Easy Questions?

[Swatopluk]

Mero, the answer is correct but not everybody would understand the "single stable state". Please explain for dummies why it exists with 3 legs but not with more.

Additional: Could there be a rigid body that is unstable independent of spatial position (in layperson's term: would fall over however you put it on a horizontal surface)?

[Sibling Zono (anon1mat0)]

Apart from a sphere? (otherwise it sounds like a perpetum mobile)

[Swatopluk]

That is indeed the most practical answer. But why exactly?

[The Meromorph]

Mero, the answer is correct but not everybody would understand the "single stable state". Please explain for dummies why it exists with 3 legs but not with more.

For stability, the 'center of gravity' must be inside the shape defined by a string runninning around theplaces where the legs touch the ground. outside of the legs (at the bottom).
With one or two lplaces touching, there is no shape to be inside. With three legs, the stool falls until the third leg touches the floor. With more than three legs the stool still falls until three legs are touching the florr, but it's possible for other legs to not be touching the floor. If the center of gravity move outside the current three-leg shape, the stool will fall again until another leg hits the floor, therfor providing a shape again.

If you say a person might not understand center of gravity, then they'll have to learn it, because it's required for any explanation of this.

[Sibling Chatty]

A "normal" person may believe a stable state is one where the auotmobile is banned.

::looks around for a 'normal person'::

All Clear!!

[Bluenose]

Three legs:  The reason a three legged stool is stable is because three points always form a plane.  So long as the lengths of the legs are approximately the same so that the centre of gravity falls within the triangle formed by the leg ends when placed on the floor (if one leg was very long for example, the stool would fall over, but I am talking about the usual case here) amd obviously also if the floor is sufficiently level that the same occurs (it wouldn't work on a 45^o slope for example) then there will allways be "matching" of the three points at the bottom of the chair leg and the three points on the floor immediately below the leg ends.  Since three points always form a plain, the two sets of three points will always coincide.

If there are more than three legs, if the legs are not all exactly the same size then the four points will form two planes and unless you are very lucky and the floor just happens to have the same two planes formed by the four points below the leg ends, which is unlikely, then the chair (or table) will rock.  Of course the same is true if the chair is even and the floor is not.  In either case, especially with tables, the usual approach to this is to adjust the length of one of the legs either with a leg end designed for this purpose or with a wad of paper (usually serviettes in the restaraunt situation).

[Sibling Zono (anon1mat0)]

Now the question is: if for a 3 dimensional environment 3 legs generate a plane (and stability) how many legs for a 4 dimension world? (or better for a n-dimensional world)? 

[Griffin NoName]

4 dimensions? don't know about legs, but sounds like a lot of serviettes.

[anthrobabe]

Because 'wobbling' (better verb - 'jiggling' more properly refers to Boobs) is oscillation between two or more stable states. Three-legged stools or table one have one stable state (all three legs on the ground).
"Jiggling' may invole 'osculation' rather than 'oscillation'.

mostly I just read this--- my siblings are way to smart for me- but I get it on an elementary level (usually)
but the word Boobs just jumps out at one
OM!

[Bluenose]

Now the question is: if for a 3 dimensional environment 3 legs generate a plane (and stability) how many legs for a 4 dimension world? (or better for a n-dimensional world)? 

Well, actually a plane is always a two dimensional construct.  Thus no matter how many dimensions for a given frame of reference, three points will still form a two dimensional object lying on a plane that intersects at least two of the dimensions of the reference (it may be more, depending on where the points are located.

[Sibling Zono (anon1mat0)]

Strictly speaking you are correct. My thoughts were regarding the fact that in a 3D space any 3 different points define a unique 2D plane within such space. If we go up to a 4D space there is an infinite amount of 2D planes with 3 points (a 4D table would tumble) therefore you need 4 points to express a unique 3D *plane*.

