From: fortunati.luigi@gmail.com
Il 02/01/2026 09:12, Jonathan Thornburg [remove -color to reply] ha scritto:
> In article I wrote
>> 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.
>>
>> [[...]]
>>
>> 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.
>>
>> [[...]]
>>
>> 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
> [[...]]
>> = 610N
>
> In article <10j2nl2$2egs7$1@dont-email.me>, Luigi Fortunati asks
>> 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.
>
> That's right.
>
>
>> But if the son's maximum force is 600N, who will help him increase it to
>> 610N?
>
> To answer this we need a finer-scale analysis. Let's idealize the son's
> body as rigid torso/legs, with arms pushing (with maximum force 600N) on
> hands. Since the father is pushing left on the hands with a force 610N,
> Newton's 3rd law does indeed stay that the son's hands push on the father
> with a force equal in magnitude (610N) and opposite in direction (pushing
> right).
>
> But what is the net force acting on the son's hands? The father is pushing
> left with a force 610N, but (we're assuming) the son's arm muscles can only
> push right on the son's hands with a maximum force of 600N, so there's a
> net force on the son's hands of 10N pushing left. By Newton's 2nd law,
> that means the son's hands must accelerate to the left.
Exactly.
In fact, the resultant force acting on the son's hands is -10 N,
resulting from the opposing forces F_father_vs_son (-610 N) and
F_son_muscles (+600 N).
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?
> In other words (assuming the son's feet stay stationary with respect to
> the ground, and the son's legs/torso stay rigid), the son's arms must
> retract, allowing the son's hands to move left (closer to the son's body).
Yes, the son's hands (and the father's) accelerate to the left because
they are both subject to the net force of -10N.
Luigi.
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