There's this very simple problem which I've never quite been able to wrap my head around, partly because different instructors have given me different answers and, further, even looking at worked problems online there seems to be no consensus.
Suppose we have a "massless" rope that is pulled at both ends by forces of equal magnitude. What is the tension in the rope?
For example, I've recently been given a problem along the lines of:
Two masses, $m_1$ and $m_2$, are given equal charges of $+Q$ and are tied together by a massless rope of length $l$. What is the tension in the rope?
(Notice that, although the masses are implied to be different, by an energy analysis there is no kinetic energy in the system thus the forces at the ends must be of equal magnitude).
According to some, we should treat the rope as being "fixed" at one end, and then the tension in the rope is given by the magnitude of just one of the forces. According to others, we should consider both forces.
In past exams, I've had points docked due to both approaches, and in each case the instructor is adamant that their approach is correct.
To me, it seems somewhat intuitive that we should consider both forces.
I'm past introductory mechanics now, but these problems still come up from time to time in some form or another.