There is always one point in introductory mechanics that has been continuously bothering me ever since I had taken my freshman physics course - Why does a massless rope have its tension equal on both ends?
I have been searching for a good explanation on various textbooks at different levels, but none of them provided me a satisfactory explanation. They either
State directly "we assume the string is massless and the pulley has no friction, therefore the tension must be the same throughout the string" without further any elaboration and continue the calculation,
Claim without any reasons that "whenever the rope and the pulley are massless and frictionless, the tension will be the same on both sides", or
Explain using a lot of hand-waving arguments regarding infinite acceleration, non-zero moment of inertia, etc. , in one line or two.
Although those hand-waving arguments of type $(3)$ do sound quite reasonable to me and will convince me for a while every time I review them. But after a while, I will start feeling not well about this whole tension concept again.
Therefore, I am looking for a rigorous (in the physicist's sense) proof of exactly why and when do the tension in rope will be uniform, and hopefully from that I can clean up all of my conceptual difficulties.
Answer
I will only consider taut, inextensible ropes. This represents an approximation regime where tension is much less than the Young's modulus of the rope material times the cross-sectional area of the rope. (As we will see this condition is enough to guarantee the conclusion for straight segments of the rope as long as they are light compared to what ever they are attached to.)
Let's start with the easiest case: there is a frame of reference in which the entire rope is still.
In this case we can reason thus: the rope is subject to net zero external force (by Newton's 2nd Law). Further any internal element of the rope must be subject to equal and opposite forces from neighboring elements. Those forces are the tension in the rope so the tension is the same throughout.
[Note that the rope need not be massless.]
Next easiest case is the rope is in motion with constant speed along it's own length, but it may pass over circular pulleys with no friction at bearing (but with enough friction at the grove that the rope does not slip) and the like so any given part of the rope might change directions at times.
The argument in part (1) applies to segments between the pulleys.
The pulleys themselves have no angular acceleration and are therefore subject to zero net torque. So, the segments on either side of the pulley have the same tension because they have to exert equal and opposite torques.
[Note that the neither the rope nor the pulleys need to be massless.]
Now we come to cases where the rope is accelerating along it's own length.
If we assume a massless rope then any change of tension along the length would cause arbitrary acceleration. That's a nasty case because it s non-physical, but it represents an approximation of a regime where the mass of the rope is much smaller than the mass of anything it is connected to. Demonstration Atwood's machines are roughly like that.
A straight, massive rope (segment) accelerating along it's own length. The segment is subject to a net external force $F_n = F_1 - F_2$ where $F_1$ and $F_2$ are the net forces at either end, so that the acceleration is $a = F_n/m$. The rope segment is assumed to have uniform mass density $\lambda = m/l$. At any fraction $f$ along the length of the rope the tension must be such that the segment on each side of conceptual divide has the same acceleration (to prevent extension without allowing slackness). So tension $\tau$ (which is the same in each direction from Newton's 3rd Law) at position $fl$ must meet the requirements $$ a_1 = \frac{F_1 - \tau}{\lambda fl} \;,$$ and $$ a_2 = \frac{\tau - F_2}{\lambda (1-f)l} \;,$$ where $a_i$ is the acceleration of the segment closer to force $F_i$, and $a_1 = a_2$ is required. Thus \begin{align*} \frac{F_1 - \tau}{\lambda fl} &= \frac{\tau - F_2}{\lambda (1-f)l} \\ \frac{F_1 - \tau}{f} &= \frac{\tau - F_2}{1-f} \\ (1 - f)(F_1 - \tau) &= f(\tau - F_2) \\ -\tau &= (f - 1)F_1 - fF_2 \\ \tau &= F_1 - fF_n \;. \end{align*} Now, this depends on $f$, which means that it is not constant, right?
OK, but the term is $fF_n$, and $F_n = ma$ where $m$ is the mass of the rope segment. So, if the rope is much less massive than the things attached to it's ends, this terms could be negligible compared to $F1$ (or, indeed $F_2$ since the labeling is arbitrary). Then we get $$ \tau \approx F_1 \,,$$ which justifies the claim in item (3) that the hypothetical "massless" condition is an approximation for "very light".
Next we could let these things run over low friction pulleys. We can recover the 'tension the same throughout' conclusion only if the pulleys are also "very light" so that we can use an argument like that in item (2) because the torque on them will arise from force small compared to $F_1$ or $F_2$.
Once the pulleys have non-trivial mass and the rope is accelerating than the segments must be assumed to have different tensions even if the rope is light.
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