I came to this thought experiment as I was pondering good teaching examples of stable and unstable systems. It occurred to me that stable systems are really quite abundant. For a shoot-from-the-hip example, the speed of a car is stable about a given speed given a constant rate of fuel injection, since any perturbation will disappear over time. To be perfectly illustrative, here is a ball on a hill. Note that the forces are balanced in both these cases, but stability is different.
Now, the above illustrates the concepts well enough, but I like to present something that's slightly more non-trivial and try to get people to exercise real physical understanding.
A submersible with a compressible cavity. A model sufficient for this will be that of a weight attached to a balloon. Saying it's an airtight bag instead of a balloon might be more straightforward, as it avoids the contribution to pressure from the balloon elasticity itself.
I want to make the qualitative argument that given the buoyancy and gravitational forces are balanced at some underwater point (obviously below the surface and above the ocean floor), this point is unstable.
The logic is that the balloon will increase in volume and decrease in density as it gets closer to the surface, where the pressure is lesser. Correspondingly, increasing depth and pressure contracts the balloon. That means that upward movement causes a net upward force and moving down causes a net downward force. Unstable.
- Does this imply that any submersible (below the surface and above the floor) must actively maintain its depth level? Is the condition associated with this claim that the submersible be more compressible than the liquid? What about a compressible atmosphere?
- Are there passive control systems that could maintain a depth setpoint?
- Would anyone like to address the problem and questions with equations, derivatives, and all that good stuff?