The balancer could be human or robot.
The beam is long, sturdy and unshakeable, with infinitessimal width and height. (The balancer gets magic shoes to avoid getting sliced and have sufficient friction with the beam)
"Maintain balance" means keeping one's center of mass higher than the beam, even in spite of small external perturbations.
Things that aren't in the spirit of "balancing" (not exhaustive):
Expelling matter irrecoverably from the balancing system doesn't count. (So throwing objects is disallowed unless the balancer also catches them)
Stuff that wouldn't work in a vacuum doesn't count. (So helium balloons or leaf-blowers wouldn't help.)
Other system-boundary violating things that I haven't thought of
Resolves to whatever I'm convinced of in 1 month (before August 19). There will also be at least M100 managrammed to the most-hearted comment.
I won't bet in this market.
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Resolving YES, and paying out M100 to @KongoLandwalker for his comment with 3 likes, a video of a cube balancing on its edges and corner. I was pretty convinced by that. It's theoretically possible that a human could accomplish this by spinning their arms in lieu of reaction wheels, but I'm unsure if it's possible for a human in practice.
@Joshua lol, you can do what you want inside the balancer, but I don't think a beam of infinitesimal mass will give you much to push on.
Hold my Kapitza's pendulum:
https://www.youtube.com/watch?v=GgYABmG_bto
https://www.youtube.com/watch?v=5uZdwxbLdJ8
https://www.youtube.com/watch?v=5oGYCxkgnHQ
(I'm not betting on this market, because I don't trust you to resolve it correctly -- sorry for that.)
@YaroslavSobolev I hope I'll earn your trust someday!
The kapitzah is very cool, but it works by moving the pivot point rapidly up and down. I don't see how that example translates to a balancer on a beam that is fixed in space.
(To be clear about my current leaning, I'm strongly leaning toward yes because of reaction wheels. I'm still curious if there's a solution without wheels)
Bounty of M500 to the first person to post a convincing physics simulation (2d or 3d) that shows a system balancing on a tiny fixed point (using human inputs or otherwise). Bounty of M100 to the first sim (from another person) showing one of the systems discussed here to be generally incapable of balancing. Consider using a low value for gravity. Algodoo is a free cross platform 2d physics playground that is one easy option (if on Linux, requires Wine).
@kenakofer I'm doing a terrible job here, but it's pretty clear in this 2d case that it's easy to move one's center of gravity from one side to the other, and that the reaction wheel won't, like, require infinite RPM.
@ShakedKoplewitz If you have to correct too much in one direction, the wheel's tensile strength will be exceeded eventually.
@osmarks You can correct to sing a bit over in the other direction so that you can slow down, you wouldn't need to go infinitely fast.
The beam can be used as the axel of an inverted pendulum. Designing a robot that balances this is a standard project, and the maths behind it already assumes a single point of contact between the balancing (human) pendulum and the axel. The center of mass is above the axel. https://www.mdpi.com/2076-3417/9/24/5279
@Ramble those control an instability around an axis orthogonal to the movement of the wheels (e.g. moving the wheels front and back to keep from tipping over front and back). But when balancing on a beam, the instability is around the beam's axis, which is also the direction you can move your contact point.
In other words, the robot thing lets you keep from falling forward and hitting your face on the beam, but it doesn't let you keep from falling sideways off the beam.
@meefburger I think you have imagined the wheels on the other way around? The wheels move orthogonal to the beam, since the beam is the axel. Balancing along the beam's axis is not harder than if the beam were wide.
@Ramble Perhaps I need a diagram to understand what you mean. I'm envisioning a segway with both of its wheels on the beam, human rider facing orthogonal to beam (and lets call the union of segway and rider the "balancer"). Start in equilibrium, then let's say the balancer is tilted backward by a gentle breeze. Now the center of mass is behind the rail. So the wheels must move backward to get under the rider. It's not clear to me that the wheels moving backward would somehow make the center of mass of the whole balancing system move forward. And moving the wheels forward would topple segway and rider off front and back respectively.
