From: fortunati.luigi@gmail.com   
      
   Il 31/12/2025 03:31, Jonathan Thornburg [remove -color to reply] ha scritto:   
   > 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   
      
   By the 3rd law, if the father's force on the son is equal to 610N, the   
   son's force on the father should also be equal to 610N.   
      
      
   But if the son's maximum force is 600N, who will help him increase it to   
   610N?   
      
      
   Luigi.   
      
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