3
$\begingroup$

In order to simulate gravity on hypothetical space stations, one approach involves rotating the space station so that a centripetal force is present. Occupants within the space station's frame of reference will then feel a pseudoforce in the form of a centrifugal force, which will simulate/create artificial gravity.

In free-body diagrams I have seen of this situation, only these two forces, and the Coriolis force, is drawn. However, as the occupants will actually be in contact with the floor of the space station, shouldn't there also be a normal force present? If so, how, within the space station's non-inertial frame of reference, is this force accounted for when considering the forces acting upon an occupant standing still on the space station? Would this not result in both a normal force and a centripetal force acting on the occupant? And then both of these forces have to be balanced by the centrifugal force for the occupant to remain still?

I know that it is probably something very fundamental I am missing here, but as I just recently started learning about these things, any clarification will be greatly appreciated!

$\endgroup$

2 Answers 2

4
$\begingroup$

If someone is at rest in an inertial frame, his acceleration is $0$. He can use $F=ma$. The total force on him is $0$.

An occupant of the non-inertial frame of a rotating space station wants to do the same. If he is at rest in his frame, his acceleration is $0$. He would like to use $F=ma$, but there is a problem.

Without anything exerting a force on him, he would follow what an inertial observer would see as an unaccelerated path. The occupant sees that path as accelerated.

To fix this, he explains the cause of the acceleration as a pseudo force that acts on everything. By using this force, he too can use $F=ma$.

If the occupant stands on the floor, he is at rest in the accelerated space station frame. The floor exerts an upward reaction force on him that opposes the pseudo force. The total force is $0$. $F=ma$ works again.

For more on this, see Coriolis Force: Direction Perpendicular to Rotation Axis Visualization


Update: Some extended comments are likely going to be moved to chat. Adding content here.

When thinking about problems like this, it is important to keep two different frames of reference straight. Often you work in an inertial frame because $F=ma$ there. Sometimes you work in an accelerated frame because you can choose one where motion is simpler. You need to think about these frames separately.

In an inertial frame, the occupant of the space station is traveling in a circle at uniform velocity. This can only happen if the total force on him is just right to deflect him from a straight line to a circle. This force is call a centripetal force. In this case, the force is the reaction force from the floor of the space station. This reaction force is the centripetal force.

In the frame that moves with the space station, motion is simpler. The occupant and the space station are at rest. In this frame, there is no centripetal force because the occupant is not moving in a circle. He is at rest.

In this frame, motion is simpler, but the laws of physics are more complex. His motion is accelerated without anything exerting a force on him. He explains these acceleration as being caused by pseudo forces. Everything in his frame has a centrifugal force acting on it. Everything moving also has a Coriolis force. With these forces, he can use $F=ma$.

So the forces acting on the occupant in the accelerated space station frame are the centrifugal force and the reaction force of the floor. Centrifugal force pulls outward on him. The reaction force keeps him from falling through the floor. These forces are equal and opposite. The total force is $0$, exactly as expected for an occupant at rest.


Note that motion is significantly simpler in the accelerated frame.

If the occupant wants to keep track of the position of a point on the ceiling, it is easy in the accelerated frame. Both are at rest. The distance and direction to the point stay fixed. He can set up x, y, and z axes with himself at the origin and easily calculate the distances and angles.

In the inertial frame, he is moving in a circle. The ceiling is moving in a circle with a different radius. He can set up the same x, y, and z axes. The calculation is simple at the instant where he passes through the origin, but both positions are functions of time. In this frame it isn't so immediately obvious that the distance and angles to the ceiling stay fixed.

$\endgroup$
7
  • $\begingroup$ Thanks. Yes, I follow your logic, and I realize that for the occupant of the non-inertial reference frame, a pseudoforce has to be included for the total force to be 0. My question, however, is: what exactly are the forces acting on the occupant standing still in his/her reference frame? Is it a centripetal force or a normal force? Or both? $\endgroup$
    – user12277
    Commented Apr 6 at 22:03
  • 2
    $\begingroup$ The centrifugal force pulls him outward. The reaction force of the floor keeps him from falling through the floor. Those add to $0$. There is no centripetal force in his frame of reference because he is not traveling in a circle. He is at rest. $\endgroup$
    – mmesser314
    Commented Apr 6 at 22:12
  • $\begingroup$ If he ran along the floor, he would be traveling in a circle. The reaction force of the floor would have to increase to supply this centripetal force. But it would also get more complicated because the coriolis pseudoforce comes into play when the velocity is not $0$. $\endgroup$
    – mmesser314
    Commented Apr 6 at 22:15
  • 1
    $\begingroup$ Almost. You are thinking about two frames at the same time in the first sentence. In the space station frame, the occupant is not moving with the space station. He and the space station are stationary. In the inertial frame, he is moving with the space station. It is a quibble, but that is what you need to keep straight to get it right. Other than that, correct. $\endgroup$
    – mmesser314
    Commented Apr 6 at 22:25
  • 2
    $\begingroup$ In the inertial frame, the occupant is traveling in a circle at uniform speed. The total force on him must be just what is required to deflect his path from a straight line to a circle. This is called a centripetal force. The only force acting on him is the reaction force of the floor. That reaction force is the centripetal force. $\endgroup$
    – mmesser314
    Commented Apr 6 at 22:34
4
$\begingroup$

Would this not result in both a normal force and a centripetal force acting on the occupant?

You are double counting. When the occupant is standing on the floor of the space station, there is centrifugal force pressing him towards the floor and the reaction force provides the centripetal force and also the normal force. They are the same thing in this situation. The reaction force balances the the centrifugal force and all is in equilibrium.

When standing in your living room, the gravitational force replaces the centrifugal force, and there is a reaction force preventing you going through the floor so everything balances. The reaction force is also the normal force in this situation.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.