Monday, 7 January 2013

Sense and instability

Figure 1. A worried little weeble.
Did you ever have set of weebles? I did. They were wonderful little toys that lived up to their advertising slogan of “weebles wobble, but they don’t fall down”. The whole premise was that you got a set of several small plastic egg shaped characters (as seen in Figure 1) that would always wobble back to an upright position, no matter what position you started them from. As you can imagine hours of fun could be had with these little things.

The toy's mechanism is very simple. The bottom of the weeble is much heavier than the rest of the body, this means its centre of gravity is very low. Due its egg shaped body the weeble will always wobble such that its centre of gravity is at its lowest point. The applet below shows how this idea works [1]. On the left shape click on the red star which is the centre of mass and move it around the shape. The effect can be seen in the shape on the right. The shape on the right can then be grabbed and started from different initial points. However, it will always evolve to a stable point.


Why am I talking about weebles? Well, they are a really nice way of demonstrating how mathematicians understand equilibrium points, stability and instability. Although theoretically if a body was placed at its equilibrium points nothing further would occur, we know that physically we cannot be that accurate. Thus, equilibrium points can then be separated (generally) into stable and unstable points [2]. Thankfully, the mathematical notion of stability accords with our everyday use of the word. A body is stable if given a small push (or perturbation) it returns back to its previous position.
Figure 2. A wobbling weeble.
For a concrete example of this consider Figure 3, which is a schematic diagram of a curved bowl containing some balls. A is a stable equilibrium point; if the ball is given a small push it will return back to A. If the ball is given a big enough push it can roll right over the dividing line into the C position and this is also a stable equilibrium point. Now consider placing a ball at B. Since the bowl is completely flat there B is also an equilibrium point. However, it is an unstable equilibrium point. This is because no matter how small a push the ball is given it will never head back to B, but rather roll to one of A or C. Importantly, stability is only defined at points A, B and C because these are the only equilibrium points.

Figure 3. Illustrating stable and unstable equilibrium points.
A similar idea is applicable to the states of the weeble. When the weeble is upright it is equivalent to the ball being at A or C; the weeble is at a stable equilibrium point. Now suppose you could balance the weeble perfectly on its head, such that the centre of gravity, G, is directly above the flat point of its head, as in Figure 4. This is equivalent to the point B in Figure 2. It is an equilibrium point and, so, theoretically, if placed there, with no perturbation, then the weeble would stay there forever. However, any perturbation, no matter how small, will cause it to flip over and turn the right way up.
Figure 4. Stable and unstable weebles.
In two weeks time I will continue this topic and see what happens when we remove the possibility of weighting the bottom and only depend on the geometry of the object.
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[1] All credit of the applet goes to this website: http://l.d.v.dujardin.pagesperso-orange.fr/ct/cusp.html

[2] Note that there are also saddle points and centres but we’re not going to talk about them as they’re special cases that few people care about. Also, the stable and unstable points could each be further split up into oscillatory and non-oscillatory points but again we’re going to ignore this complexity.


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