A perfectly valid calculation, but it’s much simpler to do it using Gauss’s Law, which works just the same way as it does for electrostatics. Really, the only place to do that integral is in a textbook or class where you’re illustrating that Gauss’s Law works.
As for the question of which frame is easier to work in, the rotating frame will always work, if you include the centrifugal and Coriolis forces, but for many problems, once the Coriolis force becomes significant, it’s easier to work in the inertial frame. If speeds are low enough for the Coriolis force to be negligible, though, the co-rotating frame is usually easier, and likewise if there’s a whole bunch of other interesting stuff that’s tied to the co-rotating frame. For instance, when discussing the weather, we don’t care where a hurricane is relative to some inertial reference frame, but we might care where it is relative to Florida.
As an aside, those discussing objects flying and/or falling in rotating space stations might be interested to know that the current arc of the webcomic SchlockMercenary deals with exactly such a situation.
If it’s helpful to anyone, here’s a list of the extra “fictitious” force components that a mass experiences as observed from a rotating reference frame. If the rotation speed is constant, or close enough, then we can ignore the third component, the “Euler force”.
The centrifugal force on a mass always acts directly away from the axis — “downward”, to those on the spinning space station — and is proportional to the distance away from it.
The Coriolis force is a little trickier. It acts in a direction perpendicular to both the rotational axis and a mass’s velocity vector. In the simple case of a ball dropped from a short height, the Coriolis force means that the ball will land slightly “anti-spinward” from where you’d otherwise expect it to.
For a jogger jogging around the station, he or she will experience the force as pushing him downward or lifting him upward a little, depending on whether he’s jogging with or against the spin.
Right, although since we live on a rotating object, sometimes you have to choose which complications you’re happier with: fictitious forces, or switching reference frames.
If you are interested in an expert opinion, Columbia physicist Brian Greene discusses the Newton’s Bucket/Weights on a String problem at length in his book “The Fabric of the Cosmos”.
My understanding is it’s considered a fairly open problem. The behaviour of the spinning weights on a string in an otherwise empty universe has implications for the physical existence of space as an entity, rather than just an absence of anything.
Ouch. This has the ring of truth. Way to take all the magic out of it, Captain Buzzkill.
So really the answer is that the net angular momentum of a closed system (even the whole universe) is always zero, and you’re always spinning (only) in relation to whatever you pushed off against in the first place.
So that should obviate the necessity of Newton’s “absolute space,” right?
I think I did it again with the too-sketchy-for-others-to-understand version of my argument. And now I don’t think anyone’s even listening any more. But oh well:
Asking what would happen if you spun the weights in an empty universe is a meaningless question. How are you going to set them spinning? You can posit a magical force that sets something in motion without Newton’s “equal and opposite reaction,” but such a force doesn’t exist.
In the real world, every action has an equal and opposite reaction. It’s part of the deal, and a thought experiment that ignores this won’t yield useful conjectures. If you accelerate your weights, there’s got to be, as Chronos points out, something physical that’s going the opposite direction-- a flywheel, or exhaust gas, or what have you.
So while you won’t see centrifugal forces in an empty universe, and you do in our “full” universe, it’s not because all the other matter in our universe creates a magical matrix of reference to measure the spin against. (Which is what I thought in my OP.) It’s just because the original acceleration had to come from two masses heading in opposite directions, even if that momentum originated in a forgotten stellar explosion two billion years ago. The spin is measured relative to that original motion, whatever it was.
You can’t make energy out of nothing. All motion is relative to its equal and opposite reaction. That’s the answer, I think, boring as it is. I wanted spooky action at a distance!
