Proof like Shell theorem for non-spherical objects?

[Zula110100100]

Is there one out there? Do we have any reason to believe we can treat other objects like point masses as well?

I ask because if you consider line-world, and there was a 4m segment with uniform density 3kg/m located with it's left end at (3), the center of mass would be at (5), and I am located at (0), the acceleration from gravity is:

$$A_{g} = \frac{Gm}{r^2{}}$$
$$A_{g} = \frac{12G}{25} = .48G$$

Now I cut the object in half, so I have two 2m segments, each still a uniform 3kg/m, and it hasn't moved, but the center of mass for the nearer segment is now at (4), and the other is at (6):

$$A_{g} = \frac{Gm_{1}}{r_{1}^2{}} + \frac{Gm_{2}}{r_{2}^2{}}$$
$$A_{g} = \frac{6G}{16} + \frac{6G}{36} = .375G + .167G = .542G$$

So then I tried to do it with an integral(not 100% sure I am doing it right) and this is what I got

$$\Delta A_{g} = \frac{G\rho \Delta x}{x^{2}}$$
$$A_{g} = G\rho \int \frac{1}{x^{2}}dx$$
$$A_{g} = G\rho \int^{7}_{3} \frac{1}{x^{2}}dx$$
$$A_{g} = 3G(-\frac{1}{7}+\frac{1}{3}) = 3G(-\frac{3}{21}+\frac{7}{21}) = 3G(\frac{4}{21}) = \frac{12G}{21}\equiv .571G$$

That's almost a fifth more than the original estimation, so one could assume if point mass fails in 1d, it could fail in some higher dimensional configuration. Unless there is an error above, or some reason why it works in 3d and not 1d...Anyone know this?

 

[Zula110100100]

In reading that now I realize that the inverse-square law is for 3D so I imagine that makes the difference here. But the question on proofs for non-spherical objects remains

 

[D H]

Proof of what? I assume you want a proof that gravitational acceleration is still given by GM/r² for non-spherical objects.

No such proof exists, because as you have found, it is not true. A similar issue arises in static electricity. Strictly speaking, Coulomb's law is only true for point charges. Newton's shell theorem also applies to this problem, so Coulomb's law is also valid for objects with a spherical charge distribution.

Just as electricity and magnetism are better described using spherical harmonics, so is gravitation. The non-spherical nature of an object leads to 1/r⁴ and higher-order terms. Note: electricity has 1/r³ terms that result from imbalances of negative and positive charge. There is no such thing as negative mass, so the first non-spherical terms in the case of gravitation are 1/r⁴. Another thing to note: These higher-order terms tend to zero much more quickly than does the 1/r² term. As distances grow large, objects tend to look more and more like a point mass. At very large distances, the simple form of Newton's law is approximately correct.

The various space faring nations take advantage of the non-spherical nature of Earth's gravitational field. It is very handy for an Earth-observing satellite if the subsatellite spot on the sunlit side of the satellite's orbit is always near local noon to minimize shadows (or always near twilight to maximize shadows). If the Earth was spherical, this would require constant orbit adjustments to compensate for the Earth's orbit about the Sun. This constant compensation is not needed thanks to the non-spherical nature of the Earth. The Earth's equatorial bulge causes the line of nodes of a satellite's orbit to precess. Place a satellite at the right altitude and inclination and this nodal precession will make the satellite be in sync with the Sun. This is called a sun-synchronous orbit.

The non-spherical nature of the Earth also causes satellites in geosynchronous orbits to migrate toward either 75.3°E or 104.7°W longitude. Except for geostationary satellites placed at these longitudes, a geostationary satellite needs to periodically perform stationkeeping maneuvers to keep the satellite at the desired location.

One final example: Read this article, http://science.nasa.gov/science-news/science-at-nasa/2006/06nov_loworbit/, on the fate of a couple of lunar satellites released in the Apollo days.