From: dr.j.thornburg@gmail-pink.com   
      
   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)   
   and hence by (3)   
      F_net_on_father_and_son < 0                                        (9)   
   and thus (by Newton's 2nd law applied to the combined father+son system)   
   both people accelerate to the left.   
      
   Since both people are accelerating to the left, we know (by Newton's   
   2nd law applied to the father) that   
      F_net_on_father < 0                                               (10)   
   and thus by (1) we must have   
      F_father_on_ground > F_father_vs_son                              (11)   
      
   Similarly, we know (by Newton's 2nd law applied to the son) that   
      F_net_on_son < 0                                                  (12)   
   and thus by (2) we must have   
      F_father_vs_son > F_son_on_ground                                 (13)   
      
   Combining (11) and (13), we have   
      F_father_on_ground > F_father_vs_son > F_son_on_ground            (14)   
   which is also consistent with (8).   
      
   In fact, we can quantify the differences between the three forces in (14).   
      
   Suppose that the father and son are both accelerating to the right   
   with an acceleration /a/ (/a = 0/ means the push-of-war is a tie,   
   /a < 0/ means the father is winning, /a/ > 0 means the son is winning)   
      
   Newton's 2nd law applied to the combined father+son system says   
      F_net_on_father_and_son = (m_father+m_son) a                      (15)   
   so that by (3)   
      F_son_on_ground - F_father_on_ground = (m_father+m_son) a         (16)   
   and hence   
      a = (F_son_on_ground - F_father_on_ground) / (m_father+m_son)     (17)   
      
   Applying Newton's 2nd law to the father, we have that   
      F_net_on_father = m_father a                                      (18)   
   so by (1) we have   
      F_father_vs_son - F_father_on_ground = m_father a                 (19)   
   i.e.   
      F_father_vs_son = F_father_on_ground + m_father a                 (20)   
   so that by (17)   
      F_father_vs_son   
         = F_father_on_ground + (m_father/(m_father+m_son))   
                                (F_son_on_ground - F_father_on_ground)  (21)   
      
   Similarly, applying Newton's 2nd law to the son, we have that   
      F_net_on_son = m_son a                                            (22)   
   so by (2) we have   
      
   [continued in next message]   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)