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I can tell you know this, I'm just tacking it on for those who don't:
Why do the nuclei stick together at all, once you've pushed them together? Because the nuclear force, which is attractive instead of repulsive, is just a little bit stronger. OK, so why then doesn't the nuclear force just pull all atoms together? Because it is short range, and only works once the nuclei are "touching."
I know that much, but actually how can a force be stronger but short range?
The repulsive force is electromagnetism while the attractive force is an exchange of mesons between the atoms.
It is different forces with different ways of interacting on objects.
This is the chatGPT answer I'm basing my answer on:
"The concept of a force being both stronger and short-range might seem counterintuitive at first, but it's a fundamental aspect of how forces operate in the quantum world, particularly within the nucleus of an atom. The force being referred to here is the strong nuclear force, also known as the strong force or strong interaction, which is one of the four fundamental forces in physics.
The strong nuclear force is indeed much stronger than electromagnetism (the force that repels positively charged protons from each other) but it operates over a very short range. Here's a more detailed explanation:
Strength: The strong force is the strongest of the four fundamental forces. Its strength ensures that it can overcome the electromagnetic repulsion between protons within the nucleus. Without the strong force, the protons would repel each other and the nucleus would disintegrate.
Range: The strong force only acts over very short distances, approximately 1 femtometer (1 fm, or (10^{-15}) meters), which is roughly the diameter of a large nucleus. Beyond this range, the force drops off very rapidly, becoming negligible compared to electromagnetic forces. This is why atomic nuclei can be stable: within the nucleus, the strong force is dominant and keeps the protons and neutrons together; outside the nucleus, its influence is minimal, so atoms do not "stick" to each other due to the strong force.
The short-range nature of the strong force is due to the mechanism by which it operates, involving the exchange of particles called mesons between nucleons (protons and neutrons). This particle exchange can be thought of as the "glue" that holds the nucleus together, but this "glue" only works over very short distances.
To understand how a force can be both stronger and short-range, it's helpful to compare it with gravity, which is much weaker but has an infinite range. Gravity affects objects no matter how far apart they are, although its effect diminishes with distance. In contrast, the strong force has a much greater effect but only over a very short range. This difference in behavior is due to the different properties and mechanisms governing these forces. "
Hope this helps, it is not really my domain so maybe someone can ELI5 better for us !