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We know a body in uniform linear motion will continue to move with uniform velocity unless any external force acts on it. Similarly, a body rotating with uniform angular velocity will continue to rotate with uniform angular velocity unless any external torque acts on it.

Occupants inside a body in uniform linear motion will experience an "artificial gravity" if an external force accelerates it, but the artificial gravity will turn off the moment the external force disappears.

But in the case of a rotating body, if I understand it right, the "artifical gravity" is present (due to centrifugal force) even when no external torque is acting on it.

In that case, since the body maintains its angular velocity in the absence of any external torque, is it correct to say that a person sitting inside a spinning spaceship will experience "artifical gravity" forever, without any intervention or additional torque required?

Assume there is just that one person sitting there, there are no other people who are walking about or moving, etc. Also, the spaceship is in empty space with no gravitational influence from any other body.

It seems that we are getting the artificial gravity "for free" without having to keep supplying an external torque, unlike in the linear case, where the artificial gravity lasted only as long as there was external force.

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  • $\begingroup$ I think some symmetry and conservation can answer. If object spinning, object can animate and input energy to system which works against the artificial gravity, that has to change the total energy of the system, either linear velocity or angular. $\endgroup$
    – marshal craft
    Apr 21, 2021 at 20:13
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    $\begingroup$ Can you define "forever"? If that's literal then no, of course not… even if the only force stopping the spaceship spinning is the teeny-weeny friction from hitting the odd atom of inter-stellar hydrogen, it will slow down with every collision and eventually come to rest. What you're asking is about like whether an infinite number of monkeys in an infinite amount of time will randomly type the complete works of Shakespeare. Of course they will: that's part of what infinity means and here, "infinity" is truly comparable to "forever." $\endgroup$
    – Robbie Goodwin
    Apr 21, 2021 at 22:22
  • $\begingroup$ I am definining "forever" in the thought experiment sense. In reality , i agree that there will be teeeny weeny atoms of inter-stellar hydrogen. But in my experiment, since it is a thought experiment, there ARE NO inter-stellar hydrogen or any atoms at all. It is a complete vaccuum. There is NO GRAVITY from any planets or stars. It is completely empty space $\endgroup$
    – silverrahul
    Apr 22, 2021 at 5:14
  • $\begingroup$ Generating force doesn't require any energy to be expended (and in fact, it can even do the reverse). It's one of the things that people often find hard to truly internalize, because we tend to imagine "man pushing against wall" rather than, say, sitting still on a chair, or a spring, or being suspended against a rubber band. Moving in the artificial gravity is a different matter. $\endgroup$
    – Luaan
    Apr 22, 2021 at 10:36
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    $\begingroup$ When someone says " A body in uniform motion will stay in uniform motion unless external force acts upon it " it is understood that we are assuming there are no such atoms colliding against it, or hull breach or leaks etc. ? right ? So, why is everyone asking me all these questions of . What if this happens? What if that happens ? Isnt that supposed to be the point of a thought experiment? To think of an idealised scenario to understand the basic concepts involved without getting bogged down with the thousand real -life what ifs involved ? How should i word my question to make this clear? $\endgroup$
    – silverrahul
    Apr 22, 2021 at 15:26

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Yes, the "artificial gravity" can last for a very long time (the other answers address the caveats to "forever"), but we are not getting anything for free: the person at the edge of this rotating spaceship has a force continuously applied to them, but as long as they stay in place (with respect to the ship) this force does not do any work, therefore there is no energy gain or loss. This is due to the fact that the force is perpendicular to the displacement at any point.

This is equivalent (locally) to gravity on Earth --- gravity is continuously pulling on you if you are stationary with respect to the ground, but you are not gaining any potential energy.

On the other hand, in an accelerating spaceship you are being moved in the direction of the force, so the work is not zero.

As @mlk notes, this is all under the assumption that you are stationary in the rotating/accelerating/surface-of-the-Earth frame.

If instead, you move vertically upward in Earth's gravitational field you can gain potential energy, and analogously if you move radially inward in the rotating spaceship you can do work against the artificial gravity. Your speed with respect to a non-rotating frame will decrease, and the rotation rate of the spaceship will increase: angular momentum is exchanged, and globally conserved.

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    $\begingroup$ Maybe worth a side-note, though it is excluded by the question: You can do work with that force. If the person sitting in the space-ship is standing up, there is work done to move the mass against the artificial gravity. This work will increase the rate of rotation slightly, because angular momentum needs to be conserved. Fundamentally this is the same effect as the classic example of figure skaters pulling their arms in to spin faster. $\endgroup$
    – mlk
    Apr 21, 2021 at 18:53
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    $\begingroup$ Fair point, I have included a paragraph to address this. $\endgroup$
    – Jacopo Tissino
    Apr 21, 2021 at 19:24
  • $\begingroup$ Even without moving, if the person simply feels the gravity, work is done at their nerves' endings and this biochemical reaction will slow down the rotation. $\endgroup$
    – Jeffrey
    Apr 22, 2021 at 13:41
  • $\begingroup$ @Jeffrey: Where would the angular momentum go? It's a conserved quantity. $\endgroup$
    – MSalters
    Apr 22, 2021 at 13:45
  • $\begingroup$ @MSalters right. I jumped a step there in concluding "will slow down the rotation". But I'm pretty sure work needs to be done for the nerve to convey the information to that persons' brain. So... All this processed information "I felt gravity for the past hour" turns into some very small amount of momentum transferred from the station to some molecules in the persons' brain? $\endgroup$
    – Jeffrey
    Apr 22, 2021 at 13:51
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It is all about conservation of angular momentum . A spinning body, a planet, in space would continue spinning for ever if it were the only body in the universe. We know that gravitational forces induce tides, in systems of planets and stars, and even though the total angular momentum is conserved individual bodies lose it.

