2010 A Space Odyssey -- Why does Discovery spin that way? (physics question)

Factual Questions
1

While this relates to the movie 2010: The Year we Make Contact this is a physics question so should be here and not on Cafe Society (I think).

This comes from another message board but need your help answering it.

Essentially the person was asking why Discovery is flipping end over end in the movie? The brief reason in the movie is it is due to the centrifuge that provided false gravity for the astronauts stopping and imparting a spin to Discovery.

Thing is the centrifuge spun perpendicular to Discovery’s main axis. Their contention is if the centrifuge stopped it should spin Discovery like a drill and not flip it end-over-end.

I had some recollection that the torque from a gyro (which this could be considered I believe) is perpendicular to the direction of spin of the flywheel so stopping the flywheel part on Discovery would want to flip it over, ass-over-tea-kettle (so to speak).

That said I really do not know hence the question.

Hopefully you all can follow what I am asking here. If not let me know and I can try to clarify.

Thanks in advance!

2

I doubt the centrifuge was the only factor. Over the 7 or 8 years between the time Bowman abandoned the ship and Floyd and company arrived, there was probably a lot of outgassing and other things that, over time, would cause a spinning like shown. In fact I rather suspect that there would be rotation along both axes, and boarding wouldn’t be nearly as easy as shown.

3

I’ve been trying to answer this, but the Board hasn’t let me yet. Maybe this time…
It’s true that you expect that , if the centrifuge stops rotating because of the bearing seizxing up (lubrication fails, or the lubricant freezes, or something) that the whole Discovery will start rotating about the long axis, and you’d think that would be the end of it.

But even in the absence of outside torques, the object can shift in interesting ways. The equations describing the motion of rotating objects are the Euler Equations* for Rotations. One of the classic examples of these equations is the rotation of a rectangular solid with differemnt dimensions along the three axes. When they demonstrate this, they use a cardboard box taped shut, or a hardcover book with a rubber band around it to keep it from opening.

You can throw the book up in the air, rotating around the axis that runs vertically through the cover, and it will simply rotate about that axis. You can also throw it into the air rotating about the axis that runs perpendicular to the cover (and the pages within), and it’ll simply rotate. But if you try to throw it up rotating about the axis that runs horizontally through the cover, it’ll flip over iin the air.

For the first two cases, if you combine the Euler equations and do a bit of algebra you get a second order differential equation that has sines and cosines as its solutions, showing that small perturbations from a simple rotation about those axes result in little oscillations with no big changes – the rotation is pretty stable. But the third equation has the sign reversed, and the solutions are hyberpoluic sines and cosines (or expoential functions, if you prefer). Small perturbations from simple spin about that one axis rapidly blow up until the small angle approximation isn’t valid anymore, and the object departs from its spin about that one axis.

The Discovery isn’t a book-shaped object. If it were really more cylindrical, like a pencil, then it would have a pretty stable rotation about that long axis, until something big perturbed it. But it’s got that wonking big AE-35 communication dish (with its two side dishes) sticking out of the center, and that’s probably throw things off and make rotation about the long axis unstable. Even absent outside perturbations, any slight wobble would quickly be magnified until it started rotating about one of the other axes. Ahnd I’d expect it to be the one running perpendicular to the plane with the antenna – which is the one they do show it flopping over.
I can’t find a webpage with the book example, but I don’t doubt it’s out there on the web. Look in a good upper-level book on mechanics.

Leonard Euler was one heavy hitter, with several sets of equations to his name. You have to specify “Euler’s Equations of Rotation” so people don’t tyhink you’re referring to Thermo, or something.

4

In addition to the CalMeacham’s excellent reply, you can also think of the phenomenon in terms of an object losing kinetic energy and “settling down” to an energy minimum. Essentially, the angular momentum of a freely rotating object is conserved; however, for a given angular momentum, the object’s rotational kinetic energy is minimized when it rotates about the axis with the largest moment of inertia. (Or, in other words, E = L[sup]2[/sup]/2 I.) This means that it’s energetically favourable for Discovery to be rotating end-over-end rather than along its long axis, as it would be under normal operation: its moment of inertia will pretty obviously be greatest when it rotates along an axis perpendicular to the primary axis of the ship.

This was actually discussed briefly in my first-year physics textbook; IIRC, it was apparently a known phenomenon with rotating satellites when the book was published (1973).

