kuruman said:
I am only expressing my doubts whether the "work done on the rope" is a useful construct when trying to explain energy transformations through work to a novice.
Let me explain my pedagogical approach before responding to some of your previous responses. Hopefully if you understand my pedagogical process you will understand why I take a different approach than yours.
A new student does not have much physics intuition. Intuition comes from experience. So my goal is to give a new student a small number of fixed rules that can simply be applied, without intuition, to consistently get the right answer. Intuition can then build as they gain experience, but they will have standard tools that they can reliably use whenever intuition fails them.
kuruman said:
the reaction to the action of the hand is exerted by the cart on the hand via the massless rope
This is, to me, problematic pedagogy because it requires intuition. Ropes are objects that have forces acting on them, not forces themselves. The hand does not touch the cart, it touches the rope. The reaction to the action of the hand is a force acting on the hand, this force comes from the rope, not the cart.
kuruman said:
The double subscript label as in ##F_{HR}## indicates the force exerted by the Hand on the Rope.
This type of subscripting is a very useful tool for teaching Newton's 3rd law. The 3rd law pair of ##F_{HR}## is ##F_{RH}##, not ##F_{CR}##. And ##F_{CH}## doesn't even exist. Whether the rope is massive or massless doesn't change any of that. It is important that it not change any of that so that the students learn just one simple set of rules and don't have to rely on intuition that they have not yet developed.
Contact forces are exerted between the things that are in contact. The hand is in contact with the rope, so there are contact forces between the hand and the rope. The rope is in contact with the cart, so there are contact forces between the rope and the cart. The hand is not in contact with the cart, so there is no contact force between the hand and the cart.
kuruman said:
To summarize, when a massless, inextensible, string is attached to a system on one end and to a force F on the other, one can always consider force F applied directly on the system in the direction of the tension. That's what is implied when we say the "tension is the same at any point along the string."
I would never teach this. What I would teach is that you can redraw the system boundaries. You can make the rope part of the cart system or part of the hand system. Since this adds no mass to the larger system, it will not change the acceleration of the larger system. But as long as you are speaking of the rope as its own system, then you need to identify the forces correctly. It is an analytical mistake to define the rope as its own system and have the forces act between the hand and the cart.
kuruman said:
The change in a massless rope's kinetic energy is zero because, well, its mass is zero therefore its kinetic energy is always zero no matter what work is done on it.
This is why I included the massless spring as another example. There, the KE is still always zero, but the PE is not. If a student has been properly instructed in analyzing scenarios systematically, then they will have no problem analyzing the massless spring case and will not get tripped up by the shortcuts used in the massless rope case. Then, they will discover that the fact that the positive and negative works are equal is not due to the masslessness of the string, but rather it is due to the string's inextensibility. The masslessness only ensures that the KE is always 0, not that the works always sum to 0.
kuruman said:
I am only expressing my doubts whether the "work done on the rope" is a useful construct when trying to explain energy transformations through work to a novice.
I assert that it is a useful construct because the novice needs simple conceptual tools that can arrive at the right answer. Not all ropes are massless or inextensible. Not all connections between hands and carts are ropes.
My preferred definition of work is that work is a transfer of energy by means other than heat. Mechanical work is a specific kind of work that can be calculated with ##dW/dt=\vec F \cdot \vec v## where ##W## is the work done on the system and ##\vec v## is the velocity of the material of the system at the point of application of the force ##\vec F##. It becomes instructive to examine the way that energy flows through a machine, which can be done with this type of simple rule. Positive work being done on the rope at the hand end and negative work being done on the rope at the cart end is just a specific case of energy transfer that can be extended to camshafts, hydraulics, and other machine parts. I choose the mental tools that I give students specifically so that they can be clearly applied to as many scenarios as possible. They can develop intuitive shortcuts as they gain experience.