Why does the the relative importance of the coulomb force compared to the strong nuclear force increase at large mass numbers?

Answer 1

Because the effect of the strong nuclear force falls (SNF) off more steeply as a function of distance than does the electromagnetic (coulomb?) (EMF) force. The SNF is an attractive force, that tends to hold nuclei together, while the EMF is an attractive/repulsive force (depending on polarity of charge) that tends to blow nuclei (at least, protons) apart. In the short distances for small nuclei, the SNF wins, but, starting at atomic number 83, bismuth, the EMF starts to win based on distance, which is why all nuclides with atomic number greater the 82 (lead) are unstable (radioactive).

Not asked, but answered for completeness sake; even for smaller nuclei, such as carbon, the proton/neutron ratio can lead to an unstable, i.e. radioactive, configuration, based on the weak nuclear force, which also enters into the picture.

Answer 2

At large mass numbers, the strong nuclear force becomes less effective at overcoming the repulsive Coulomb force between protons due to the increased electrostatic repulsion. This makes the Coulomb force relatively more important in determining the stability of the nucleus, leading to a higher likelihood of radioactive decay in heavy nuclei.