Tension in a string with unequal forces

Question

Let's suppose that there is a massless string. If we apply unequal forces along the rope along a straight line, let's say we apply 60N and 70N forces, then what is the tension in the rope? It would be of great help if you lead my way with both mathematics and intuition.

Answer 1

If you apply different forces at each end, you will set the string in motion (relative to an inertial frame). In this case, the string will suddenly start to move with infinite velocity since it is massless, which is a way of saying that such a massless string is unphysical.

If you have a massive string, you will still set the string in motion, but it will gradually accelerate with an acceleration equal to the difference in forces divided by the mass of the string (Newton's 2nd Law).

If we can assume the speed of waves in this massive string are sufficiently high that we can think of each piece of the string as moving with the same velocity, then the tension in the string will vary linearly from one end to the other: in this case, from 60 N to 70 N. The linear increase is required by the fact that each piece of this string has the same acceleration. By Newton's 2nd Law, for a piece of the string of length $\Delta x$ and mass per unit length $\lambda$, $$ \Delta F = \lambda \Delta x a $$ or $$ \frac{dF}{dx} = \lambda a = \lambda \frac{\Delta F}{M} = \lambda \frac{\Delta F}{\lambda L} = \frac{\Delta F}{L}$$ where $F$ is the tension at a given point and $a$ is the acceleration of the string, $M$ is the total mass of the string, $L$ is the length of the string and $\Delta F$ is the difference between the forces being applied at the ends of the string. The rate of change of $F$ being constant means the tension varies linearly along the string.