How is the singularity in Newtonian gravity resolved?

Question

In Who's afraid of a Black Hole? at time ~20:38, Michio Kaku makes a claim that $1/r$ when $r=0$ equals $\infty$ and when physicists see the $\infty$ they see a monstrosity. Coincidentally, Newton's gravitational equation $$PE=-\dfrac{GMm}{r}$$ also has a $1/r$ relationship, which might be problematic, except that in Newtonian theory there is no maximum velocity $c$ we would need to be concerned about. Of course it can be observationally verified that the potential energy of two objects, colliding inelastically, is not $-\infty$.

I suspect this same question of singularity must have bothered Newton as well, since we can find Newton working to resolve singularities through the development of the Puiseux series in 1676, which is a generalization of the Laurent series, which itself can be turned into a Fourier series.

The question I am having a hard time to resolve is whether there a pedagogical approach to explain how to address Newtonian gravity when $r=0$ ?

Answer 1

Having $r = 0$ is only a poblem if you have infinitely small, infinitely dense bodies. For any body with a finite radius, at $r = 0$ you will be inside the body and the gravity will be finite. In fact for a spherical body like a planet or star the gravity is zero when $r = 0$.

So the problem only arises if you're prepared to accept other infinities like infinite density. Since this doesn't occur in the real world (except possibly at the centre of a static black hole) the problem never arises.