From: dr.j.thornburg@gmail-pink.com   
      
   We're considering a tug-of-war where (a) there's no rope, i.e., where   
   two people push/pull directly on each other, and (b) we've switched   
   the direction of forces, so each person is now *pushing* on the other.   
   This "push-of-war" has a father (the stronger of the two people) on   
   the right and a son (the weaker of the two people) on the left, each   
   pushing on the other, so the the father pushes *left* on the son, and   
   the son pushes *right* on the father.  Luigi specified that both people's   
   feet are planted solidly on the ground, and don't slip, and we're   
   assuming the ground to be a Newtonian inertial reference frame.   
      
   I analyzed this system in the article   
      
     Newsgroups: sci.physics.research   
     Subject: Re: Tug of War   
     From: "Jonathan Thornburg [remove -color to reply]"    
     Date: Wed, 24 Dec 2025 23:28:13 PST   
     Message-ID:    
      
   and here (below) I'll refer to some of the numbered equations from that   
   article.   
      
   In my analysis, I made the simplifying assumptions that   
   (a) we're only considering horizontal forces & motions (in particular,   
       the forces between father and and son are purely horizontal, with   
       no vertical components), and   
   (b) both people's bodies are approximated as rigid, with no relative   
       motion between different body parts (so we can unambiguously refer   
       to "the (horizontal) acceleration" of father and son, with no   
       ambiguity about different body parts having different accelerations).   
      
   In article <10imfhe$2q8c6$1@dont-email.me>, Luigi Fortunati asked   
   > When the father increases his force F_father_on_ground, not only does   
   > this force increase, but his force F_father_vs_son also increases.   
   >   
   > The father cannot increase his push to the right (against the ground)   
   > without also increasing his push to the left (against the son)!   
   >   
   > Or not?   
   >   
   > But the son can't further increase his force F_son_vs_father because it   
   > was already at its maximum!   
   >   
   > And so, the force F_father_vs_son becomes greater and no longer equal to   
   > F_son_vs_father, contrary to what Newton's third law states.   
      
   Let's look at this situation a bit more:   
      
   We're starting with both people stationary (so the push-of-war is a tie).   
   Then my previous analysis leading up to the previous article's equation (7)   
   applies:   
      F_father_on_ground = F_father_vs_son = F_son_on_ground             (7)   
      
   To make this concrete, I'll consider the case   
     m_father = 100 kg   
     m_son = 50 kg   
     F_father_on_ground = F_father_vs_son = F_son_on_ground = 600 Newtons   
      
   Now suppose the father increases /F_father_on_ground/ (pushing right   
   on the ground), say to   
     F_father_on_ground = 630 Newtons.   
      
   Assuming that the son can't increase /F_son_on_ground/ (pushing left   
   on the ground) above the stationary value of 600 Newtons, then by   
   equation (3) of my previous article, the net force acting on the entire   
   father+son system is   
      F_net_on_father_and_son   
             = F_son_on_ground - F_father_on_ground                      (3)   
             = 600 N - 630 N = -30 N   
   i.e., 30 Newtons pushing to the left.   
      
   By Newton's 2nd law, this means that the father+son center of mass will   
   now accelerate to the left with an acceleration of   
     a = F_net_on_father_and_son / (m_father+m_son)   
       = -30N / 150 kg   
       = -0.2 m/s^2   
   In other words (still assuming both bodies stay rigid), the son would   
   be pushed backwards, and his feet would start skidding backwards (left)   
   on the ground.  (Actually, the father's body would have to distort a bit   
   (e.g., bending at knees or hip joints) in order to push his hands left   
   while keeping his feet fixed on the ground, but that's a relatively   
   small effect that we can reasonably neglect here.)   
      
   Applying Newton's 2nd law to the father, equation (21) of my previous   
   article gives   
     F_father_vs_son   
         = F_father_on_ground + (m_father/(m_father+m_son))   
                                (F_son_on_ground - F_father_on_ground)  (21)   
         = 630N + (100/150)(-30N)   
         = 610N   
   or equivalently (applying Newton's 2nd law to the son) equation (25) of my   
   previous article,   
      F_father_vs_son   
         = F_son_on_ground - (m_son/(m_father+m_son))   
                             (F_son_on_ground - F_father_on_ground)     (25)   
         = 600N - (50/150)(-30N)   
         = 610N   
   We see that the direct force /F_father_vs_son/ that each person exerts   
   on the other has in fact increased increased a bit (from 600N to 610N).   
   Notice, however, that it hasn't increased as much as /F_father_on_ground/   
   increased (from 600N to 630N), so there are still net forces on each of   
   father and son   
     F_net_on_father = F_father_vs_son - F_father_on_ground   
                     = 610N - 630N   
                     = -20N   (this is of course equal to /m_father a/)   
     F_net_on_son    = F_son_on_ground - F_father_vs_son   
                     = 600N - 610N   
                     = -10N   (this is of course equal to /m_son a/)   
      
      
      
   So far we've assumed that the son is able to keep his body rigid despite   
   the increased force from the father.  But what if that's not true?  E.g.,   
   what if the son's feet have enough friction with the ground to stay fixed,   
   but the son isn't able to resist the father's increased /F_father_vs_son/,   
   i.e., the son's arms can't apply more than 600N pushing right on the son's   
   hands?  In this case there will be a net force to the left on the son's   
   hands, so the son's hands will move left, and the son's arms will be   
   forced to retract.   
      
      
      
   Finally, it should be noted that I made use of both Newton's 2nd law   
   and Newton's 3rd law in my analysis.  In particular, I used Newton's 3rd   
   law to infer that /F_father_on_son/ and /F_son_on_father/ are equal in   
      
   However, I didn't actually need to use Newton's 3rd law to infer this.   
   In an article back in October,   
     Newsgroups: sci.physics.research   
     Subject: derivation of Newton's 3rd law from 2nd law (was: Re: The   
   experiment)   
     From: "Jonathan Thornburg [remove -color to reply]"    
     Date: Wed, 01 Oct 2025 22:45:04 PDT   
     Message-ID:    
   I showed that if we have *any* pair of rigid bodies applying forces   
   to each other, we can use Newton's 2nd law to prove that derive Newton's   
   3rd law must apply to the forces the bodies exert on each other:   
      
   In that October article I wrote:   
   # In the context of Newtonian mechanics, consider the 1-dimensional motion   
   # of two rigid bodies |X| and |Y| which are either touching, or directly   
   # (rigidly) attached to each other, subject to the external forces   
   # |F_ext_on_X| applied to |X| and an external force |F_ext_on_Y| applied   
   # to |Y| (and to no other external forces).   
   #   
   # [[...]]   
   #   
   # using *only* Newton's *2nd* law, we have worked out that   
   # the action-reaction pair of forces |F_X_on_Y| and |F_Y_on_X| at the   
   # |X/Y| interface are equal in magnitude and opposite in direction, just   
   # as Newton's 3rd law predicts.   
   #   
      
   [continued in next message]   
      
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