Newton computed the gravitational attraction of a planet as a spherically symmetric distribution of matter, to be equal to that of a point mass of the same total mass at the center of the planet, away from the planet.
This made it possible for Newton to model gravitational interaction of planets as gravitational interaction of point masses, as a much simpler problem from computational point of view.
Newton thus could simplify the computationally impossible problem of instant gravitational interaction at distance of all the individual atoms of one planet with all the individual atoms of another planet, to the interaction between two point masses of the same total mass. Genial and absolutely necessary to make the theory useful and thereby credible.
But let us reflect a bit about the physics of instant individual interaction at distance of each atom of one planet with each atom of another planet, which we agreed is computationally impossible. We now ask if it is physically possible?
Is it thinkable that each atom can instantly at distance exchange details about position and mass with all others atoms by using some form of world wide web? Think about it!
The answer can only be NO. Atoms cannot have access to such technology. It is unthinkable.
The result is that we have to view gravitation in a different way, not as individual instant attraction between small pieces of matter at distance, and then why not in the other way as suggested in recent posts:
What is primordial is then a gravitational potential $\phi$ with associated gravitational force $\nabla\phi$, to which matter density $\rho$ is connected by $\rho =\Delta\phi$ through the local operation in space of the Laplacian $\Delta$.
With this view there is no instant action at distance between atoms to explain, but instead local production of matter without demand of atomistic resolution into pieces, which at least is thinkable.
It would be interesting to listen to Newton's reaction to this idea.
PS Business Insider reports that:
PS Business Insider reports that:
- Earth's core is 2.5 years younger than its crust due to some eerie physics
the eerie physics being Einstein's general theory of relativity claiming that clocks slow down with increasing gravitation. Yes, maybe your feet are a bit younger than your head...
This post must be a joke?
SvaraRaderaThe derivation of the equation for the center of mass is probably done in week one in most introductory mechanics courses. It is a one-liner and a direct consequence of Newton's third law after applying Newton's second law to the definition of the center of mass.
In the case of gravity over long distances. Taylor expand a perturbation in r for 1/r^2.
1/(r+dr)^2 ~ 1/r^2 - 2dr/r^4
That is really negligible.
It is not so trivial but requires integration over spherical surfaces, in one way or the other, to sum contributions from all little pieces of say a homogeneous sphere
SvaraRaderato the gravitational potential outside the sphere.
The center of mass (R_{center}) fulfills (you can of course take an integral if you want to do a continuum)
SvaraRaderaM R_{center} = \sum_{i} m_{i} r_{i}.
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N2 gives:
M d^2(R_{center})/dt^2 = \sum_{i} m_{i} d^2(r_{i})/dt^2 = [N2] = \sum_{i} F_{i,external} + \sum_{i} F_{ij}
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N3 gives that the sum over internal forces vanishes since F_{ij} = -F_{ji}, so
M d^2(R_{center})/dt^2 = \sum_{i} F_{i,external} = F_{total}
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This is a really nice result. The collection of particles (or a continuum if you so like) acts as a particle with mass equal to the sum of all contributions and that the particle is located in the center of mass.
Nice, but the connection is missing to the gravitational force generated by a spherically symmetric mass distribution, as far as I can see.
SvaraRaderaWell, if you take Newton's laws seriously that resulting force must have a reactive companion. Equal in magnitude but opposite in direction. Wonder were that will act? ;)
SvaraRaderaOr just use the shell theorem. Originally derived by Newton
SvaraRaderaen.wikipedia.org/wiki/Shell_theorem
This is precisely what I am talking about.
SvaraRadera