derivation of Newton's 3rd law from 2nd law (was: Re: The experiment)

In article <101qb1s$11u5c$1@dont-email.me>, Luigi Fortunati asked

What are these experiments on which Newton based his third law and what
are the subsequent ones?

In a moderator's note to that same article, I commented

There are lots of experiments which support Newton's *2nd* law.
Given Newton's 2nd law, there's a gedanken-experiment which lets us
derive (or at least strongly argue for) Newton's *3rd* law.  Briefly,
the gedanken-experiment has 3 bodies touching each other
   A B C
with an external force pushing right on A (which then pushes right
on B, which then pushes right on C).  We apply Newton's 2nd law to B,
and then consider the limiting case where B becomes very thin in the
horizonal direction (e.g., maybe B is a sheet of aluminum foil oriented
vertically) and B has very small mass.

It turns out I was slightly wrong in what I wrote above -- we don't
actually need the limiting case, but we need to apply Newton's 2nd law
multiple times (though never actually to B alone).  Here's a more
detailed explanation:

As noted, consider the 1-dimensional motion of 3 (rigid) bodies touching
each other (A on the left, B in the middle, C on the right), with an
external force F_ext pushing right on A.  Because the 3 bodies are
touching each other and are rigid, they all share a common acceleration
(with respect to some inertial reference frame), which by Newton's 2nd
law applied to the entire compound body A+B+C is  

a = F_ext/(m_A + m_B+ m_C)                                        (1)

Let's enumerate the forces acting on each body:

  A: F_ext pushing to the right
     some (as-yet-unknown) force F_B_on_A from B pushing to the left

  B: some (as-yet-unknown) force F_A_on_B from A pushing to the right
     some (as-yet-unknown) force F_C_on_B from C pushing to the left

  C: some (as-yet-unknown) force F_B_on_C from B pushing to the right

Applying Newton's 2nd law to A, we have
  F_ext - F_B_on_A = a * m_A                                         (2)
Substituting in the acceleration from (1), this becomes
  F_ext - F_B_on_A = (F_ext / (m_A + m_B + m_C)) * m_A               (3)
so that
  F_B_on_A = F_ext - (F_ext / (m_A + m_B + m_C)) * m_A               (4)
           = F_ext * (1 - m_A/(m_A + m_B + m_C))                     (5)
           = F_ext * ((m_B + m_C)/(m_A + m_B + m+C))                 (6)

Now consider the compound body A+B.  The only external forces acting
on A+B are F_ext pushing to the right and F_C_on_B pushing to the left,
so Newton's 2nd law gives
  F_ext - F_C_on_B = a * (m_A + m_B)                                 (7)
Substituting in the acceleration from (1), this becomes
  F_ext - F_C_on_B = (F_ext/(m_A + m_B + m_C)) * (m_A + m_B)         (8)
so that
  F_C_on_B = F_ext - (F_ext/(m_A + m_B + m_C)) * (m_A + m_B)         (9)
           = F_ext * (1 - (m_A + m_B)/(m_A + m_B + m_C))            (10)
           = F_ext * (m_C/(m_A + m_B + m_C))                        (11)

Now consider the compound body B+C.  The only external force acting
on B+C is F_A_on_B pushing to the right, so Newton's 2nd law gives
  F_A_on_B = a * (m_B + m_C)                                        (12)
Substituting in the acceleration from (1), this becomes
  F_A_on_B = (F_ext/(m_A + m_C + m_C)) * (m_B + m_C)                (13)
           = F_ext * ((m_B + m_C)/(m_A + m_B + m_C))                (14)

Applying Newton's 2nd law to C, we have
  F_B_on_C = a * m_C                                                (15)
Substituting in the acceleration from (1), this becomes
  F_B_on_C = (F_ext / (m_C + m_B + m_C)) * m_C                      (16)
so that
  F_B_on_C = F_ext * (m_C/(m_C + m_B + m_C))                        (17)

So far we've used Newton's *2nd* law 5 times (once on A+B+C, once on A,
once on A+B, once on B+C, and once on C), but we haven't used Newton's
*3rd* law at all.

So, we can use the results of the above calculation to *check* if
Newton's 3rd law is valid:

The action & reaction forces across the A/B interface are F_A_on_B and
F_B_on_A.  Newton's 3rd law says that these are equal in magnitude and
opposite in direction.  Looking at the above calculation (which, recall,
were worked out using only Newton's *2nd* law), we see that F_A_on_B
(given by (14)) is indeed equal in magnitude and opposite in direction
to F_B_on_A (given by (6), i.e., Newton's 3rd law is confirmed.

The action & reaction forces across the B/C interface are F_B_on_C and
F_C_on_B.  Newton's 3rd law says that these are equal in magnitude and
opposite in direction.  Looking at the above calculation (which, recall
were wokred out using only Newton's *2nd* law), we see that F_B_on_C
(given by (17)) is indeed equal in magnitude and opposite in direction
to F_C_on_B (given by (11), i.e., Newton's 3rd law is confirmed.

In conclusion, by applying Newton's *2nd* law to various parts of the
compound system A+B+C, we've shown that the action/reaction forces
across each interface are in fact equal in magnitude and opposite in
direction, i.e., we've shown that Newton's *3rd* law holds across each
interface.

ciao,
--
-- "Jonathan Thornburg [remove -color to reply]" <dr.j.thornburg@gmail-pink.com>
   (he/him; currently on the west coast of Canada)
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