@kenakofer Ok, we have basically the same setup. The wheels indeed move backwards in this case - they don't significantly move the center of mass, but they move the pivot point to behind the center of mass (until the balancers stops rotating backwards and are in danger of falling forwards) then roll slightly forwards to put the pivot point exactly underneath the center of mass as the system comes to a halt
https://youtube.com/watch?v=mH8a_UaOFHY&feature=share9
@KongoLandwalker Yep, I think either or both of gyroscopes and reaction wheels could absolutely allow for balance on an immovable and thin wall.
Would @WinstonOswaldDrummond or others like to make the case to not be convinced by this self-balancing cube that Kongo linked? Looks to me like it can balance on a single point just fine, and can even let its reaction wheels slow down somewhat while maintaining balance. Just chop off one of the cube corners a bit so it can perch atop our infinitely narrow beam.
I believe this works, and I think it's equivalent to what @ShakedKoplewitz was getting at.
Here's an argument for why the problem is confusing and why a wheel solves the problem:
What makes it seem like balancing on a perfectly rigid infinitely narrow beam hard is that, once you're tipping one way or the other, you have some unwanted angular momentum (because you prefer to be stationary at a particular position, but you're tipping away from that position). But the only thing you can push on directly is the beam, and that will give you torque=0 since r=0. The other source of torque is gravity, but you can't change the direction of that without getting your CM on the other side of the beam, and you can't do that without torque.
The wheel solves the problem because you don't care about your angular momentum, you just care about your angular position. And motion of your CM isn't the only way you can have angular momentum. So if gravity starts adding or subtracting angular momentum by accelerating your CM, you can dump it into the wheel or take it from the wheel.
At a more granular level, the way to see how the torque is applied to you when you spin (or slow) a wheel, suppose you hold the axle of the wheel with one hand so its rotational axis is parallel to the beam at distance d from the beam, and apply a force to the rim at the top of the wheel a distance (d+r) from the beam. You apply equal and opposite forces at the axle and the rim (because you're not accelerating the wheel's CM), but they're applied at different distances from the beam, so the net torque on you is non-zero. [Or maybe it's easier to think about holding the axle and pushing down on one side of the wheel, so that one force applies zero torque and the other applies non-zero torque]
I think you can get the same thing by holding a very heavy bar, so long as you have room to rotate it far enough around.
@meefburger Yeah, an interesting side question would be: is this feasible for a human in Earth's gravity? I wonder how heavy a bar would have to be and fast a human would have to angularly accelerate it.
.
Imagine a 2-D balancing robot whose body is massless and who has only one point of contact with the point beam, but who holds two weights, one in either hand. The robot only applies force to either weight in a direction radial from the beam (because the beam has no radius, it would be impossible to apply torque to one weight without an equal and opposite torque to the other).
Both weights are above the beam, one 45 degrees off vertical to the left and one 45 degrees off vertical to the right. When the robot starts to lean left, we move the right weight outward and the left weight inward. This increases the gravitational clockwise torque, because we now have a longer lever arm on the right. We therefore have control over the torque, and as long as the perturbations are small enough that our maximum robot arm reach can counteract them, and we never allow both weights to end up on the same side of the y axis, we balance.
Edit: let me head off a couple potential objections.
How do we know we can apply radial force to the weights? Imagine that the robot consists of two tracks attached to a rigid frame, with each track being oriented radially, one on either side. The weights sit on the tracks and slide along them on wheels that clamp them to the tracks, and powerful motors are attached to the wheels that let the robot have precise control of where on the track the weights are at any given time.
How do we know that if we apply torque to the left, we wont go too far and fall to the right? I might invoke control theory, but to be very simple, if the angle of the frame is on the left but we are moving to the right, we can control the torque to be near zero, and see if we tilt past center. Only if we are on the right and moving right or on the left and moving left do we apply a big torque.
I can draw a picture or fill in more details if that helps.
@BoltonBailey Interesting side comment, this would be impossible with one weight and a frictionless beam, for the reason that you cannot torque the robot and therefore cannot correct the bot as it tips over. Edit: Though as @ShakedKoplewitz points out, you could also rotate the weight.