# What is the tension on a string with unbalanced forces on each side?

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 $$F_1 + F_2$$.

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.
• 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? Commented Sep 30, 2023 at 11:43