This question arose from my sci-fi discussions with my friend. It concerns artificial gravity through centrifugal Force. Partly inspired by the Artificial Gravity created on the Hermes ship from the Martian.
Imagine there's a hollow torus (donut), in space, without any external forces acting upon it, which is rotating (around the axis through the middle).
Will an object which you place inside the 'pipe' of the torus experience the centrifugal force due to rotation and be attracted to the edge of the torus? Or would it only experience the force if it was originally touching one of the walls? And would the results be different if the torus was filled with a gas (air)? And if the inside of the torus was divided into sections (like the ship)?
Best Answer
The answer here is yes. In the reference frame of the spinning torus, the object in the tube receives the expected amount of centrifugal force towards the outer rim ... but, it also receives a coriolis force of even greater magnitude towards the axis of rotation.
Let's say that the frame of reference (and the torus) is rotating with angular velocity $\omega$, and the object of mass $m$ is "stationary" (in the stationary frame) at a distance $r$ from the axis of rotation.
Imagine viewing your object from the rotating frame. This object is not "stationary" -- that's actually whizzing along the tube magically in an apparent perfect circle. From the perspective of the tube frame, this object is actually constantly accelerating inwards with $a = v^2 / r$. (If anything is experencing circular motion, its acceleration is $a = v^2 / r$). $v = \omega r$, so the object is whizzing around the tube, constantly accelerating inwards with acceleration $a = \omega^2 r$, somehow. So, to the rotating frame, the object is experiencing some magical force of magnitude:
$$F_{net} = m \omega^2 r$$
Towards the axis of rotation. Where is that coming from?
Well, like you mentioned, the object is expected to experience a centrifugal force away from the axis of rotation. The centrifugal force $F_c$ on a body with a given mass in a given rotating frame is:
$$|F_c| = m \omega^2 r$$
Going away from the axis. But! Our object is moving in this rotating frame, and all moving objects in a rotating frame also experience a coriolis force $F_C$:
$$F_C = - 2 m (\omega \times v)$$
For our object, $\omega \times v$ points radially outwards, so $|F_C|$ points radially inwards (because of the negative sign), and remembering that $v = \omega r$, we have:
$$|F_C| = 2 m \omega^2 r$$
Going inwards. Adding it all together (and considering inwards forces to be positive), we get:
$$F_{net} = F_C + F_c$$
$$F_{net} = 2 m \omega^2 r - m \omega^2 r$$
$$F_{net} = m \omega^2 r$$
Which is exactly the force that we should expect to see to explain the apparent motion of our object!
So, in summary:
YES, the object does experience a centrifugal force away from the center, even though it never touches any walls. BUT, because it is moving with respect to the frame of reference, it also experiences a Coriolis force, towards the center, that's actually even bigger. The two forces act together to create a net force towards the center, which causes circular motion in the reference frame. To observers in the rotating frame, it would appear as if the object was orbiting around the axis of rotation, as if gravitationally attracted to it.
EDIT Just to clarify some issues that have come up in the comments.
The centrifugal force arises mathematically from the coordinate transformation of moving from a stationary frame to a rotating frame. It doesn't come from any physical interaction. The walls and friction do not provide centrifugal force. Even if it was just a single object in an isolated system, as soon as you move to the rotating frame, it is influenced by a centrifugal force (as long as it is not sitting exactly on the axis of rotation). Centrifugal force is the mathematical result of the coordinate transformation, not a physical force created from physical interactions.
What some might be mixing up is the perception of being pulled outwards. But remember, being influenced by a force is very different than perceiving the force as a human being. Astronauts in orbit are moving under the influence of gravity, even though they feel weightless. This is because they're in "free-fall" -- they are being pulled by gravity, but nothing is preventing them from falling their free-fall path. As soon as something (like a couch or a chair) resists your freefall path, you perceive being pulled down by gravity.
So there's a bit of an irony here -- the only reason you know that gravity is pulling you down is because you feel a force pushing you upwards, provided by something that isn't gravity.
The object in the torus (stationary in the stationary frame, flying around in the rotating frame) is in "free-fall" in the rotating frame. It's moving under the influence of the centrifugal force (like the astronaut moving under the influence of gravity), but it feels weightless because nothing is resisting its freefall path.
So yes, the object is under the influence of the centrifugal force, and experiences it (just like an astronaut is under the influence of gravity, and experiences gravity's pull), but if it was a human, it would not "perceive" being pulled outwards (just like an astronaut does not "perceive" being pulled by gravity). Not until there is something to impede its freefall motion.
To gain an intuition, let's imagine a scenario: The torus (and our reference frame) is spinning. In the rotating frame, the object is whizzing around the torus under the influence of some forces that add up to have it attracted to the center of the torus. Now, imagine that we add a partition suddenly in the tube. The object is whizzing around, and eventually, it hits the partition! The partition is made out of cotton so the object doesn't bounce off. What happens?
The partition is stationary in the rotating frame, so now the object is stationary in the rotating frame too.
Because the object is now stationary in the rotating frame (it's pinned against the partition), it no longer is influenced by the Coriolis force! And now, the only force on it is the centrifugal force. Now, the object will start being pulled towards the edge of the torus, because the centrifugal force is the only force, so the net force pulls it outwards. It will look like the object is "sliding down" the partition to the outer edge.
Now, because the object is sliding towards the edge, it once again experiences a Coriolis force! (remember, all moving objects are influenced by the Coriolis force). This Coriolis force is actually directed towards the wall/partition, so the object will actually perceive the Coriolis force pinning it to the wall (because the wall pushes back).
The object continues to slide/roll and eventually reaches the outer edge. Now, the object is no longer moving, so there's no Coriolis force. The only force again is the centrifugal force, and now that force is being resisted by the outer edge. So the object will perceive a centrifugal force pulling it out because of the outer edge of the tube pushing back.
Now, because there is no Coriolis force, there is no force pushing the object either way (back or forth) up or down the tube. The partition could actually be removed, and the object will naturally stay in place because its only net force is directly radially outwards.
Note that friction never comes into play in any of these situations, too :)