# [Physics] How does tension in a rope change when you cut it

continuum-mechanicsforcesnewtonian-mechanicsstring

Suppose we have a mass $m$ dangling from the ceiling on a vertical rope of length $\ell$ with uniform mass density $\lambda$ per unit length. The weight of the mass $mg$ is balanced exactly by the tension in the rope.

Now, suppose the rope is cut at the top of the ceiling. After a moment, the tension in the rope will be (pretty much) zero throughout and the mass will be in free fall. But, presumably the process is actually continuous, and over some period of time the tension in the rope will decrease from its initial value $T(y)$ (depending on the distance $y$ from the ceiling). How does $T(y)$ evolve over time?

But, presumably the process is actually continuous, and over some period of time the tension in the rope will decrease from its initial value $T(y)$ (depending on the distance $y$ from the ceiling). How does $T(y)$ evolve over time?
A single value changing over time won't be a useful model. In a static or slowly evolving situation, we can model the string as massless and perfectly rigid. In this case, a single value for $T$ throughout the item is reasonable. If you continue to use this ideal model when the rope is cut, then we would consider the tension goes to zero immediately.
But if this model is insufficient, then assuming it has a single $T$ throughout is also insufficient. Instead, changes in the forces on the rope propagate from one part of the string to another at a finite speed (often very close to the speed of sound in the material). In your example, if the rope is light, then immediately after the cut, regions of the rope near the cut will have a tension near zero, while regions far from the cut will have a tension equal to $T$. Rather than a single value in the material smoothly changing over time, different portions will differ dramatically.