How Does the Force Between Two Spheres Relate to Their Potential Energy?

[lark]

Something I realized the other day - trying to figure out what the gravitational (or electrostatic) force would be between two spheres -
the force ON a sphere from another sphere, is the same as if the other sphere's mass were concentrated at its center.

So what is this force? It turns out that the force on a sphere FROM a point mass is the same as the force a sphere exerts ON a point mass - that is, the sphere is attracted to another point as if the sphere were a point itself! (from Newton's third law or whatever - action = reaction)

So the force between two spheres is the same as if the mass in the two spheres were concentrated at their centers.

In general if potential energy is 0 at infinity, the potential energy of object 1 from the gravitational field of object 2 is the same as the potential energy of object 2 in the grav. field of object 1 ...

laura

 

[arildno]

Something I realized the other day - trying to figure out what the gravitational (or electrostatic) force would be between two spheres -
the force ON a sphere from another sphere, is the same as if the other sphere's mass were concentrated at its center.

So what is this force? It turns out that the force on a sphere FROM a point mass is the same as the force a sphere exerts ON a point mass - that is, the sphere is attracted to another point as if the sphere were a point itself! (from Newton's third law or whatever - action = reaction)

So the force between two spheres is the same as if the mass in the two spheres were concentrated at their centers.


laura

Congratulations!
You have just shown one of the most important discoveries Sir Isaac Newton made with his theory of gravitation.
this was not meant ironic, it is indeed a pleasing and elegant result.

 

[Meir Achuz]

Poor Newton didn't know Gauss's law (He didn't even know Gauss), so he had to work that out in a very complicated derivation.

 

[rbj]

You have just shown one of the most important discoveries Sir Isaac Newton made with his theory of gravitation.

just want to reiterate to the OP what Meir said, that this can be shown pretty clearly using spherical symmetry and Gauss's Law.

 

[arildno]

Is it Newton's fault he didn't know Gauss, and essentially proved this, if I recall correctly, within the context of Euclidean geometry?