To explain what kept atoms together, other forces were needed, and in the 1930s two were discovered: the strong nuclear force and weak nuclear force. The strong force binds atoms together; its what allows protons to bed down together in the nucleus. The weak force engages in more miscellaneous tasks, mostly to do with controlling the rates of certain sorts of radioactive decay.
The weak nuclear force, despite its name, is ten billion billion billion times stronger than gravity, and the strong nuclear force is more powerful still—vastly so, in fact—but their influence extends to only the tiniest distances. The grip of the strong force reaches out only to about 1/100,000 of the diameter of an atom. Thats why the nuclei of atoms are so compacted and dense and why elements with big, crowded nuclei tend to be so unstable: the strong force just cant hold on to all the protons.
The upshot of all this is that physics ended up with two bodies of laws—one for the world of the very small, one for the universe at large—leading quite separate lives.