From what I learned in chemistry, the protons in the nucleus pull the electrons in and push on each other through electromagnetic forces, but are held in place by the strong nuclear forces in its gluons. Not much was said, however, about what keeps the electrons orbiting. I've always just assumed it was other electrons that prevented an electron from becoming part of the nucleus. In the form of Hydrogen that only has one electron, what keeps that electron from being pulled completely into the nucleus?
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This is indeed a problem but not for the reason you think.
If the electron obeyed classical mechanics and it was only subject to electrostatic attraction to the nucleus, it would never fall into the nucleus despite the fact that it would be constantly attracted to it. This is exactly analogous to why the Earth doesn't fall into the Sun: it has too much angular momentum, so by the time the Sun has made it "fall" significantly, it is already on another part of its orbit. Thus the Earth (like the prospective electron) keeps "falling" in circles around the Sun.
Electrons, alas, do not feel only electrostatic forces but must comply with the full electromagnetic theory of Maxwell, which dictates that accelerating charges (like circling electrons) must radiate their energy as electromagnetic waves. This energy is taken out of the orbital motion, which would steadily collapse into the nucleus. And within a fraction of a second, too.
This was a puzzle for a very long time and it was the glaring flaw in the planetary model of the atom when it was first proposed by Rutherford and Moseley. You can only cure it by making the electron a quantum-mechanical beast, a weird hybrid between a particle and a wave.
Essentially, the electron fails to fall into the nucleus because its position, like any wave, cannot be tightly confined without giving it a very small wavelength (and that would confer it a large momentum, allowing it to break out of the nucleus). Electron waves, like all waves, like to spread out, and they can also interfere with themselves to make complicated interference patterns around the nucleus. It was then a huge triumph of theoretical physics when Schrödinger proposed an equation describing the way in which these electron waves can add with themselves constructively to make standing wave patterns whose energies exactly matched those of the planetary Bohr model and therefore experimental facts. These standing waves are the only stable states of the electron waves in the atom, which is why it doesn't collapse.