From: dr.j.thornburg@gmail-pink.com   
      
   In article <10jageu$12sh6$1@dont-email.me> (Sat, 03 Jan 2026 22:22:58 PST)   
   Luigi Fortunati wrote:   
   > And the same resultant force of -10 N also acts on the father's hands,   
   > resulting from the external force F_son_vs_father (+600 N) and the   
   > internal force F_father_muscles (-610 N).   
   >   
   > This shows that the force F_father_vs_son (-610 N) is greater than, and   
   > not equal to, the force F_son_vs_father (+600 N).   
   >   
   > If there's a mistake in all this, where is it?   
      
   Sorry for the delayed reply -- I've been sick.  Catching up now.....   
      
   To respond to Luigi's question properly, we need to analyze the   
   biomechanics a bit more carefully.  In particular, we need to drop my   
   previous assumption that the father's body stays rigid, and go back and   
   redo the analysis without that assumption.   
      
   Let's now model the father's body the same way we're already modelling   
   the son's body, namely, as rigid legs/torso with arms pushing on (i.e.,   
   applying a force on) on hands.   
      
   That is, we now have   
   * father's feet are assumed to be fixed on the ground   
   * father's legs/torso are assumed to be rigid   
   * father's arms push left on father's hands   
     with a force of magnitude /F_father_arms_on_father_hands/   
   * father's hands push left on son's hands   
     with a force of magnitude /F_father_hands_on_son_hands/   
   * son's feet are assumed to be fixed on the ground   
   * son's legs/torso are assumed to be rigid   
   * son's arms push right on son's hands   
     with a force of magnitude /F_son_arms_on_son_hands/   
   * son's hands push right on father's hands   
     with a force of magnitude /F_son_hands_on_father_hands/   
      
   Notice that I am *not* assuming that /F_son_hands_on_father_hands/   
   must necessarily be the same as /F_father_hands_on_son_hands/ (which   
   is what Newton's 3rd law would say).   
      
   One complication in this analysis is that if a person's hands move,   
   then necessarily their arms must also be in motion, but not all of   
   their arms have the same acceleration.  The easiest way to model this   
   is to treat the hand-arm system as having an "effective mass" which   
   includes all of the hands but only a fraction of the arms, and say that   
   that the effective mass is the only part of their body that accelerates.   
   I'll do this from now on.   
      
   Applying this to the son and father, let's take   
   * effective mass of son's hands + arms = 5kg   
   * effective mass of father's hands + arms = 10kg   
      
   When the push-of-war is tied, we have   
     F_son_arms_on_son_hands = 600N   
     F_son_hands_on_father_hands = 600N   
     F_father_arms_on_father_hands = 600N   
     F_father_hands_on_son_hands = 600N   
   and clearly the net force on the hands is zero.   
      
   Now if the father increases /F_father_arms_on_father_hands/ to 630N,   
   but the son doesn't (can't) increase /F_son_arms_on_son_hands/ above 600N,   
   what happens?  (This is basically the situation Luigi was asking about.)   
   We have   
     F_son_arms_on_son_hands = 600N   
     F_son_hands_on_father_hands = don't know yet   
     F_father_arms_on_father_hands = 630N   
     F_father_hands_on_son_hands = don't know yet   
      
   If we look at the son's and father's hands (and the moving parts of their   
   arms), their combined effective mass is 15kg, and the net force acting on   
   them is   
     F_net_on_combined_hands   
             = F_son_arms_on_son_hands - F_father_arms_on_father_hands   
             = -30N   
   Applying Newton's 2nd law to the two hands together, we see that they   
   accelerate with an acceleration of   
     a_hands = F_net/m   
             = -30N / 15kg = -2 m/s^2   
   (This is to the left, which is what we expect since the father is winning   
   the push-of-war.)   
      
   Now let's apply Newton's 2nd law to the son's hands:   
     F_net_on_son_hands = m_son_hands a_hands = 5kg (-2 m/s^2) = -10N   
            = F_son_arms_on_son_hands - F_father_hands_on_son_hands   
            = 600N - F_father_hands_on_son_hands   
   so we must have F_father_hands_on_son_hands = 610N.   
      
   Now let's apply Newton's 2nd law to the father's hands:   
     F_net_on_father_hands = m_father_hands a_hands = 10kg (-2 m/s^2) = -20N   
            = F_son_hands_on_F_father_hands - F_father_arms_on_father_hands   
            = F_son_hands_on_F_father_hands - 630N   
   so we must have F_son_hands_on_F_father_hands = 610N.   
      
   So, using *only* Newton's 2nd law, we conclude that in fact   
     F_son_hands_on_F_father_hands = F_father_hands_on_son_hands = 610N   
   i.e., Newton's 3rd law does indeed hold here.   
      
   ciao,   
      
   --   
   -- "Jonathan Thornburg [remove -color to reply]"    
      (he/him; currently on the west coast of Canada)   
        Alberto Moreira (comp.arch, Jan 1999):   
          "Cache is, basically, a kluge generated by technological restrictions."   
        Donald C. Lindsay (also comp.arch, Jan 1999):   
          "No. It's a kluge generated by deeply fundamental restrictions,   
           like the speed of light."   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)