If the space ship is far away from gravitational bodies and such a mechansm can be ignored within lifetimes and errors, there are people in the spaceship. People will be walking and generally moving around affecting the total angular momentum with their motion due to friction with the floors. Angular momentum would escape the isolated system with the black body radiation radiating in preferential directions, so unless a motor to correct for (even if small) loss of angular momentum is included it would stop at some future time.

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  • $\begingroup$ " People will be walking and generally moving around affecting the total angular momentum with their motion due to friction with the floors. " In my thought experiment, there are no people walking around or moving. There is just one guy, sitting on a wall of the spaceship, feeling the artificial gravity due to the rotation $\endgroup$
    – silverrahul
    Apr 21, 2021 at 8:55
  • $\begingroup$ @annav Correct me if I'm wrong, but won't the angular momentum eventually be radiated away as gravitational waves? $\endgroup$
    – Martin C.
    Apr 21, 2021 at 9:20
  • $\begingroup$ @MartinC. gravitational waves need assymmetric bodies, If the spaceship is built symmetric the rotationl acceleration will not radiate gravitationally ( in any case the asymmetries introduced by humans moving with the gravitational coupling being so small would deplete angular momentum very much later than the electromagnetic waves from friction) $\endgroup$
    – anna v
    Apr 21, 2021 at 10:32
  • $\begingroup$ @silverrahul uour title says "forever", can even one man sit motionless in one place forever? $\endgroup$
    – anna v
    Apr 21, 2021 at 10:36
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    $\begingroup$ @silverrahul I think I have answered this, for a completely gravitationally isolated body any rotation because of conservation of angular moentum lasts forever. $\endgroup$
    – anna v
    Apr 21, 2021 at 10:56
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"is it correct to say that a person sitting inside a spinning spaceship will experience "artifical gravity" forever, without any intervention or additional torque required. ?"

Without any external torque, yes. But not without any intervention. The intervention needed is the centripetal force.

In any spinning body, a particle at a distance $r$ from the axis of rotation is under the influence of an external force of magnitude $mr\omega ^2$, and whose direction is towards the center of rotation. This force does not result in any torque on the particle because this force makes an angle of 0 with the line joining the particle and the axis. This force only serves to change the direction of motion of the particle to make it go in a circle, without affecting the particle's angular velocity.

Because of this force, the frame of an spinning body is an accelerated frame, which means a pseudo force is experienced in this frame.

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  • $\begingroup$ " The intervention needed is the centripetal force. " I meant intervention as in anything external. The person is sitting inside, and will continue to sit inside. No external intervention is involved. $\endgroup$
    – silverrahul
    Apr 21, 2021 at 8:31
  • $\begingroup$ @silverrahul Suppose you're standing on top of Earth, and Earth is going around the sun in a circle. Assume Earth and yourself to be a point particle here. There clearly is a centrifugal force in the Earth frame. Do you consider this centrifugal force to be "for free" or "not free, because of the external gravitational force from the sun"? Keep in mind I'm only talking about the frame of Earth, as a point particle, revolving around the sun. I'm not talking about the frame of a spinning spherical Earth $\endgroup$
    – Ryder Rude
    Apr 21, 2021 at 8:44
  • $\begingroup$ But in that case, there is an external gravitational influence of sun. In my example, there is only a spinning spaceship and a person sitting on a wall of that spaceship . Will he keep experiencing that artificial gravity for ever ? $\endgroup$
    – silverrahul
    Apr 21, 2021 at 8:53
  • $\begingroup$ @silverrahul I'm trying to say that both cases are identical. The only thing that changes is the source of the external force on the surface you're standing. In case of the spinning body, consider that small surface area on which you're standing to be a point particle. This point particle is going in a circle. There's an external centripetal force on it. The source of this centripetal force, however, is the electrostatic force of attraction from the other particles of the spinning body. $\endgroup$
    – Ryder Rude
    Apr 21, 2021 at 9:00
  • $\begingroup$ @silverrahul In both cases, the small surface that you're standing on is under the influence of external forces. In the first case, the source the gravity from the sun. In the second case, it's the electrostatic forces from the neighbouring particles. $\endgroup$
    – Ryder Rude
    Apr 21, 2021 at 9:01
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Yes.

Absent external forces working on the spaceship, the sum total of angular momentum of the spaceship and all its contents will remain the same.

No action taken within the spaceship, and confined to act only on the spaceship or contents, can change this. Moving stuff inside, spinning up a wheel etc. could redistribute the angular momentum, but the sum will remain the same.

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Yes, the system (ship + person) rotates around its center of mass forever.

Any rotating rigid body has internal stresses, that is exactly what allows it keeps being a rigid body. Otherwise its different portions would fly away following its instantaneous (tangent) velocity.

One of the internal stress of that system is the radial compression inside the person's body. On the other hand, there are tensile stresses in the structural columns that join the periphery to the center.

That is a difference compared to "natural" gravity, where the buidings columns and all bodies are under vertical compression.

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