Gravitation law equation as R approaches zero

[pconstantino]

Hello,

in the equation which describes the force as a function of the radius:

F = GMm/R^2


What happens as R approaches zero? or even when R is less than the radius of the planet.

mass m will be inside the planet so this formula seems to break down because m will be pulled from both sides.

how can this be handled?

thank you !

 

[zhermes]

That's an excellent question. For a detailed answer, try looking at http://en.wikipedia.org/wiki/Gauss's_law_for_gravity and the references it cites.

If you have two massive point particles (infinitesimally small) which are gravitationally interacting with each-other, the force becomes undefined as the separation approaches zero. This is a non-physical scenario however, as two particles can never perfectly overlap.

If you're talking about a planet (or other large body) you're exactly right---the equation has to be modified. What "Gauss' law" (the link I posted) states, is that only the mass within the radius you're interested in, matters. (This is an oversimplification, but its the basic idea). If you are half way to the center of the earth, you'll only need to consider the mass within a sphere of radius = half the Earth's radius (approximately 1/8 the mass of the earth). Thus, as your distance from the center approaches zero, the mass enclosed approaches zero, and the force goes to zero.

Does this make sense?
We can go into more math/details if you're interested.

 

[Raama]

Two objects cannot match exactly. The distance is calculated between point of gravity of both objects. So in 3d 2 objects cannot overlap exactly on their point of gravity. Then why thinking about it?

 

[Lsos]

Hello,

in the equation which describes the force as a function of the radius:

F = GMm/R^2


What happens as R approaches zero? or even when R is less than the radius of the planet.

Of course there at objects which are VERY massive and their radius is in fact considered to be 0. What happens then is that gravity gets larger and larger as you approach them, until it goes to infinity.

 

[zhermes]

Two objects cannot match exactly. The distance is calculated between point of gravity of both objects. So in 3d 2 objects cannot overlap exactly on their point of gravity. Then why thinking about it?

Because non-physical questions are still often good ones both per se and for elucidating physical ones. Additionally, as 'objects' are actually waves, they can overlap largely.

Of course there at objects which are VERY massive and their radius is in fact considered to be 0. What happens then is that gravity gets larger and larger as you approach them, until it goes to infinity.

That's not accurate. In pure gravity theories one would simply say the gravitational force becomes undefined. Otherwise, most unifying theories always preserve a finite radius (e.g. string theory).