Re: [Phys-l] definition of gravity

[John Denker <jsd@av8n.com>]

On 11/08/2011 02:27 PM, John Mallinckrodt wrote:

> Suppose you have two non-rotating Earth's with a center to center 
> distance of 3 Earth radii.  What gravitational field (in terms of g,
>  the gravitational field on an isolated non-rotating Earth) would be
>  determined by a person standing at the point on the surface of one
> of the Earths that is closest to the other Earth?  Is it the same as
> the acceleration that person observes for an object dropped to the 
> ground?

The question is seriously ill-posed.

There are innumerable possible interpretations.  The three most
symmetrical possibilities are:

a) Static:  The planet-planet separation vector is held constant, in
 both direction and magnitude ... somehow.

b) Smash:  Each planet is freely falling toward the other.

c) Orbit:  Each plant is in orbit around the other, such that the
 planet-planet separation vector is changing as to direction but
 not magnitude.

 Note:  The statement of the problem rules out rotational motion,
 but not orbital motion.  They can orbit with out spin, dos-à-dos.

> Thanks for biting.  My guess is that some folks would claim that the
> gravitational field "at that position" is 3/4 g toward the ground.

That is the correct answer in case (a).

> The observed acceleration of freely falling objects in a frame
> attached to that position however is 31/36 g toward the ground.  In
> my world that means that the gravitational field in that frame is
> 31/36 g.  

That's the correct answer in case (b).

> Note that I didn't say and you certainly shouldn't assume
> that the Earths are (somehow!?) *held* with their centers 3 Earth
> radii apart.

Ahemmmm.  The statement of the problem doesn't say that the situation 
is static ... but it doesn't say that it's not.  Assuming case (b) is 
no more correct than assuming case (a).

In my world, when confronted with an ill-posed question, the crucial
first step is to clearly identify it as ill-posed.

=========

I also remark that in most cases other than (a), any observer with 
the slightest amount of sense would have noticed that the gravitational
acceleration was time-dependent, and would report the time-dependence 
explicitly (rather than merely quoting the instantaneous value).