[extropy-chat] Juicy Classical Physics Problem Involving Gravitational Potential

[The Avantguardian]

--- Lee Corbin <lcorbin at rawbw.com> wrote:
> Newton---but not I---was able to calculate the
> spherical oblateness of
> the Earth by carefully regarding this notion of
> gravitational potential. (He'd
> be wonderfully jazzed by the generalization to
> relativity theory spoken of
> above.)
> 
> But my Newtonian calculation---problematical in some
> way---is off by a
> factor of two! Perhaps someone here who really likes
> classical physics
> can see what is going wrong.  Of course, I *could*
> find some alternate
> derivations on the web, or find some in a book
> somewhere, but that
> would take all the fun out of it.  Nothing like
> trying to work through
> something yourself to really master the concepts.
> 
> 
> Let  r  be the equatorial radius and R the polar
> radius.  The facts are
> that R is 6357000 meters (or 6357 km), and r is 6378
> km, a
> twenty-one kilometer difference.
> 
> But my calculations below show that r "should" be
> only 6368 km,
> that is, just 11 km more than R instead of the
> correct value of 21 more.

I did not actually do the calculation myself however
there are two possible errors which may have led to
the discrepancy. 

1. Did you remember the factor (1/2) in your kinetic
energy term? i.e. KE = (1/2)mv^2 for your particle.

2. The mass of the earth that contributes to the
gravitational field at the poles is LESS than the mass
contributing to the field at the equator. The reason
for this is Guass' Law in gravitation.

At any point, a given distance from the barycenter of
an extended body, the only mass that contributes to
the gravitational field at that point is that portion
of the mass that is closer to the barycenter.

In other words, the mass of the earth that is included
as M in GM/R should be the portion of the mass that is
contained within a perfect sphere of radius R. If R is
the polar radius this will be the total mass of the
earth minus the mass of those parts of the earth that
bulge out past the polar radius at the equator. Now
mind you the mass of the bulge will be very small
compared to the total mass of the earth. But 10 km is
not a very large error at the scale of distances you
are considering.

> The next equation, and what I'm trying to go by, 
> reads "the sum of
> the K.E. of a particle at the equator  and  the
> remaining energy it
> needs to get to infinity must equal the energy that
> the particle at the
> pole would need to acquire to get to infinity".   Is
> there a problem
> with that? It certainly raises questions in my mind,
> and one's that
> I don't have clear answers for.

I am curious as to what those questions are. As far as
whether your equation is problematic, I don't think
so, at least not classically. What you stated is
pretty much the mathematical definition of a
gravitational equipotential (guassian) surface.

Now I have a puzzle for you guys: What is the maximum
value of g, the acceleration of earth's gravitational
field, and where is this maximum g located?

Hint: Don't get stuck in two dimensions. :)



Stuart LaForge
alt email: stuart"AT"ucla.edu

"Believe nothing. No matter where you read it, or who said it, even if I have said it, unless it agrees with your own reason and your own common sense."- Siddhartha Guatama aka Buddha.


 
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