Massless String Having Different Tensions

Question

I'm a student fairly new to physics, and I was working through a textbook (this is not for homework) when I came across a problem involving

Two masses, $m$ and $2m$, hang over a pulley with mass $m$ and radius $R$ (and $I = \frac{mR^2}{2}$), as shown. Assuming that the (massless) string doesn’t slip on the pulley, find the accelerations of the masses.

I was able to deduce on my own that there must be a net torque on the pulley, for it to accelerate along with the string of the pulley. I believe this also implies a static friction force between the string and the pulley, for the torque to be applied from the string to the pulley, and prevent slipping of the string.

However, the next natural assumption this led me to is that the string must have different tensions throughout itself. But, everything I read in the book, and online, has told me that massless strings must have a constant tension throughout (which I understand, using the argument that a small piece of string cannot have net force, as that would yield infinite acceleration). However, everywhere I read online implies that this is an absolute rule $-$ several forums, including discussions on Stack Exchange seem to imply that a massless string has zero acceleration, regardless of the setup. I understand why the deduction about the net torque on the pulley leads to the result that the string must have different tensions throughout itself, but can't intuitively grasp how such a massless string can exist.

I was slightly illuminated by this Wikipedia article, which states that a massless string only has uniform tension if the string has no bends (i.e. it is perfectly straight). However, I can't convince myself of this fact, and would like help (an explanation) for why this is true $-$ in particular, why can't we just, once again, use the argument where we consider a piece of the massless string, and show the tensions on either side of it, along the string, must be equal to prevent infinite acceleration? Another reason I'm not fully convinced that bending a massless string allows it to exist with non-constant tension is that in problems with a massless pulley and a massless string, the string having non-constant tension is never possible, even though it's bent around the pulley.

Though I found a post on Stack Exchange with a nearly identical question from 11 years ago, it was never really resolved. The textbook's solution does not address this, but it mentions that the accelerations of the two masses are equal, despite it also saying that the tensions on either side of the pulley are different, which confuses me even more.

I would really like to understand why a massless string can exist with varying tensions along its body in such a setup, as well as why such a setup can coexist with the masses on either side of the pulley having equal accelerations.

Thanks!

Answer 1

The reason that an isolated massless string cannot have a net force on it is that if there was a net force the string would suffer an infinite acceleration.

System: string and pulley
In the case you have written about the massless sting is physically connected, via frictional forces, to a pulley which does have mass and so any forces you are dealing with are acting on the string and the pulley together.
Thus there can be a net force on an element of the string and the pulley which will cause a finite acceleration of them both.

System: small element of string
Consider a small element of string with forces $T_1$ and $T_2$ acting on opposite ends of the string due to adjacent parts of the rest of the string with $T_1>T_2$.
In this situation those are not the only forces acting on the string as there is a frictional force $T_1-T_2$ acting on the string due to the pulley such that the net force on the string is zero.

The other Newton third law force is the frictional force which acts on the pulley due to the string of magnitude $T_1-T_2$ and in the opposite direction to the frictional force on the string and it is this frictional force which causes the acceleration of the pulley.

Answer 2

We often assume the tension in a string, particularly a massless string, is the same throughout but we have to be careful that we are considering one string, not two. If you think of the classic toy train problem where you have a string connecting the first and second train cars and another string connecting the second and third train cars, the tension is not the same in these two strings, even though the acceleration of the cars is the same, because they are two different strings.

This situation with a massive pulley like that that train setup. With the pulley acting as the middle train car.

The string in causing the pulley to accelerate (angularly) which, since the pulley is not massless, means the string must be exerting a force on the pulley. Newton's Third Law tells us this means the pulley must be exerting a force on the string.

Even though there is just a single string we have to treat it as if it were two strings because of the force acting on the string in the middle.