From: fortunati.luigi@gmail.com   
      
   Il 25/12/2025 08:28, Jonathan Thornburg [remove -color to reply] ha scritto:   
   > In article <10i5vnm$1t2p1$1@dont-email.me>, Luigi Fortunati asked   
   > about the forces in a tug-of-war where there's no rope, i.e., where   
   > two people push or pull directly on each 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   
   > with outstreatched arms (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.   
   >   
   > To properly understand this system, we need to consider *all* the   
   > forces acting on the father and the son, including the forces their   
   > legs/feet exert on the ground, and the Newton's-3rd-law reaction   
   > of the ground on their legs/feet.   
   >   
   > To keep things simple, let's treat the ground as a Newtonian   
   > inertial reference frame.  Let's only consider horizontal forces   
   > and motions, and let's ignore the changes in shape of the people's   
   > bodies, i.e., let's treat father and son as having the *same*   
   > (horizontal) acceleration with respect to the ground.   
   >   
   > Let's list all the (horizontal) forces acting in this system:   
   >   
   > The son is trying to push the father's body to the right.  To resist   
   > this, the father's feet must push *right* on the ground, with with a   
   > force (applied by the father's leg and hip muscles) of magnitude   
   > /F_father_on_ground/.   
   >   
   > By Newton's 3rd law, this means that the ground must react on the   
   > father's feet with a force of equal magnitude but opposite direction,   
   > i.e., the ground exerts a force on the father's feet of magnitude   
   > /F_father_on_ground/ pushing *left*.   
   >   
   > Similarly, the father is trying to pull the son's body to the left.   
   > To try to resist this, the son's feet must push *left* on the ground,   
   > with with a force (applied by the son's leg and hip muscles) of   
   > magnitude /F_son_on_ground/.   
   >   
   > By Newton's 3rd law, this means that the ground must react on the   
   > son's feet with a force of equal magnitude but opposite direction,   
   > i.e., the ground exerts a force on the son's feet of magnitude   
   > /F_son_on_ground/ pushing *right*.   
   >   
   > And finally, the father and son also push directly on each other:   
   > The father pushes left on the son with a force of magnitude   
   > /F_father_vs_son/.  The *son* pushes right on the father with a   
   > force which, by Newton's 3rd law, must be of equal magnitude   
   > (/F_father_vs_son/) but opposite direction (pulling right).   
   >   
   > Now let's collect all the (horizontal) forces acting on each person:   
   >   
   > Forces acting on the *father*:   
   > * The son pushes right on the father with a force of magnitude   
   >    /F_father_vs_son/.   
   > * The ground has a reaction force on the father, of magnitude   
   >    /F_father_on_ground/ pushing left.   
   > So, the net force to the right acting on the father is   
   >     F_net_on_father = F_father_vs_son - F_father_on_ground            (1)   
   >   
   > Forces acting on the *son*:   
   > * The father pushes left on the son with a force of magnitude   
   >    /F_father_vs_son/.   
   > * The ground has a reaction force on the son, of magnitude   
   >    /F_son_on_ground/ pushing right.   
   > So, the net force to the right acting on the son is   
   >     F_net_on_son = F_son_on_ground - F_father_vs_son                  (2)   
   >   
   > Finally, the net force to the right acting on the two people is just   
   >     F_net_on_father_and_son   
   >            = F_net_on_father + F_net_on_son   
   >            = (F_father_vs_son - F_father_on_ground)   
   >              + (F_son_on_ground - F_father_vs_son)   
   >            = F_son_on_ground - F_father_on_ground                      (3)   
   >   
   > Let's first consider the case where the pull-of-war is a tie, with   
   > both father and son stationary (and hence unaccelerated horizontally).   
   > We'll apply Newton's 2nd law three times:   
   >   
   > Applying Newton's 2nd law to the father, we see that the fact that   
   > the father is unaccelerated (horizontally) means /F_net_on_father = 0/,   
   > so by (1) we have   
   >     F_father_vs_son = F_father_on_ground                               (4)   
   >   
   > Applying Newton's 2nd law to the son, we see that the fact that   
   > the son is unaccelerated (horizontally) means /F_net_on_son = 0/,   
   > so by (2) we have   
   >     F_father_vs_son = F_son_on_ground                                  (5)   
   >   
   > Applying Newton's 2nd law to the combined father+son system,   
   > we see that the fact that the combined system is unaccelerated   
   > (horizontally) means /F_net_on_father_and_son = 0/, so by (3) we   
   > have   
   >     F_father_on_ground = F_son_on_ground                               (6)   
   >   
   > Combining (4), (5), and (6), we have (still for the case where the   
   > pull-of-war is a tie)   
   >     F_father_on_ground = F_father_vs_son = F_son_on_ground             (7)   
   >   
   > Now let's say the father wants to win for a while.  Since the father   
   > is stronger than the son, the father can increase F_father_on_ground   
   > so that   
   >     F_father_on_ground > F_son_on_ground                               (8)   
      
   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.   
      
   Luigi Fortunati   
      
   --- SoupGate-DOS v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)