Re: Falling Objects, How They Fall


Joe Fischer wrote:
> On 28 Dec 2005 "PD" wrote:
>
> >Joe Fischer wrote:
> >> I said in another message it was a brain slip,
> >> it should have read 1.012 * 981 / r2
> >
> >Uh, no. Where on earth did you come up with that conclusion?
> >PD
>
> The formula F = GMm / r^2 is an Earth centric
> formula, _simplified_ to be valid in all situations, for
> the F _between_ two objects.

No, it is quite general. It works just fine, for example, between the
Sun and Mars and between the Sun and your comet of choice.

>
> ____ IF ____ , both the Earth and another object
> are free to move,

....which they are...

> both objects do accelerate toward the
> center of mass of the two body system.

.... yes, that's right.

>
> Any falling problem can be worked as a two-body
> problem, but for an object with a mass much less than
> the moon, nothing is gained, and it can be worked as
> a one-body problem.

It can be worked out for the Moon just as easily.
The acceleration of the Moon towards the Earth is F/m = [GMm / r^2]/m =
GM/r^2.
The acceleration of the Earth towards the Moon is F/M = [GMm / r^2]/M =
Gm/r^2.
The relative acceleration of the two is the sum of these: G(M+m)/r^2.

This analysis works if the Moon is replaced by a brick, or by Jupiter.

>
> The F in the Newtonian formula is often called
> a "rubber band" force, because it is "exerted" in
> Newtonian gravitation on both bodies.

Well, actually, it's exerted on each separately, but Newton's 3rd law
says the two are equal and opposite.

>
> So while the F = GMm / r^2 is correct and
> of better accuracy than can be measured for any
> object smaller than the moon, it is not precise,
> even in the model Newtonian gravitation.
>
> The term "reduced mass" is used in formal
> presentations of two bodies free to move, and I
> have not had a formal class that covered reduced
> mass, so I may be mistaken about certain assumptions,
> sorry.
>
> I have seen on the www a statement that the
> reduced mass is always less than the mass of
> both objects (combined?), but I think that is
> wrong.
>
> If anybody really knows reduced mass concepts
> really well, I would like some comments.

The term reduced mass is just a shorthand for a collection of terms.
Replacing the collection of terms by the shorthand term makes the math
look similar to the extreme case of one very mass and one very light
object, but there is nothing intrinsically different about the physics.

>
> Even "double precision" computer math is
> not precise enough to handle this kind of problem,
> and there is considerable simplifying done because
> better than 4 digit precision is rarely needed.
>
> Two-body problems can be worked with full
> precision within the Newtonian gravitation model
> using acceleration instead of force, but there
> needs to be a + sign in the formula to cover the
> acceleration of both the Earth and the other object.

Yes, as shown above. But that's only if you want to know the *relative*
acceleration.
This + sign appears, by the way, for *every single force known*,
because the same principles apply:
1. The force between two objects always appears as a pair of forces on
the two objects, and this pair is equal and opposite.
2. Each object has a unique acceleration as a result of that force.
These two accelerations will both be on a line between the two objects.
3. The relative acceleration is the sum of the two accelerations.

Thus, gravity is in no way special to this.

>
> This avoids the "rubber band" concept that
> is inherent in the F = GMm / r^2 formula.
>
> I admit I am stating my impression of what
> E. G. Valens states in "The Attractive Universe",
> which is a K-12 level book at best, but is very
> complete in the discussion of "attractive" gravity.
> It is filled with numerous multiple exposure
> high speed time lapse photography by an award
> winning photographer which has made it a very
> desirable item for musuems and photographic
> collectors, with www prices of the 1969 first
> edition running as high as $300.
> I was lucky to get a copy around 1971,
> and it has been the most interesting book
> I have ever owned, I like it better than MTW.

I'll bet. I'll wager you ran aground in chapter 4 of MTW.

>
> I can't be sure that Valens is correct,
> or that there are no printer's typos, but it
> all sounds perfectly logical and like he
> did a lot of research and had peer review
> before publishing.
>
> I have a couple of extra second edition
> copies, and may offer one for sale on ebay
> next year so anybody really interested in
> gravity will be able to afford it.
> When I do, I will offer a scan of the
> page that covers "unequal falling" and
> post the ebay link here and you will be
> able to make comments on what Valens
> wrote and my stated impression of it.
>
> Joe Fischer

.

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