Hi, Yesterday I was reading the wikipedia entry for Newton's law of universal gravitation <http://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation>, and under "Bodies with spatial extent" it says:
"If the bodies of question have spatial extent (rather than being theoretical point masses), then the gravitational force between them is calculated by summing the contributions of the notional point masses which constitute the bodies. In the limit, as the component point masses become "infinitely small", this entails integrating the force (in vector form, see below) over the extents of the two bodies. In this way it can be shown that an object with a spherically-symmetric distribution of mass exerts the same gravitational attraction on external bodies as if all the object's mass were concentrated at a point at its centre.^[2] <http://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation#cite_note-Newton1-1> (This is not generally true for non-spherically-symmetrical bodies.)" It is interesting to continue reading the explanation, for the force exerted on points inside the sphere. Regards, Mauro On 10/24/2010 03:38 PM, David Jonsson wrote: > Hi > > No problem with hijacking. Your subjects are related and I read some > of it but I realize it is too much to read for me right now. Maybe > your programs can be adjusted with the force formula for > spherical distributions. > > Actually I just typed the ACSII version of the integral into Wolfram > Alpha, a fantastic web resource, and after several seconds I got the > answer > http://www.wolframalpha.com/input/?i=Integral+from+-r0+to+%2br0+of+(r0^2-r^2)/(R0-r)^2+dr&incParTime=true > <http://www.wolframalpha.com/input/?i=Integral+from+-r0+to+%2br0+of+%28r0%5E2-r%5E2%29/%28R0-r%29%5E2+dr&incParTime=true> > > A bit down it lists the indefinite integral as (R0^2-r0^2)/(r-R0)-2 R0 > log(r-R0)-r+constant > > But the result dimensions do not fit... Something is wrong. > > For the spherical mass distribution I assume per particle > F=GMm/R² > > and for a mass differential dM in the star (see attached picture) it > becomes > > dF=G dM m / R² = G rho dV m / ( R0 + r ) ² = G rho pi (r0² - r²) dr m > /(R0 - r)^2 > > Each dM is a vertical slice in the star. Each slice weighs dM = rho pi > (r0² - r^2) dr. So I don't calculate to the highest precision. I just > assume that the distance to all particles in the slice is R0-r > > and if I integrate to get the total it would become > > F = Integral from -r0 to r0 of G rho pi (r0² - r²) dr m /(R0 - r)^2 > > which is the integral I initially asked aboutand that Wolfram Alpha > gave the indefinite form of. > > I can't see why using spherical distribution would make the > computation much more complex? Computers can handle almost anything. > > Regards, > David > > David Jonsson, Sweden, phone callto:+46703000370 > > > > On Mon, Oct 18, 2010 at 8:03 PM, OrionWorks - Steven V Johnson > <[email protected] <mailto:[email protected]>> wrote: > > >From David > > > 12 replies to my question is not bad but the integral is > actually about what > > the gravity force is to a spherical mass distribution compared > to a point > > mass. The so called center of gravity can not be used as a > center of gravity > > since matter closer to a body attracts more than what the remote > parts do. > > How big can this effect be? > > Can anyone solve the integral? I haven't even tried, yet. Can > Maxima solve > > it? > > David > > David, > > I must apologize as well. Guess you could say I intentionally > "hijacked" your thread. In your original question you brought up > interesting concepts that were related to a branch of mathematical > study that I've been exploring for years. I only hope the tangential > aspects of what has been discussed in your hijacked thread has been be > of some interest to the readers, including you. > > Following up on some of the tangential aspects, the physics text books > state that the force known as Gravity is considered to be several > orders of magnitude weaker than the strong and weak nuclear forces. > This is basic high school physics. > > In the meantime, David brings up an interesting concept that I > consider related to a similar discussion pertaining to whether it is > (legally) appropriate to computer model the effects of gravity using a > point mass, or whether one should model the effect as a spherical mass > distribution. From my own POV, and I'm speaking strictly from a > computer programmer's POV, it is FAR more convenient in the heuristic > sense to use a centralized point position in order model/generate > orbital simulations based on the so-called laws of Celestial > Mechanics. If one models one's algorithms using a point mass concept, > it is important to "play god" and summarily change the rules > so-to-speak where appropriate, particularly when the orbiting > satellite approaches too close to the main orbital body. To do so > introduces bizarre/chaotic orbital behavior. While it would probably > be more accurate (or realistic) to employ a spherical mass > distribution formula, to approach the problem as a computer > programming exercise, would increase the complexity of the algorithms > to the point that it would quickly become impossible to code. > > After reading just a sprinkle of Miles Mathis's papers, a novel > concept recently dawned on me pertaining to the fact that we could > speculate on the premise that the force of gravity may not necessarily > be as weak as the text books have always claimed the force to be. What > if we looked at the manifestation of gravity as emanating from the > "center" of each sub-atomic particle, what then? What if we were to > move the ground rules for "spherical mass distribution" away from the > surface of typical macro bodies, like stars, planets, or moons, and > scale it all the way down to the surface radius of protons and > neutrons... how strong would the "point mass" force of gravity > manifest at quantum-like distances? Obviously at that scale of > distance the effects of gravity would be several magnitudes stronger > that what is experienced within the familiar macro world! After all, > we are told neutron stars are held together by the crushing force of > the star's own gravity! A neutron star is essentially a gazillion > sub-atomic "point mass" neutron particles collectively behaving as if > they were all just one massive spherical mass distribution set as > perceived on the macro scale. I suspect that in some of Miles Mathis' > paper he is hinting at something akin to this. I suspect Miles is also > hinting at the premise that gravity, just like all the other forces, > are essentially one and the same "force" manifesting in different ways > and/or scales of distance. > > Regards, > Steven Vincent Johnson > www.Orionworks.com <http://www.Orionworks.com> > www.zazzle.com/orionworks <http://www.zazzle.com/orionworks> > >
