Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: [Phys-l] Sharing a problem for students



Ludwik wrote:

2) Let us go to a simple case of static stable equilibrium. I am
thinking about an equal-arms balance. The center of mass of the
suspended frame is below the point of suspension. The weight indicator
is vertical when two weights are equal. Put an extra weight, dW, on one
of the plates and the indicator shifts accordingly. Here many
equilibrium positions are possible, one for each dW. In the ideal case
this is an example of a "stable system." Suppose we begin with dW=0;
this is one equilibrium position. We disturb the system, by adding a
dW, and a new tilt of the indicator is established, perhaps after some
oscillations. Here the term "stable" is used to say that the original
orientation is recovered when the disturbance is removed.

Stability generally refers to a system's tendency to return to an equilibrium configuration if the *configuration* is changed slightly. In the case of the scale that means that you physically move scale away from its equilibrium configuration, let it go, and see what happens.

When you change the mass as indicated above, you are changing the system itself, not merely its configuration. Since the system is different, the equilibrium configuration is also different.

I was not aware that the term "dynamic equilibrium," implies that the
same is true for a planet orbiting the sun. That is why I removed it.

I don't understand this.

In the first sentence of your reply (see above) you are saying that my
electrical "system is highly unstable." The same is true for our
solar system. Is this what you have in mind? If so then I agree.

Yes and no. To first order, the solar system could hardly be more different from your system. It consists of a central body that is a thousand times more massive than all of the rest of the bodies in the system put together. That body is surrounded by negligibly massive objects in enormous circular orbits with widely varying radii (again to first order) so that they hardly interact with each other at all. As far as I know, the stability of the solar system may be an open question, but it is most certainly not "highly unstable."

On the other hand, the solar system certainly is unstable in the sense that, for instance, it includes comets which do occasionally interact significantly with other low mass objects in the system and that may be flung out of the system (or crash into the Sun (or Jupiter (or even Earth!))) as a result.

To avoid possible confusion I will start using the term "durable,"
instead of "stable" or "dynamically stable." But I suspect you have
something different in mind. Perhaps you think that, unlike a solar
system, my electric system will not be durable. If so then please
explain. Keep in mind that my model ignores gravitational forces and
emission of electromagnetic waves.

The reason your system is SO unstable has mostly to do with the fact that it is irreducibly a three body system. We know quite a lot about the detailed motions of arbitrary two interacting body systems, but the moment you want to be able to predict the detailed motions of a system of three or more bodies, then, with a few interesting but rare exceptions, one of them had better be *far* more massive than any of the others.

John