Q: Why do electrons fly around the nucleus of an atom instead of getting sucked in?
A: The same reason the planets don't fall into the sun. The planets are always "falling" towards the sun due to the gravitational force but the fact that they are moving perpendicular to the sun means they "miss" and so a balance is setup between the speed in which they orbit and the distance they are from the center. In the case of electrons and nucleus it is not the gravitational force that sets up this equilibrium but the electrostatic interaction between the negative electrons and the positive protons in the nucleus. The gravitational force does act but is negligible compared to the nucleus.
This is an over simplification, in actual fact the electrons don't fly round the nucleus. Due to the Heisenberg uncertainty principle we can't know exactly where the electron is, and so what comes out from the maths and quantum mechanics are lobes of electron density around the nucleus where calculations place the electron within most of the time. So the electron can be considered smeared across these lobes, or orbitals to give them their proper name. So they don't fly around the nucleus as in the pictures and in the case of the 1s orbital the electron could be within the nucleus. But the same principle applies as in the case with the planets. The angular momentum balances out with the interaction with the nucleus and stops the electron being "sucked in".
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