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If a string is pulling on an object of mass $M_1$ with force $F_1$, and the other side is pulling an object of mass $M_2$ with force $F_2$, what is the tension in the rope?

Intuitively, it makes sense to me that the tension in the rope would be the sum of the forces which it exerts (and thus is exerted on it), and so the tension would be $F_1 + F_2$.

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If an inextensible string is massless, and with unbalenced forces, it is accelerating infinitely fast. If it is inextensible, but not exactly massless with total mass $m$, then the acceleration at every point on the string is is $(F_1-F_2)/m$ and the tension is $F_1$ at the $F_1$ end, and $F_2$ at the $F_2$ end. The exact value of the tension at intermediaate points along the string depends on the density profile along the string.

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  • $\begingroup$ The string can't be accelerating infinitely fast; there are non-massless objects on either side. $\endgroup$
    – Thuong Vo
    Commented Sep 30, 2023 at 7:38
  • $\begingroup$ The string only cares about the forces on it. In $F=ma$ it is the external forces is $F_1-F_2$. What part of this do you not understand? $\endgroup$
    – mike stone
    Commented Sep 30, 2023 at 11:43
  • $\begingroup$ The fact that it is impossible for the string to accelerate infinitely if the things it is attached to cannot? $\endgroup$
    – Thuong Vo
    Commented Oct 2, 2023 at 10:36
  • $\begingroup$ So we conclude that the things that it is attached to cannot apply different forces at the ends -- as was said right at the begining. $\endgroup$
    – mike stone
    Commented Oct 2, 2023 at 12:20
  • $\begingroup$ i don't see why not. perhaps there's a tension force, or an electromagnetic force, pulling on those, which exerts tension force onto the rope. $\endgroup$
    – Thuong Vo
    Commented Oct 3, 2023 at 13:07

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