[Alan McDougall (OP)]

In an atom, why do the negatively charged particles (Electrons) orbit the atom instead of joining with the protons at the nucleus of the atom?

Of course orbit is not really an accurate reflection of quantum reality, if you like?

Alan

[chiralSPO]

This is a topic that has been discussed at length in other threads on this forum. Long story short:

It is possible for electrons to be "captured" by a nucleus, effectively combining with a proton to make a neutron. However, this is typically a very unlikely transformation, and is only favorable for isotopes that have "too many" protons for the number of neutrons. (And the reverse process is a common mode of radioactive decay, wherein nuclei with too many neutrons will release an electron, forming a new nucleus with one more proton and one less neutron.)

As you pointed out, "orbiting" is not an accurate description of the behavior in an atom. Instead, one can think of a hydrogen atom as an electron that is "stuck to" a proton. The center of mass of the electron is in the same place as the center of mass of the proton, but the proton is localized to a radius about several hundred times smaller than the radius of the electron density. This disparity in radius of localization is because electrons have so little mass, they naturally spread out over a larger volume than heavier particles such as protons and neutrons (a proton is 1836 times as massive as an electron). The same approach applies to heavier atoms with multiple electrons, and can even be applied to molecules with multiple nuclei, but the theory gets more complicated as the system is more complex.

[PmbPhy]

Simply put: Particles such as electrons and protons can't be treated classically so classical ideas of motion simply don't apply. A complete explanation is given here: http://www.chem1.com/acad/webtut/atomic/WhyTheElectron.html

[evan_au]

Electrons orbit the atom instead of joining with the protons at the nucleus of the atom?

It's not quite right to say that an electron+proton=neutron.

There has to be another particle involved before an electron can disappear from its orbit and be confined within the relatively tiny volume of the nucleus: a ghostly particle called the "neutrino". Quite often the neutrino (or its anti-particle) carries away a considerable amount of energy from such interactions.
https://en.wikipedia.org/wiki/Neutrino