Can you balance any object?

[ChrisBooth12]

What i mean is say i was strong enough, could i balance the empire state building on my finger, without it tipping over? Assuming it would not collapse under its own weight by lack of foundation or whatnot. Could i do this with any object? what about the bat bridge?

[Squink]

Stable Balance Points:

Domokos and Várkonyi managed to prove mathematically that for any flat shape, there are at least two stable balance points and at least two unstable balance points.

Next, the pair began to investigate whether all three-dimensional shapes have at least two stable and two unstable balance points. They tried to generalize their two-dimensional proof to higher dimensions, but it didn’t hold up. Therefore, it seemed possible that a self-righting three-dimensional object could exist. Such a shape would have only one stable and one unstable balance point.
…
Eventually, the team managed to construct an object mathematically that has just one stable and one unstable balance point.
…
Once the pair had built their self-righting object, they noticed that it looked very much like a turtle.

So every two dimensional object has two or more balance points, and every three dimensionally object has at least one.

[Valgard]

Generally speaking any rigid object should be balanceable, given that it will hold its own weight, can be supported and that we ignore other forces (such as wind). By this I mean that since the object has a center of mass, you can draw a straight line between that COM and the earth’s COM and then rotate the object so that a support point is directly on that line, that’s your balance point.

Using your example of the Empire State Building, sure it could be balanced - conceptually it’s no different than balancing a stick on end.

I can’t think of any shapes that are “unbalanceable” offhand but someone who’s good at topology may be able to point them out.

[OldGuy]

@Squink

Stable Balance Points: So every two dimensional object has two or more balance points, and every three dimensionally object has at least one.

It needn’t follow of course that the OP could balance it on his finger. We might well be able to construct an object whose balance point(s) was in a recessed (concave) portion too narrow for a finger to be inserted into. It would be tricky to determine if you could then prove that you could suport enough points around the balance point to work, I’d think.

[j_sum1]

All interesting stuff. And it shows just how much I assume intuitively without thinking about it.

Just read the link.
Let me see if I can wrap my head round what is being said…
A balance point (both stable and unstable) is a point on the surface such that a line connecting that point to the centre of mass is perpendicular to the surface at that point.
An unstable point is one where a small movement willcause the object to topple to a new position.
A stable point is one where a small movement will result in the shape righting itself back to the original position.
Yep. It seems pretty logical that a simple closed object should have at least a stable balance point. And it fits with experience.
If an object had no stable balance points then it would roll continually if you put it down. I smell a violation of a physical law.
I would have thought that a sphere or a cylinder would have no unstable points, but I am guessing that these were probably covered in the study and classified as neutral. I’d like to find out more.

[Colophon]

But surely you could come up with a shape where the “balance point” was actually in space. E.g. take a flat ring with a large hole in the middle, the balance point is in the middle of the hole. OK, so if it was symmetrical you could balance it edge-on, but I suspect you could tweak the shape so that that wouldn’t work either, couldn’t you?

[Mangetout]

I don’t think so - however you tweak it, you’re only moving the centre of mass and/or the parts by which it can be supported, and it would always be possible to shift the object slightly so that the centre of mass was above something one of the parts by which it could be supported.

Even with an asymmetrical ring, it could be balanced edge-on, as long as the centre of mass is above the point where it contacts the supporting object

[Infinite_Levels]

Maybe slightly off topic. but you can make any four-legged table balance on an uneven surface, like a tiled patio, by rotating it. At the current wobble point, two legs touch and the othe two don’t, while rotating 90[sup]o[/sup] switches the pair of legs. At some point between 0[sup]o[/sup] and 90[sup]o[/sup] they are about to trade places, and there all four are touching the ground. Of course there’s no guarantee that the tabletop is level enough at that point.

[Chronos]

Eventually, the team managed to construct an object mathematically that has just one stable and one unstable balance point.

Begging their pardon, but it’s really easy to construct such a shape. Take a sphere and give it an axisymmetric density which is greater at one end than at the other. Heavy end down is a stable balance, and heavy end up is an unstable balance. Any other orientation, and it’ll roll towards the stable position.

Meanwhile, if we go with actual physics, it is not guaranteed that an object be balanceable in a particular orientation (a pencil on its point, for instance). The Heisenberg Uncertainty Principle is actually enough to guarantee that things won’t stay balanced at an unstable point for long.

[brad_d]

Eventually, the team managed to construct an object mathematically that has just one stable and one unstable balance point.

Reinventing the Weeble, so to speak?

[Squink]

@Chronos

give it an axisymmetric density which is greater at one end than at the other…

The linked article didn’t say, but Várkonyi and Domokos were interested in homogeneous bodies. Here’s the pdf Mono-monostatic Bodies: The Answer to Arnold’s Question (Mathematical Intelligencer 28)

We consider bodies resting on a horizontal surface in the presence of uniform gravity. Such bodies with just one stable equilibrium are called monostaticand they appear to be of special interest. It is easy to construct a monostatic body, such as popular children’s toy called “Comeback Kid” (Figure 1A). However, if we look for homogeneous, convex monostatic bodies, the task is much more difficult. In fact, in the 2D case one can prove [1] that among planar (slab-like) objects rolling along their circumference no
monostatic bodies exist.
…
Not surprisingly, the 3D case is more complex.

[Santo_Rugger]

@Squink

Stable Balance Points: So every two dimensional object has two or more balance points, and every three dimensionally object has at least one.

Seems to me that an egg would satisfy the requirements of two unstable and two stable points. Although there may be an infinite number of stable points, or only one.

[Chronos]

An egg shape of uniform density would have one point which is definitely unstable (the pointy end), one which may be stable or unstable depending on proportions (the round end), and a one-dimensional infinite set of points which are neutral along one axis and unstable or stable on the other axis, depending on the stability of the round end.

Which is of course irrelevant to the question of whether every shape has such points.

[Mangetout]

I assume we’ve been talking about rigid objects all along? I think it would be quite hard to balance a glassfull of water, without the glass. (At standard temperature and pressure)