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Let us say you are in a giant bucket on the end of a long rope attached to a big motor spinning horizontally, swinging the bucket around fast enough that there is a force equivalent to 6 gravities pushing the contents of the bucket outwards. If I understand correctly, the bucket and rope would tilt outwards to an angle of about 75 degrees, balancing the outwards force with the downwards pull of gravity.

Could a pole balance if it was placed exactly perpendicular to the flat floor of the bucket?

schematic of situation described

This is actually regarding a discussion about artificial gravity installations on the surface of the Moon. I contend a person could not move normally in a centrifuge simulating 1 g because the 1/6 g of the Moon would pull you off balance if you tried to stand up, you'd have to constantly compensate. The question I think boils down to the above question.

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  • $\begingroup$ Does the pole balance on Earth without the centrifuge? What difference do you think the centrifuge makes? Are you assuming that the floor of the bucket is vertical (or horizontal)? (the bucket and rope would tilt outwards to an angle of about 75 degrees) A diagram would be helpful here. $\endgroup$ – sammy gerbil Feb 24 '18 at 12:40
  • $\begingroup$ @sammygerbil alright, you have your diagram. See, the bucket and the pole are subjected to two forces - centrifugal force, pushing them outward from the center, and gravity, pulling them down. The angle of the floor of the bucket depends on how great the outward force is. The question is if something can balance in that situation. $\endgroup$ – kim holder Feb 24 '18 at 18:30
  • $\begingroup$ Sorry I meant a Free Body Diagram which shows the forces and torques. Your argument applies also to the bucket, which is subject to the same forces. If the axis of the bucket remains in line with the rope (it can pivot about the handle), why should the rod (which can pivot about its base) behave any differently? $\endgroup$ – sammy gerbil Feb 24 '18 at 19:25
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    $\begingroup$ @sammygerbil oh, my diagram isn't good enough for you, is it? Anyhow, I'd say my uncertainty came from the difference that the bucket is fixed to the rope, and short of tearing itself apart, will orient itself in the way that corresponds to the sum of the vectors, while the pole isn't fixed at all, and so I thought it would respond to the greater force by moving in that direction as far as it can before being stopped by the bucket. $\endgroup$ – kim holder Feb 24 '18 at 19:35
  • $\begingroup$ Are you asking whether it's stable? Does it have a finite width? $\endgroup$ – Ben Crowell Feb 25 '18 at 2:20
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There are two relevant cases:

1. The rotation is in a horizontal plane

In this case, the pole would be able to balance (at least to the extent that that unstable equilibrium is possible on Earth to begin with, and adding on any vibrations or other imperfections of the motion), and artificial gravity would indeed work within those confines. The direction of this artificial gravity would be at an angle to both the surface and its normal, which might make for some awkward geometry when constructing the station, but there's nothing wrong with the principle ─ the (constant!) direction of the acceleration in the rotating frame would just be identified as the local version of "down" in that frame, and that's that.

(Or, at least, that's the case in the limit where the length $\ell$ of interest (the height of the pole and/or human) is much smaller than the radius $R$ of the centrifuge. If it's not, as pointed out by Duncan, then there will be noticeable variations in the outward acceleration (outward component of gravity) at different radii, which will mean that the local direction of 'down' (i.e. the direction of $\vec g$ in the rotating frame) will change from your feet to your head. That'll be nauseating and distracting, most likely, but you'll still be able to balance a rigid pole ─ the only change will be that the angle of equilibrium will depend on the pole's length, and you'll need to hold the base to stop it from slipping.)

2. The rotation is in a vertical plane,

I.e. the axis of rotation is horizontal. In this case, the pole would not be able to balance, because the acceleration due to gravity in the rotating frame would constantly oscillate, including phases where it points partly to the side (so the pole would topple).

Artificial gravity in such a situation might partly work, in the sense that you'd be held down to the ground, but anything that wasn't firmly held down would jitter around, and it would be extremely nauseating.

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  • $\begingroup$ It occurred to me that asking about exactly equivalent forces, downwards and outwards, was a mistake. I can see that in that specific situation, but the one on Moon would never be balanced, it is 1 g outwards and 1/6 g downwards. I guess I should change the question. $\endgroup$ – kim holder Feb 23 '18 at 19:24
  • $\begingroup$ "1g outwards and 1/6g downwards" just adds vectorially to a constant $\sqrt{37/36}$g pointing in the "down" direction as experienced in the rotating frame. The pole would be able to balance on your Moon scenario. $\endgroup$ – Emilio Pisanty Feb 23 '18 at 19:27
  • $\begingroup$ Rotation does not create a uniform acceleration through space, and neither a pole nor a human occupy a single point. See my answer for the (very important) consequences of this. $\endgroup$ – Duncan Harris Feb 23 '18 at 19:38
  • $\begingroup$ But how can the pole stay balanced if there are forces acting on it in two directions at the same time, with two different strengths? I can understand the bucket stabilizing at an orientation that balances the forces, because it is fixed to the rope. But I can't understand the pole balancing. I would have thought it would fall against the wall of the bucket. Surely it doesn't experience one net force that is the result of the two vectors, it continues to experience two. $\endgroup$ – kim holder Feb 23 '18 at 19:40
  • $\begingroup$ @kimholder How can you stay balanced when standing on a slope when there are normal and friction forces acting on you in two directions at the same time? The forces just add vectorially to make a single resultant and in the rotating frame you have no way of knowing how that came about. That resultant is then just cancelled out by a single contribution from the normal force exerted by the floor, which must obviously be at an angle. $\endgroup$ – Emilio Pisanty Feb 23 '18 at 19:43
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If the pole (or person) is very short in comparison to the radius of the centrifuge, then they will balance and walk as usual. Strange effects are noticeable in smaller centrifuges, though. A human standing in a 3.5 meter centrifuge on the moon would experience an acceleration on their upper body which would be about 0.6 g and at an angle compared to the 1g acceleration at their feet. So your contention is valid for relatively small centrifuges. My seat of the pants estimate for the case of a lunar centrifuge is that a human could stand with effort even in a very small centrifuge, but that the effect would be distracting while walking all the way up to around a 20 meter radius.

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Gravity is equivalent to an acceleration in the opposite direction. There is no way of telling the difference. And two accelerations can be added as vectors.

So the resultant of the vertical downward acceleration due to gravity $g \hat{j}$ and the horizontal outward centrifugal acceleration $a\hat{i}$ of the rotating bucket is equivalent to a new acceleration due to gravity $$\vec{g'}=g\hat{j}+a\hat{i}$$ acting in a direction which makes an angle $\theta$ with the vertical, where $\tan\theta=a/g$.

The bucket can pivot about its handle and will align its axis with the direction of $\vec{g'}$, just as it aligns with $\vec{g}$ when hanging but not rotating.

Neither the bucket nor the pole can tell the difference between centrifugal and gravitational accelerations. Both bucket and pole will align themselves with the total acceleration vector $\vec{g'}$. If the pole can balance inside the bucket when the bucket is hanging vertically on the rope, then it can also balance inside the bucket when the bucket (and the pole itself) are rotating at the end of the rope.

enter image description here

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