Conservation of Momentum Paradox Thought-Experiment (Please Explain)
The Device: Imagine a physical system involving two circular rings of distinct inner and outer radii, but equal in inertia. The smaller ring is superimposed inside the inner radius of the larger ring. These two rings have negligible thickness and are allowed to freely rotate about an axis going through their center of mass, lying on the horizontal x-y plane. The axis of rotation is orthogonal to the face of the rings.
Now, imagine a long rod along the z-axis attached to both rings, connecting them together by acting as the "spine" of this ring-system. This rod is attached symmetrically through their center of masses along the z-axis. When the rod moves along the z-axis, it moves both rings translationally along with it. The rings are still allowed to freely rotate as mentioned previously, but their faces can no longer rotate because of the rod.
On the other end of the rod, some distance away, is attached a large square plate whose length or width (being the same) cannot be less than the outer radius of the largest ring on the other end of the rod; it is at least slightly longer. Both faces of the two-ring system and the square plate are orthogonal.
The Phenomenon: Imagine that potential energy is introduced into the system. After the full amount of potential energy is stored into the system, we will call this system a close-system for energy whose total energy will be equal to the stored potential energy.
The energy is stored by, e.g., two loaded mechanical box springs. One is placed sitting tangent outside of the smaller inner ring to the right and the other is placed tangent outside of the larger ring to the left. Suppose that each ring has at least one groove or a tiny paddle that the springs are sitting on. 782
There are two balls sitting on the other end of each spring of same mass.
Now the system is given energy by compressing the springs together, pushing the ball down closer the groove of each ring. Then, in an instant, the springs are released turning all the potential energy into kinetic energy. (The springs had negligible mass and no longer become relevant once released).
All net forces and torques in this system during the release were zero, e.g., the force on the ball was the same yet opposite in direction to the tangential force on the ring. The torques on both rings were also the same but opposing in direction. Their new angular momentum is the same in magnitude but are anti-parallel because they have the same inertias, consistant with <PERSON>'s Laws. Energy and Angular Momentum are both conserved in this thought-experiment
Has Linear Momentum Been Conserved?
The whole two-ring, rod and plate system is NOT moving translationally because no force had acted through it's center of mass.
Up to this point, in terms of linear momentum, only the balls carry linear momentum. The whole system, if you include the balls is net positive for linear momentum, which is perplexing considering that it started with zero linear momentum before the energy release. This was true for angular momentum--it started with zero and ended with net zero, but not linear momentum. Now let us continue this thought-experiment because the balls are moving in space right now.
To show what this means fully, the balls are headed for a collision course to the square plate at the end of the rod. Let's suppose that they make an "inelastic collision" with the plate, such that linear momentum is conserved. The balls are now stuck to the plate and all their momentum has been transferred to the entire mass of the contraption. The entire system is now slowly moving forward with net linear momentum!
Please tell me what went wrong in this experiment 782