johannes

03-18-2010, 08:22 AM

Hi there,

The Millennium Universal Law of Gravitation is given by

(1) F_m = Gm[M+m]/r²

So the force that the Earth would feel caused by the Sun would be

(2) Gm[M+m]/r²

where m is the mass of the Earth and M is the mass of the Sun and r is the distance between the Earth and the Sun.

Now imagine that the Earth is composed by atoms. Say that there are N atoms and the mass of each atom is a.

The force on each atom is given by

(3) Ga[M+a]/r²

Then the total force is given by

(4) Σ Ga[M+a]/r²

Now this can be written as

(5) Σ GaM/r² + Σ Ga²/r²

Or

(6) G(Σa)M/r² + G(Σa)²/r² + G(Σa²)/r² - G(Σa)²/r²

As m=Σa this is

(7) GmM/r² + Gm²/r² + G[(Σa²) - (Σa)²]/r²

Thus the total force is given by

(8a) Gm[M+m]/r² + G[(Σa²) - (Σa)²]/r²

While it should be

(8b) Gm[M+m]/r²

Now if we for simplicity say that all the atoms are the same - then m=Na.

And

(9) G[(Σa²) - (Σa)²]/r² = Gm²[1/N - 1]/r²

So the total force can be written as

(10) GmM/r² + Gm²[1/N]/r² ≈ GmM/r²

As there are many atoms (N is large)

The force acting on a body is given by

(8b) Gm[M+m]/r²

But viewed as if the body is composed by atoms the force is given by

(8a) Gm[M+m]/r² + G[(Σa²) - (Σa)²]/r²

How does nature know which form to use?

It looks like the Millennium Universal Law of Gravitation contradicts itself.

- johannes

The Millennium Universal Law of Gravitation is given by

(1) F_m = Gm[M+m]/r²

So the force that the Earth would feel caused by the Sun would be

(2) Gm[M+m]/r²

where m is the mass of the Earth and M is the mass of the Sun and r is the distance between the Earth and the Sun.

Now imagine that the Earth is composed by atoms. Say that there are N atoms and the mass of each atom is a.

The force on each atom is given by

(3) Ga[M+a]/r²

Then the total force is given by

(4) Σ Ga[M+a]/r²

Now this can be written as

(5) Σ GaM/r² + Σ Ga²/r²

Or

(6) G(Σa)M/r² + G(Σa)²/r² + G(Σa²)/r² - G(Σa)²/r²

As m=Σa this is

(7) GmM/r² + Gm²/r² + G[(Σa²) - (Σa)²]/r²

Thus the total force is given by

(8a) Gm[M+m]/r² + G[(Σa²) - (Σa)²]/r²

While it should be

(8b) Gm[M+m]/r²

Now if we for simplicity say that all the atoms are the same - then m=Na.

And

(9) G[(Σa²) - (Σa)²]/r² = Gm²[1/N - 1]/r²

So the total force can be written as

(10) GmM/r² + Gm²[1/N]/r² ≈ GmM/r²

As there are many atoms (N is large)

The force acting on a body is given by

(8b) Gm[M+m]/r²

But viewed as if the body is composed by atoms the force is given by

(8a) Gm[M+m]/r² + G[(Σa²) - (Σa)²]/r²

How does nature know which form to use?

It looks like the Millennium Universal Law of Gravitation contradicts itself.

- johannes