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While reading about the normal reaction for different scenarios, I faced minor confusion regarding their origin.

Suppose a mass is kept on a surface. The mass exerts a downward force indirectly due to its weight, and the surface would exert a force back on the mass. I wonder where this force is coming from.

One explanation claims that this force comes from the electrostatic repulsion between the two masses and the exclusion principle, which sounds right, obviously.

Another explanation uses the fact that the mass compresses the surface a bit, which tries to restore itself by pushing the mass upwards i.e. against the compression. This is the source of the normal reaction force.

My doubt is, isn't it a bit of both of the above effects? I've seen all sources use one of the two explanations above, but never both of them together. Shouldn't both of these be responsible for the normal force ?

Imagine the surface is perfectly rigid, there is no compression. It would still exert a normal force on the mass. But this force would be purely electrostatic, as there is no compression on restoring force involved.

In the general case however, the mass applies a force on the surface, greater than the normal force due to pure electrostatic repulsion. Hence the surface gets compressed, and the restoring force steps in the balance the forces to bring both of the bodies to rest.

So, am I correct in claiming the following :

$N_{net} = N_{electrostatic} \space+\space N_{restoring}$

In case of a perfectly rigid floor, only the first one works. For a general floor, the first one is not strong enough to resist compression, so the second restoring force walks in to balance the system. If both the forces are not enough, the surface breaks or gets punctured or something.

But is it correct to stay, that there are these two distinct effects that give rise to the net normal force. That the force is a result of electrostatic repulsion and the restoring force due to compression ?

Or are both of these effects the same thing, which doesn't seem to make sense intuitively, as they have different origins, but contribute to the net normal force. Using only one of the explanations above is incomplete. For example, using the restoring force explanation, fails to mention, why a perfectly rigid body would exert a normal force. On the other hand, using the purely electrostatic expression fails to mention, why a heavy object would bend a surface. We clearly need both of them.

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  • $\begingroup$ I think you need to distinguish between theoretical models and what happens in reality. Many theoretical models are coarse representations of reality, but if they are sufficiently useful then they are used for their simplicity. $\endgroup$
    – Evan
    Aug 26, 2021 at 19:02
  • $\begingroup$ Suggest you read the following, then specifically indicate what it is you don't understand about the origin of normal reactions. energyeducation.ca/encyclopedia/Normal_force $\endgroup$
    – Bob D
    Aug 26, 2021 at 19:29
  • $\begingroup$ @BobD I've read this, and this claims that the normal force arises out of electromagnetic repulsion. However, this is true for only rigid surfaces. This doesn't explain why the table or the surface bends when the object is heavy. Kleppner says that the forces arise due to compressions of the surface, that tries to restore itself. These are two alternate explanations. I think both of them are true, that both these things happen. Only electromagnetic repulsion cannot explain normal reaction, you also need the compression of the surface. That is my doubt. Am I correct in this assertion ? $\endgroup$ Aug 26, 2021 at 19:34
  • $\begingroup$ If you are going to delve into the bending, compression, etc. of materials, you are then in the realm of mechanics of deformable solids since there are no such things as rigid, non deformable bodies. $\endgroup$
    – Bob D
    Aug 26, 2021 at 19:51

2 Answers 2

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The two explanations of a repulsive force (electrostatic and arising from Pauli repulsion) and a restoring force (arising from compression) are functionally identical; the two descriptions simply emphasize different aspects. Calling them two different effects and adding them together is not useful.

It sounds like you aren't clear on how electrostatic repulsion changes with spacing and surmise that repulsion alone is an independent component that can levitate an object and avoid any compression. This is incorrect. A weight (in the context of this question) always leads to compression of the support.

Arguments to the contrary based on a perfectly rigid material can be discarded*, as such a material doesn't exist.

Your remaining question is why a heavy object would bend a surface, and how this can be explained through electrostatic interactions. The answer is that the system finds an energy minimum in which the sum of the gravitational potential of the object and the strain energy stored in the molecular bonds is minimized. Put another way, all the forces end up balanced; if a heavier object is used, the molecular spacing is smaller and the repulsive force larger (as quantified by pair potential models).

*Why is this? Why can't we simply idealize one part of the problem for simplicity to gain insight? We can, but we must then commit to the idealization; we can't mix and match. If you say, "Let's idealize the support as perfectly rigid," you can't then say, "Now, the electrons within the support are repelling with force $F$..." because you've already replaced the support with something nonphysical—a magical block—that doesn't contain electrons. If we agree that the support doesn't deform, we can't then talk about its constitutive atoms, because bonded atoms do deform. We can only investigate the physics of the parts of the system that remain realistically modeled. Another option is to calculate the results for an enormous support stiffness of $E=10^{20}\,\mathrm{Pa}$, say. This will yield negligible deflection but still allow calculation of the strain energy. It also underscores that the restoring force is the repulsive force; they cannot be decoupled.

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The effects are not distinct. When the surface is not distorted there is zero force between molecules (if we forget about thermal vibration). The repulsive force arises when you compress the surface. The more you compress it the greater the repulsive force between molecules becomes.

It all becomes much clearer if you recall the interatomic force curve. When two atoms are at their equilibrium separation there is no force between them (otherwise they would be accelerating!). When they are pushed closer together than their equilibrium position they repel each other. A hand-waving explanation is that the electron cloud between the nuclei is partly pushed out of the way, so the nuclei are not wholly screened from each other's repulsion. If the atoms are pulled further apart (as when a wire is slightly stretched) the force between them is attractive. [The intermolecular force curve is qualitatively similar to the interatomic curve.]

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