We have another problem with atoms, though. What binds the nucleus together?

The nucleus of an atom consists of a bunch of protons and neutrons crammed together. Since neutrons have no charge and the positively-charged protons repel one another, why doesn't the nucleus blow apart?

We cannot account for the nucleus staying together with just electromagnetic force. What else could there be? Gravity? Nope! The gravitational force is far too weak to overpower the electromagnetic force.

So how can we account for this dilemma?