This is a situation that I've been sharing and discussing with colleagues over the past few months:
Imagine you and friend are holding a hula hoop. Your friend grasps one part of the hula hoop in his hand (not so tight that it won't move through his hand), and you start spinning the hula hoop around until it seems to reach a constant rate of rotation. At this point, I just want to consider what's going on while the hula hoop is and continues to rotate at a seeming constant rate.
Now, based on my description, the hula hoop system can be described as having a constant influx of energy (from you pushing). That rate of energy in to the hula hoops is equal to the rate of energy being lost into your friend's hands. The equality of inflow and outflow rates seems consistent with the idea that the hulahoop is moving with a constant speed, and thus has a constant kinetic energy.
Energy would seem to flow into system at your hand, and flows out at your friends hand. But your hand and your friend's hand are spatially separated. This leads us to question one: How would you explain how energy gets from one side of the hula hoop to the other?
Once again, energy is being lost at your friend's hand. But the speed of the hoop seems to be the same everywhere. More specifically, the speed of hula hoop pieces would seem to be the same on both sides of the hand. This leads us to question two: How would you explain how energy is lost at your friend's hand, while, at the same time, the kinetic energy remains the same throughout the process of moving through the hand?
The hula hoop is not a rigid object. Every time you pass the hula past you, you compress a piece of the hula hoop. With your friends hand pushing back, one side of the hula hoop is actually in compression. (We'll ignore for the moment whether or not the other side is in tension or not)
The compressed pieces of the hula hoop act as a energy storage mechanism. Your hand does work on pieces of hula hoop and that work goes into increasing the potential energy stored in the hula hoop. Alternatively, as pieces of hula hoop move across your friends hand, this potential energy is released as those pieces decompress. Thus, the energy lost at the hand is not the kinetic energy of hula hoop; rather it is the potential energy that was stored in the compressed parts of the hula hoop.
The compressed pieces of the hula hoop are necessarily more dense than the pieces that are uncompressed (i.e., the compression forces the atoms closer together). Since the mass of the hula hoop must be conserved at each point in the circle, this requires that the less dense pieces move faster than the dense pieces (which move slower). This leads us to this questions: If your friend's hand is pushing back on the hula hoop pieces that move through it, how would you explain how the hula hoop pieces end up moving faster on the other side?
The piece of hula hoop right in your friend's hand is actually sandwiched between two different regions with distinct mass densities. Behind your friend's hand, the hula hoop is squished up like a spring. This "spring" creates a force which accelerates the hula hoop piece through your friend's hand, leaving it with a faster speed than before. This faster speed is consistent with the fact that the atoms are more spaced out. The faster speed allows it to get further away from the pieces behind it, which are still moving at the slower speed.
Intuitively, your friend's hand would seem to the agent slowing things down. On the other hand, as defined by the original problem, the hula hoop seemed to be moving at a constant speed through out the whole process. Through the reasoning we've walked through, we're concluded that pieces of the hula hoop actually speed up through this region.
Loose ends and questions:
#1: It only really makes sense to describe the hula hoop as having a single rotation rate if it is a rigid body. Given that we've concluded it can't be a rigid body, is there a single quantity which describes the flow rate. Is it momentum? Is it kinetic energy? Is it mass current? Does this necessitate a change to the chain of reasoning anywhere?
#2: Is the other side of the hula hoop in tension? Is there any reason to think the hula hoop arc length is longer than, shorter than, or the same as it's resting arc length?
#3: How quickly does energy propagate from your hand to your friend's hand? How does this compare to the rate at which hula hoop pieces make the same journey?
#4: What's going on during the initiation stage before and as the hulahoop reaches steady state? Is this consistent with our stead state solution?
#5: What does this have to do with an electric circuit with a bulb, battery, and wire?
#6: Could you explore the validity of my story experimentally? How would you do it? Could you explore the validity of my story with a simulation? How would you do it? With either, what assumptions or approximations would you need to make?
#7: What parts of my story seem wrong? What assumptions have I made? Are they reasonable assumptions? What aspects of the situation am I ignoring? Is it reasonable? Overall, is this a viable model? How could you tweek it or refine it?
#8: Typically, we use energy to tell stories about initial and final states. Have we gained anything by trying to tell a spatially and temporally continuous energy story? Why is it so hard to tell such stories?