Was the membrane at a stable equilibrium before you pushed it? How could you tell?

Well, first let's define the difference between stable and unstable equilibrium.

Stable: After a little perturbation, the equilibrium state is retaken.

Unstable: After a little perturbation, the state changes to one more stable.

An example of this is:

Imagine a pencil, when the pencil lays down in the table, it is in stable equilibrium because you can touch it and it will return to the same initial state.

Now suppose that the pencil is standing up (we can think this as a more energetic state). The pencil is in equilibrium, it does not move and there is no force acting on it.

Now you can touch it (a small perturbation) and the pencil will likely fall down, to the stable state of equilibrium.

That is the difference between stable and unstable (There is also an image below, where you can see it in a more physical way, where the vertical axis represents energy and the horizontal represents a given variable or set of variables).

Then, how to know if the membrane was at a stable equilibrium before you pushed it?

Did it return to the initial state? then yes, it was a stable equilibrium

Did It change to a less energetic state? Then no, it was in an unstable equilibrium.

the denominator has a very specific role.

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