastically deformed body consists of zillions of atoms. Yet, for sufficiently big deformation Hooke's law breaks: first the deformation becomes non-linear, then plastic (i.e., the object does not return to its initial state anymore), and eventually it breaks.

Wikipedia figure of deformation stages

Roger V.
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  • Isn't that supposed to be for each and every material? But rubbers do elongate a lot and returns to its original shape(almost), how could rubber sustain such deformation and still get back to its original shape(alnost)? – MSKB Oct 04 '21 at 05:46
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    @MSKB this is for bulk materials. Rabbers are made of polymers that are coiled - like a ball of twine. When uncoiled it gives long elongation, but deformations at the atomic level are still very small. – Roger V. Oct 04 '21 at 06:05
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    Ohh I didn' know that. Learnt a lot from this question – MSKB Oct 04 '21 at 06:08