Potential Energy vs. Position Graph

[Bashyboy]

I attached a graph of a potential energy vs. position graph. My question is, the relative minimum is characterized as a equilibrium point, but what is, specifically, is this type of equilibrium point--a stable, unstable, or neutral one?

 

[Bashyboy]

I am reading this article, http://home.comcast.net/~sharov/PopEcol/lec9/equilib.html , regarding potential energy and equilibrium, and I am rather bewildered by the article's use of the word asymptote. The sentence containing it is, "An equilibrium is considered stable (for simplicity we will consider asymptotic stability only)..." I tried to look up the word in the dictionary, but it came up with only the mathematical definition. What do they mean by asymptotic stability?

 

[Nugatory]

I attached a graph of a potential energy vs. position graph. My question is, the relative minimum is characterized as a equilibrium point, but what is, specifically, is this type of equilibrium point--a stable, unstable, or neutral one?

Stable - if you were to set a ball there it would tend to stay there unless disturbed strongly enough to push it "out of the valley".

 

[Nugatory]

I am reading this article, http://home.comcast.net/~sharov/PopEcol/lec9/equilib.html , regarding potential energy and equilibrium, and I am rather bewildered by the article's use of the word asymptote. The sentence containing it is, "An equilibrium is considered stable (for simplicity we will consider asymptotic stability only)..." I tried to look up the word in the dictionary, but it came up with only the mathematical definition. What do they mean by asymptotic stability?

In this context, asymptotic stability means that if you drop a little ball into the "valley", it will roll back and forth around the bottom for a while before it comes to rest at the bottom.

 

[Bashyboy]

So, does a relative minimum always correspond to a stable equilibrium? At this point, the system possesses kinetic energy and potential energy, but that seems odd. Could you give me an example of a system that is in stable equilibrium that possesses kinetic energy and potential energy? Also, I understand that the slope of the potential energy vs. position graph is force, but why does it have to be the negative of the slope in order for it to be force?

 

[haruspex]

So, does a relative minimum always correspond to a stable equilibrium? At this point, the system possesses kinetic energy and potential energy, but that seems odd. Could you give me an example of a system that is in stable equilibrium that possesses kinetic energy and potential energy? Also, I understand that the slope of the potential energy vs. position graph is force, but why does it have to be the negative of the slope in order for it to be force?

In normal usage, stable equilibrium assumes there is little or no KE. If a body enters that position with significant KE, it might very well fail to stay there. But that does not alter the fact that it would have stayed there if the KE had been sufficiently small. If sufficiently small is still nonzero, it is a position of stable equilibrium.