[Swatopluk]

The explanation I'd give for the hypothetical object that is unstable in any position would be:
In a stable (or at least metastable) state the center of gravity is in a position that any shift of the object would move the center of gravity upward (i.e. requiring a nonnegligible force). Because in a rigid body the center of gravity is fixed as related to the body there must be a spatial orientation where the center of gravity lies lowest => stable state. The lone exception would be the sphere* because here the center of gravity does not shift because all points on the surface have the same distance to it.

*or an object in the same symmetry group like hollow spheres, concentric hollow spheres etc.

-----------------------------------------

Next questions:
1. Why are modern submarines much faster in the submerged state than on the surface?
2. Could a hollow metal sphere be made that would float in the air when evacuated?**

** a macroscopic one, not one of a size lower than 1 mm

[Bluenose]

I'll take on the submarine one.

When on the surface of the water, a moving body will create surface waves, expanding from the object exactly like ripples on a pond from when you throw a stone into it.  Because of the size of a ship (or boat in the case of a submarine) the amplitude of the waves is much greater than those from the stone.  However, the velocity of any powered vessel is usually (almost always) greater than that of the waves themselves, and so the vessel is always moving faster then the lead waves and crashing into undisturbed water.  This is entirely analagous to the shock wave of a supersonic aircraft, and like a supersonic aircraft it takes a considerable amount of energy to do it.  This is the reason why modern powered vessels have sharp bows, just like supersonic aircraft to minimise the drag caused by shockwave and to keep the hull of the ship, within the cone of the shockwave (called a wake in this case) so as to minimise the energy requirement.

On top of that there is the drag caused by the shape of the hull of the vessel and this is equivalent to what in aerodynamic terms is called form drag.  Once submerged the submarine only has this type of drag and the drag induced by the surface wake (shockwave) is no longer present and only the form drag exerts itself.  The most efficient aerodynamic (or hydrodynamic) shape is the so-calles teardrop shape which modern submarines make a reasonable approximation of.  Similar effects to supersonic flight will only occur once submerged as the vessel approaches the speed of sound, but since the speed of sound of water is higher than in air, (ie > 700 knots or so) it is is not an issue for any submarines I am aware of, including torpedoes and so on.  BTW, modern surface ships, particulalry very large bulk carriers where efficiency equates to dollars, often have large bulbs below the water line at the bows to gain benefit from this more effient subsurface shape, but they still have to cut through the wake.

Hope this is a bit clrearer than mud.

As for the metal globe, in prinicple I believe that this could be done, so long as there is an alloy of sufficient strength to be able to withstand atmospheric pressure.  This metal would need to be light enough that the volume of air displaced has a lower mass than the shell of the metal of a thickness sufficiently strong to withstand the pressure.  I do not think that there is such a metal, but I could well be wrong.

[Swatopluk]

I am not completely sure about the sphere either but I think/guess above a certain size the strength of the wall can be kept constant, and then it is just a matter of increased diameter. For "practical" reasons a half-sphere/ellipsoid with flat bottom may be preferable to keep the structure from collapsing under its own weight with a minimum of internal stabilizing structures.

In a related area, would a gas-filled or evacuated tube of high diameter submerged in a high density incompressible fluid (i.e. lower pressure on the top, significantly higher on the bottom) still have the circle as ideal profile or would it go towards an ovaliform shape? In the latter case, would the "sharp" end be top or bottom?
That's one I simply do not know the answer for.

Now something simply crazy. How fast would a rigid spherical body of earth mass and diameter have to rotate to cancel gravity on the surface (i.e. centrifugal force = gravitational force towards core)?

[Sibling Zono (anon1mat0)]

For a small sphere the material could theoretically be a fullerene (like carbon nanotubes and buckyballs) although I don't know if it is possible to make it as a macroscopic structure.

About the earth like planet, wouldn't the core be spinning even faster to make the surface spin that fast? It would create a huge amount of heat because in the equator it might cancel the forces but the closer you go to the poles, you'll have less centrifugal force. My guess is that it would tear the planet apart in no time.