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My intuition says that person 1 exerts a force $F_{1}$, but it's not exerted on person 2, rather, it is exerted on the rope in between. So far so good. From Newtown's third law, the rope exerts a force of the same magnitude in the opposite direction on person 1. If the ground is a frictionless ice, then this is actually the whole story, no other forces should be acting on person 1 other than the force the rope exerts. However, I have seen other solutions that doesn't make sense at all. For example, one of the solutions that I have seen somewhere was one that says that there is a net force equals F1−F2 acting on person 1. Another one says that the tension is constant all along the rope and both bodies move with the same acceleration in the same direction given that there is a net force acting on person 1 equals to $F_1−T$ and a net force equals to $T−F_{2}$on person 2. So, which is which?

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2 Answers 2

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In the game of tug and war all the forces between persons 1 and 2 and the rope are internal forces, being equal and opposite per Newton's third. Call all these force $F$. See FIG 1.

But the internal forces exerted by the rope are external forces with respect to person 1 and person 2, Neglecting the mass of the rope, and given no friction (or other external horizontal force) acting on each person, the acceleration of each person is determined by $F$ divided by the persons mass, per Newton's 2nd law. Since in this example the mass of person 1 is greater than person 2, person 2 will have a greater acceleration than person 1. See FIG 2.

Finally, since there are no external horizontal forces acting on the combination of the two persons the center of mass (COM) is stationary with person 1 moving towards it faster than person 2, until both meet at the COM. See FIG 3.

Hope this helps.

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  • $\begingroup$ I am stating already that each one exerts a different force, so what're saying is that wrong to begin with? $\endgroup$
    – Jack
    Commented May 19 at 16:01
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    $\begingroup$ @Jack Yup. If they exerted different forces the center of mass of the two persons and rope would accelerate, which is not possible without a net external force on the two persons and rope system. $\endgroup$
    – Bob D
    Commented May 19 at 16:48
  • $\begingroup$ I still can't see why can't I say that $F_{1R}=F_{R1}$, and the same for person 2, and proceed with Newton's second law even if $F_{R2}≠F_{R1}$ ? $\endgroup$
    – Jack
    Commented May 19 at 17:25
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    $\begingroup$ @Jack I will add a diagram to my answer. $\endgroup$
    – Bob D
    Commented May 19 at 19:57
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    $\begingroup$ @Jack See revision to my answer $\endgroup$
    – Bob D
    Commented May 19 at 22:43
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If there is no friction between the surface and the person, pulling the rope will simply cause the person to accelerate forward. If two persons are pulling the rope, The two people will accelerate towards each other and collide.

The force that person 1 exerts is on the rope and the rope also pulls person 1 iwith same force. If the rope pulls person 1, it means there is tension in the rope, which means that the rope will also pull the 2nd person. If the 2nd person is not holding the rope, then, there is no force on the 2nd person.

Can you please specify what are the "other" solutions in the question?

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  • $\begingroup$ One of the solutions that I have seen somewhere was a one that says that there is a net force equals $F_{1}-F_{2}$ acting on person 1. Another one says that the tension is constant all along the rope and both bodies move with same acceleration in the same direction given that there is a net force acting on person 1 equals to $F_{1}-T$ and a net force equals to $T-F_{2}$ on person 2 $\endgroup$
    – Jack
    Commented May 19 at 6:50
  • $\begingroup$ @Jack I'll vote to leave open for now but please edit them into your question. And make clear what it means by frictionless. Tug of war works using friction on the ground. $\endgroup$ Commented May 19 at 11:39
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    $\begingroup$ @Jack As F1 and F2 are reaction forces, they are equal in magnitude. The Net force on the man is the tension force exterted. If the man does not hold the rope tightly enough, rope will slip and that will reduce the tension. Also if the rope has considerable mass and is accelerating, then the tension at different points will be different, as there has to be some force of pull the rope which has mass. $\endgroup$ Commented May 19 at 12:09

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