B Artificial gravity rotating on two axes

[wrobel]

That means that there must be a vector ω→ such that

That actually means that for each $\boldsymbol r_i$ there must be a vector $\boldsymbol \omega_i$ . A correct argument you have already brought in #35 and it is not a high school

 

[vanhees71]

The proof that it must be the same $\vec{\omega}$ for all $\vec{r}_i$ is the same as the one given that it is the same $\vec{\omega}$ for different choices of the reference point $\vec{R}$. You can apply it to the constraint that for any two body-fixed points
$$|\vec{x}_i-\vec{x}_j|=|\vec{r}_i-\vec{r}_j|=\text{const},$$
i.e.,
$$\dot{\vec{r}}_i-\dot{\vec{r}}_j=\vec{\omega}_{ij} \times (\vec{r}_i-\vec{r}_j) = \vec{\omega}_i \times \vec{r}_i -\vec{\omega}_j \times \vec{r}_j$$
can only be fullfilled for all $\vec{r}_i-\vec{r}_j$ if
$$\vec{\omega}_{ij}=\vec{\omega}_{i}=\vec{\omega}_{j}.$$
Since this holds for all pairs of points labelled with $i$ and $j$, one must have $\vec{\omega}_i=\vec{\omega}$ for all $i$.

I still don't see, why this is not understandable by an interested high-school student. At least in Germany 3D Euclidean vectors are still in the curriculum.

 

[wrobel]

can only be fullfilled for all r→i−r→j if
ω→ij=ω→i=ω→j.
Sin

I think that this implication still needs a detailed explanation

I still don't see, why this is not understandable by an interested high-school student.

Perhaps this is understandable. But I think an angular velocity of a rigid body is nowhere included to the curriculum of a high school. By my own experience I see that this is not a simple topic for university students.

 

[wrobel]

At least in Germany 3D Euclidean vectors are still in the curriculum

a lot of very complicated things can be produced from the 3D Euclidian vectors :)

 

[WernerQH]

Sure it is. It can be answered simply, without having the math as the answer.But the end result is not. At their simplest, the OP question(s) could be answered with a yes/no.

Yes, an object can rotate around two axes simultaneously. But these axes cannot remain "fixed".

It strikes me that nobody has yet mentioned the Earth as an example. The south pole (i.e. the momentary rotation axis) does not remain at the same spot in Antartica. It moves semi-regularly about some "mean" pole at a distance of several meters with a "period" of about 430 days (the Chandler period). This causes variations of geographic latitude and longitude of a fraction of an arcsecond. For a perfect rigid body the instantaneous rotation axis is neither fixed with respect to the body, nor the direction of angular momentum, but moves simultaneously on two cones surrounding both.

I still don't find this easy to visualize. It is an interesting topic that features in vol.1 of the Berkeley physics course, as well as in Sommerfeld's course on theoretical physics.

 

[wrobel]

an object can rotate around two axes simultaneously.

the phrase "can rotate around two axes simultaneously" is incorrect by itself: there is continuum ways to present the angular velocity as a sum of two vectors.

ps In my opinion by its initial post itself this thread was promised to become odious. I quit it.

 

[DaveC426913]

then why do you ask questions when you know answers?

I don't know the answer.

 

[DaveC426913]

Reporting to have this thread closed, as it seems to be unanswerable at the education level specified.

 

[Rive]

there is continuum ways to present the angular velocity as a sum of two vectors.

From the simple POW of vectors, that's absolutely true.

But exactly because of that from the POW of a complex body, sometimes it may be more practical to use (body) locked directions (with special significance for that complex body) as references.

Yet, the original question was a bit further down the rabbit hole.

...what would occupants experience as gravity? Would it change over time?

I do admit that I have only an intuitive answer for that, but I'm too interested in an actual answer.

 

[jbriggs444]

I do admit that I have only an intuitive answer for that, but I'm too interested in an actual answer.

"What would the occupants experience as gravity?"

The [inverse of the] acceleration of the floor under their feet.

"Would it change over time?"

Yes. As has been described, the instantaneous axis of rotation will not remain fixed relative to the body. Points at the instantaneous axis have zero g. Points away from the axis have non-zero g. It follows that the g for body-fixed points can change over time.

 

[DaveC426913]

"What would the occupants experience as gravity?"

The [inverse of the] acceleration of the floor under their feet.

"Would it change over time?"

Yes. As has been described, the instantaneous axis of rotation will not remain fixed relative to the body. Points at the instantaneous axis have zero g. Points away from the axis have non-zero g. It follows that the g for body-fixed points can change over time.

Cool. That's half the answer.
Magnitude would change over time. Would direction?

 

[jbriggs444]

Cool. That's half the answer.
Magnitude would change over time. Would direction?

Yes. A little bit of reasoning shows that it must be so.

Consider two points on the body with the instantaneous axis somewhere between them. "Gravity" at both points is away from the direction of the other.

Now let the rotational axis move so that gravity stays in the same direction at both points. That means [some point on] the rotational axis moves along a line between the two points.

Now consider a third point not located on the line connecting the first two. "Gravity" there also points away from the rotation axis. But the direction will be changing over time. This is not quite bulletproof, as it stands, but the basic idea holds -- if the rotational axis wanders around and gravity points away from the rotational axis, gravity cannot point in the same direction forever for everyone.

Edit: Had to swap "away from" for "toward" due to brain error.

 

[vanhees71]

I think that this implication still needs a detailed explanation



Perhaps this is understandable. But I think an angular velocity of a rigid body is nowhere included to the curriculum of a high school. By my own experience I see that this is not a simple topic for university students.

I didn't say, it's simple, I only said it's understandable by a motivated enough high school student.

 

[vanhees71]

Reporting to have this thread closed, as it seems to be unanswerable at the education level specified.

I have answered it at the education level specified. You only have to think hard enough about it!

 

[DaveC426913]

You only have to think hard enough about it!

'Think hard enough...' I can't even decipher it. I didn't learn this in HS.

 

[OmCheeto]

View attachment 301959'Think hard enough...' I can't even decipher it. I didn't learn this in HS.

It's possible you did but are just too old, like me. I've just finished going through my college level first year math textbooks* and much of the vector notation used by vanhees71 is nowhere to be found.

*
ISBN 0-87150-283-6 Functions and Graphs 3rd Edition Stokowski circa 1980
ISBN 0-15-505731-6 Calculus With Analytic Geometry 2nd Edition Ellis and Gulick circa 1982

 

[berkeman]

Thanks for an entertaining and interesting thread folks.