Question

Why does the nucleus of an atom become radioactive?

Answer 1

As the nucleus of an atom gets bigger, the outer protons would be relatively loosely bound.
In other words, as no. of protons increases the nucleus becomes unstable.
Thats precisely why radioactivity occurs.

Answer 2

the nucleus of an atom becomes radioactivity when the number of neutron becomes more than the number of proton i.e. the neutron , proton ratio becomes more than 1.5
In such atoms the nucleus is unstable. to make the nucleus stable they undergo nuclear changes which is termed as radioactivity
Radioactivity is a phenomenon mainly shown by heavy metals and the isotopes of some light metals.