Newton's Universal Law of Gravitation doubt

Question

The Universal Law of Gravitation states that the module of the force, $F$ is $$F = \frac{GmM}{r^2},$$

where $m$ and $M$ are the mass of the two objects and $r$ is the distance between the two objects. From this, it is also derived that the gravitational potential energy is equal to $$ E = -\frac{GmM}{r}$$

So does this mean that when the two objects are together, i.e. $r=0$, the force and the gravitational energy is infinite? How is it possible that energy and force be infinite?

Please can you explain my doubt?

Answer 1

At $r=0$, the force is $+\infty$, and the potential energy is $-\infty$. This arises if you consider point masses, which we model mathematically using Dirac-delta distributions. Physically, we cannot have two point masses overlapping ($r=0$) since this would require an infinite amount of energy. Why? Well, fix one mass, and bring the other mass from $r=\infty$ towards $r=0$. Then, as the masses get closer, more and more energy is required to bridge the gap. So there is no contradiction, because you can never construct any physical system with overlapping point masses. Furthermore, in any real system, the objects are not point masses, but have some finite volume. Even if you consider an electron as a point mass, the electromagnetic force would far outweigh the gravitational force.

Answer 2

The thing is, that, as every equation, these are based on model views. The model in this case includes the representation of the mass object as a point mass. Hence we are able to attribute to it some coordinate and measure distances between objects, which, in fact, consist of several gravitating "sub"objects. One of the properties of the point mass is that it can't be overlapped with another object (including another point mass). That is, the equation, which represents the model, includes a restriction on r = 0. Thus, in frames of this model (or equation), it is improper to consider r = 0 - as, obviously, model wouldn't work in this case.