Relative Centripetal Acceleration

[Ryan Reed]

Let's say you have two rings. Both rings have the same radius and are aligned so that the holes are perfectly parallel to each other and a straight line can be drawn through them without interference. Both rings spin along the same axis with the same speed, but in opposite directions.

If you put a person on ring A and a person on ring B, they both experience the centripetal force of acceleration that pushes them towards the outside of the ring.

Here is what I cannot understand.
From the perspective of person A, he is not spinning. Both ring A and himself are stationary with some force that is pushing himself away from the center of his ring. But in reality, ring A is spinning at a speed of V, and to person A, person B should be spinning with a speed of 2V as he is spinning in the opposite direction.

If person A knows of the centripetal acceleration formula, they would calculate person B's acceleration as (2V)^2/R.

Person B's actual acceleration is only V^2/R, just like person A's, but person A would see the other's acceleration as (2V)^2/R, which is 4 times what it actually is.

Is there an error to my thought process, and if so what am I missing?
In this universe there are no stars, no planets, just the two rings and the two people.

 

[Orodruin]

If you put a person on ring A and a person on ring B, they both experience the centripetal force of acceleration that pushes them towards the outside of the ring.

Let us start by getting the concepts clear. It is the centrifugal force (i.e., the inertial force in the respective rotating frames) that pushes the persons towards the outside of the ring. The centripetal force is the force from the rings on the persons and that is a very real force that is present regardless of looking at it in a rotating frame or not.

Is there an error to my thought process, and if so what am I missing?

You are missing the fact that (seen from the rotating reference frame of A), B is moving. As such, there will be a Coriolis force on top of the centrifugal force.

Your situation can actually be simplified with the same kind of "paradox". Consider B to be stationary in the inertial frame. There are then no forces acting on B, but in the rotating frame of A, B is subject to a centrifugal force that is exactly canceled by the Coriolis force.

 

[Ryan Reed]

I do not think I fully understand. The Coriolis Force is caused by the difference of the change in angle as you travel farther from the center point right? So the head would experience different centrifugal forces than the feet.

And you are right, I did not account for those forces when thinking about this.
But I am still missing something because I can't see how the Coriolis force could counteract the centrifugal forces as they operate in directions that are perpendicular(centrifugal pointing from the center of the ring, and Coriolis pointing tangent to the circumference). Would this be different if the rings were large enough to make the Coriolis forces negligible?
I also forgot the mention that both person A and person B are already up to speed with the ring and are planted firmly on its surface and are not falling towards the outside.

 

[Orodruin]

The Coriolis Force is caused by the difference of the change in angle as you travel farther from the center point right?

No, this is not correct. It is caused by a velocity vector that is not parallel to the angular velocity.

and Coriolis pointing tangent to the circumference

This is not correct either. It is only correct when the velocity is radial. When the velocity is tangential, the Coriolis force is radial.

 

[pervect]

Let's say you have two rings. Both rings have the same radius and are aligned so that the holes are perfectly parallel to each other and a straight line can be drawn through them without interference. Both rings spin along the same axis with the same speed, but in opposite directions.

If you put a person on ring A and a person on ring B, they both experience the centripetal force of acceleration that pushes them towards the outside of the ring.

Here is what I cannot understand.
From the perspective of person A, he is not spinning. Both ring A and himself are stationary with some force that is pushing himself away from the center of his ring. But in reality, ring A is spinning at a speed of V, and to person A, person B should be spinning with a speed of 2V as he is spinning in the opposite direction.

If person A knows of the centripetal acceleration formula, they would calculate person B's acceleration as (2V)^2/R.

Person B's actual acceleration is only V^2/R, just like person A's, but person A would see the other's acceleration as (2V)^2/R, which is 4 times what it actually is.

Is there an error to my thought process, and if so what am I missing?
In this universe there are no stars, no planets, just the two rings and the two people.

Suppose person A has a gyroscope. The gyroscope points in the same direction in an inertial frame of reference. But in your "perspective of person A", the gyroscope doesn't point in the same direction, it's axis of rotation precesses. How does person A explain this? It's not due to "centrifugal force". That force doesn't cause a gyroscope to precess. This is an important difference from "the perspective of person A" as compared to an inertial frame of reference. Clearly, there is something different about person A's perspective, and it's not just "centrifugal force". One way of describing the difference is to introduce the coriolis force, but the basic issue is assuming that the "perspective of person A" is the same as an inertial frame.

Gyroscopes can be physical, or implmented with lasers as in a ring laser gyroscope. It's hard to describe the details of the behavior of "the perspective of person A" without mathematics, but these simple thought experiments can show the need for thought on the matter.

Note that we have by no means completely described "the perspective of person A". Notably missing is a discussion of the behavior of clocks according to "the perspective of person A", which don't all tick at the same rate, unlike an inertial frame.

 

[Ryan Reed]

So the gyroscope, if in an inertial frame, would keep its angle the same, but in this case it would precess? And that an inertial frame is different from a perspective? Is this correct?

And are you saying that if we include time dilation, it would help solve this problem?

 

[Orodruin]

So the gyroscope, if in an inertial frame, would keep its angle the same, but in this case it would precess? And that an inertial frame is different from a perspective? Is this correct?

What do you mean by "a perspective"? An inertial frame is different from a non-inertial frame. In the typical non-inertial frame in classical mechanics, you both have the possibility of the frame origin accelerating as well as the basis vectors rotating. It is common to refer to a non-inertial frame as an object's rest frame if its position and orientation relative to that frame remains the same. Such a frame is not necessarily inertial.

And are you saying that if we include time dilation, it would help solve this problem?

No, this is a purely classical issue as long as you do not start rotating at relativistic speeds. I believe @pervect is going beyond your question due to your post being in the relativity forums (the classical physics forum would have been a better placement).

 

[pervect]

So the gyroscope, if in an inertial frame, would keep its angle the same, but in this case it would precess? And that an inertial frame is different from a perspective? Is this correct?

And are you saying that if we include time dilation, it would help solve this problem?

Basically yes, an inertial frame is different from what I think you mean by perspective.

As far as the time dilation goes, I'm basically thinking that your question is purely Newtonian

THe thought occurs to me that perhaps you are talking about rotation being relative, rather than the special theory of relativity. The short answer to that revised question is that rotation is not relative, one can tell by experiments (as with the gyroscope, for instance) if one is rotating or not.