Gravitational acceleration towards and through an object

[jameswhite]

Take a hypothetical object with an infinite density (purely so that a particle can get incredibly close to it). Furthermore imagine that this object has no close range repulsive force, or in other words, particles can pass right through it. The only thing is does is exert a standard gravitational force.

Now, if we drop a particle in the vicinity of this mass, it will accelerate towards it as always. However, due to the infinite density (r = 0) the force will eventually become infinite (F = GMm/r^2 and r=0) right? Now, I am fully aware of the implications of special relativity here but I would like to discuss this from a purely classical and non-relativistic perspective, it's a thought experiment, nothing more. The particle would feel an infinite acceleration and by taking the derivative of the acceleration function it is clear that the velocity is also inversely proportional to the distance, r. So it seems like the particle would reach and infinite velocity, BUT surely it would take an infinite amount of time for anything to reach and infinite speed...? So what really happens? In a purely classical, non-relativistic setting.

Furthermore, what happens to the particle after it passes through the object? Does it slow down again and achieve a harmonic motion? Or does it fly through the object so fast that it just keeps going?

Normally we don't need to worry about these things because celestial bodies like the Earth have a surface, and sooner or later any particle falling towards it would hit the surface. What happens when there is no surface and all the mass is concentrated at a point?

 

[Arnoldas]

I think we should firstly determine if F really becomes infinite. The mathematical equation you propose is a limit and does not necessarily need to be infinite. By this i mean: Let's just say that radius of your object is 'a'. The distance between cm of that object and the particle is 'x', thus x>a always. The mass of the particle is m(constant) and the object has M, which is a function of density ro, where ro goes to infinity. M is thus proportional to Volume function times density function. Volume function is proportional to a^3 (3 dimensional of course). If we put all this crap together we get that F is proportional to (a^3/x^2)*ro. Since a < x that means that a^3 < < x^2 as a aproaches 0 which in turn means that (a^3/x^2) aproaches zero. We get kind of a 0 multiplied by kind of an infinity situation and that could really be any number on number line. Does this make sense? I only got more confused about the property of an object that has the density that goes to infinity and radius that goes to zero. I think the result could be a lot of things depending on situation?