Re: Newton's Gravity

[Moderator's note: That's a repost of the previous contribution of the
author with a slight slight edit]

In article <vm2d31$1mbqr$1@dont-email.me>, Luigi Fortunati wrote:

The consequence of all this is that the gravitational force of the
larger body of mass M acts on the entire mass <m> of the smaller body
and this justifies the product m*M of Newton's formula, which
corresponds to the force exerted by the larger mass M on the entire
mass <m>.
 
Instead, the gravitational force of the smaller body of mass <m> cannot
act on the entire body of mass M because M is larger and therefore acts
only on a part of body A of size compatible with <m> and, therefore,
the force of body B on body A is not proportional to m*M but to m*m.
 
Consequently, the total gravitational force is proportional to the sum
of m*M plus m*m (mM+mm=m(M+m)).
>
Newton's formula should contain this small change: from F=GmM/d^2 to
F=Gm(m+M)/d^2 (with m<=M) [[...]]

In article <vm536m$2acss$1@dont-email.me>, Thomas Koenig pointed out
a crucial ambiguity with Luigi's suggested formula.  His argument may
be easier to follow if we consider a simple special case: Suppose we
have 3 similar masses A, B, and C, arranged like this:

                      B
     A
                      C

with B and C actually touching (hard to represent in ASCII-art) so as to
form a compound object B+C.  What is the horizontal gravitational force
between A and the compound object B+C?

If we follow Luigi's formula, we'd get
  G m_A (m_A + m_BC)/d^2 = G m_A (m_A + m_B + m_C)/d^2             (1)

But another way to calculate this same force is that it's just the
sum of the horizontal gravitational force between A and B, and the
horizontal gravitational force between A and C.  (The vertical
gravitational forces between B and C don't matter.)  Again using
Luigi's formula, this gives
  G m_A (m_A + m_B)/d^2 + G m_A (m_A + m_C)/d^2
              = G m_A (2m_A + m_B + m_C)/d^2                      (2)

Clearly, calculating the horizontal gravitational force via (1) gives
a different answer from calculating it via (2).

In other words, if we consider decomposing the larger mass into pieces,
Luigi's formula gives two different results for the same quantity, i.e.,
the formula is (depending on your taste in words) either ambiguous or
self-contradictory.

Here's a related problem: what if m_B = m_C = m_A/2 so that
m_A = m_B + m_C, i.e. A has the same mass as B+C?  How do we decide
which body (A or B+C) should be "m" and which should be "M" in Luigi's
formula?

Newton's formula always gives the same result (G m_A (m_B + m_C)/d^2)
now matter how the masses are decomposed.



Luigi also asked:

Is it possible to carry out an experiment to verify which of the two
formulas (F=GmM/d^2 and F=Gm(m+M)/d^2 with m<=M) is more adherent to
reality?

Yes.  With a torsion pendulum it's fairly easy to directly measure
the gravitational forces between laboratory masses.  See the Wikipedia
article
  https://en.wikipedia.org/wiki/Cavendish_experiment

Here's nice collection of reprint articles on these and similar
measurements:
 
  G. T. Gillies, editor
  "Measurements of Newtonian Gravitation"
  American Association of Physics Teachers, 1992
  ISBN 0-917853-46-6

I should also note the conference "Testing Gravity 2025" being held
Jan 29-Feb 2 in Vancouver, Canada,
  https://www.sfu.ca/physics/cosmology/TestingGravity2025/
I'll be attending this conference, and I'll try to post a synopsis
of some of the presentations to s.p.r.  The conference program includes
a talk by someone from the Eot-Wash group discussing tortion-pendulum
and similar exeriments (Michael Ross, "New experimental tests of gravity
from Eot-Wash group").

--
-- "Jonathan Thornburg [remove -color to reply]" <dr.j.thornburg@gmail-pink.com>
   on the west coast of Canada
       The Three Laws of Thermodynamics:
       1) You can't win, only lose or break even.
       2) You can only break even at absolute zero.
       3) You can't get to absolute zero.