From: fortunati.luigi@gmail.com   
      
   Il 12/01/2026 04:51, Jonathan Thornburg [remove -color to reply] ha scritto:   
   > 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.   
      
   How can the son increase his force on his father to +610N if the force   
   he's exerting (+600N) is already equal to his maximum capacity?   
      
   Ciao, Luigi.   
      
   [[Mod. note --   
   The son can't increase /F_son_arms_on_son_hands/.  But the son doesn't   
   directly control /F_son_hands_on_F_father_hands/: the additional 10N by   
   which /F_son_hands_on_F_father_hands/ exceeds /F_son_arms_on_son_hands/   
   is essentially due to the inertia of the son's hands (which are accelerating   
   to the left with an acceleration of /a_hands/).   
      
   This is similar to how /F_father_arms_on_father_hands/ is 630N, but   
   /F_father_hands_on_son_hands/ is only 610N -- the 20N difference is due   
   to the inerta of the father's hands (which are also accelerating to the   
   left with an acceleration of /a_hands/).   
   -- jt]]   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)