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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 $r=0$ ?

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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.

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Thanks, I would accept that except if Newtonian gravity admits infinite velocities, then it has to allow for infinite kinetic energy for an object, which would mean it has infinite energy density. –  Hal Swyers Sep 26 '12 at 12:54
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In Newtonian mechanics energy doesn't gravitate. Energy only gravitates if you switch to general relativity. GR does actually have a problem with infinite gravitational forces at points called singularities. The centre of a black hole is an example. However it's believed that some future theory of quantum gravity will eliminate the singularities. –  John Rennie Sep 26 '12 at 13:00
    
Good points, and I will accept the answer, however Newton was smart enough to build the relationship between kinetic and potential energies and equate potential energy to gravitational force. So even if he didn't have a clear equivalence of mass to energy, he certainly had enough knowledge to parametrically manipulate the relationships. I just suspect that he was aware of some of these issues and was actually investigating them. –  Hal Swyers Sep 26 '12 at 19:01
    
@HalSwyers: you might be interested in the Liebniz-Clarke coorespondance. Clarke was one of Newton's students, and they engaged in a debate via mail for several years. The most interesting part is that it appears that Newton was vociferously NOT interested in going down a path that would have lead to relativity. –  Jerry Schirmer Nov 10 '13 at 2:59
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Well, the answer given by John Rennie is very good but I think there's still a problem. The r in the equality is actually the distance between two particles that is $r= |\overline{r_1}-\overline{r_2}|$. If it weren't its gradient wouldn't give the gravitational force.

We can only use r itself, i.e $r= |\overline{r_1}|$ when one body is assumed to be not moving, when one body's mass is comparably higher than the other which is not the whole story. So the real problem is how close can two bodies approach?

If we assume the bodies are structureless point particles there's a real problem which I think can only be overcome by quantum mechanical concepts, uncertainty being the most reasonable.

If we assume the bodies have structure and cover a volume then we have to look at points on bodies to determine if they actually touch. So there's again the problem of distances between points.

To sum up, interestingly Newtonian Mechanics doesn't seem to give a proper answer for the singularity in gravitation.

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It's the distance between the centers of two bodies. In a Newtonian picture, it's easy enough to say that all bodies are finite in extent, and when two bodies (or in his language, "corpuscles") touch, they will still be a distance $r_{1} + r_{2}$ apart. –  Jerry Schirmer Nov 10 '13 at 3:01
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Yes, quantum mechanics comes to the rescue where infinities are the limits in classical physics. Sea my answer here to a similar question physics.stackexchange.com/questions/83396/… –  anna v Nov 10 '13 at 3:51
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