5

MikeS – that’s a seductive answer, and I’ve almost fallen into it myself, but there’s no mechanism that changes the system energy – the Discovery has the same amount of kinetic energy whether it’s spinning about its long axis or flipping end over end. It’s awfully tempting to say that the latter is the lower energy state, and that the system tends to fall into it, but we’re talking about a free object rotating in space with no interactions – nothing can introduce torque or withdraw energy. The system has conserved momentum (the center of mass stays where it is), conserved angular momentum (something has to keep spinning) and energy (there’s no friction in the system and nothing to transfer the energy to, and no storage to transform any into potential energy).

6

I think that in the original, the centrifuge wasn’t coaxial with the ship: That is to say, the ship’s long axis was in the plane of the centrifuge’s rotation. Most of the circles in the design of the front section of the ship seem to be oriented “horizontally”.

So when the bearings seized up, this would lead naturally to the ship tumbling, from conservation of angular momentum.

As mentioned above, I don’t think it’s relevant, but an object will stably rotate about the axis with the largest moment, or the smallest moment. So spinning about the long axis like a drill would also be stable.

7

True, but only if we have a perfectly rigid object or one which is tidally locked with the central body; otherwise, wouldn’t there be a loss of energy due to the object’s deformations under the time-varying tidal stresses? Granted, this is perhaps more of an “orbital engineering” effect than a “physics” effect, but Clarke was never one to keep the two separate.

8

For this I will use airplane terms…hopefully it makes sense here:

  • Yaw: Point the “nose” left/right (a flat spin)
  • Pitch: Point the “nose” up/down (spin end-over-end as in the movie)
  • Roll: Roll along the long axis (spin like a drill bit)

Looking at the picture I suppose it does seem the centrifuge could fit below the flight deck and above the pod bays. I thought it was perpendicular to the long axis of the ship behind the flight deck but upon looking again there is a side-door (famous in 2001) which I think precludes that.

So, with a horizontal centrifuge if it stopped wouldn’t it induce a flat (yaw) spin?

Seems to me if you want an end-over-end (pitch) spin the centrifuge would need to be aligned up/down (not flat or horizontal) and be aligned edge on with the spine of the ship.

I guess I am unsure again, assuming this can be viewed as a gyro, where the forces are all being applied. I looked at the Wiki page in gyros and it just confused me more as it seems there are force components to deal with on all the axis.

CalMeacham seemed to say above that a “drill” spin would be unstable because of the ships odd shape (notably the antenna) which would wobble it out of the drill spin and into a stable end-over-end spin. If I followed what he said correctly.

9

Yes, it’d be a yaw, rather than a pitch (in the coordinate system in which the ship is usually pictured), but with a basically dumbell-shaped object in zero G (with no obvious reference for “up” or “down”), I think it would be fair to refer to either a yaw or a pitch as “end over end”.

Ever object, no matter how odd its shape, has a set of three perpendicular principle axes. The long principle axis of the Discovery might not line up exactly with the long girder (we don’t know all of the details of the mass distribution), but it’d probably be pretty close. If you start off spinning along an axis that’s close to either of the extreme principle axes, you’ll stay spinning close to that principle axis. It’s only if you start off on an axis close to the middle principle axis that you start wobbling unstably (and if there’s anything to dissipate your rotation, eventually end up close to one of the stable axes).

10

As I said, if your object had its weight pretty close to the axis, then it would, in fact, be stable, and its wobbles around the axis wouldn’t cause it to flip. But the fact that they depict the Discovery flipping end over end suggests that this isn’t the case, and that the moment of intertia along the “up” direction of the Discovery (along the direction from the center through the dish antenna) is sufficiently larger than that along the “left-right” direction that the rotation is, in fact, not stanle, so the rotation changes to the one shown in the film. We don’t know the mass distribution, as Chronos says, but if the craft really behaved like that, then I’d have to say there’s not an inconsiderable amount of unbalanced mass.

I think that over as short a period of time as the nine years between the two books/films, you can pretty much ignore any interactions with the gas giants or their moons, or dissipation due to “working” of the structure. For practical purposes, the Discovery is a pretty rigid body isolated in space.

11

These are all plausible explanations. Remember that the American space agency (the NCA?) hadn’t adequately accounted for the complexities of the Jupiter-Io system, which over the span of nine years caused Discovery to begin to fall from orbit and to tumble in ways perhaps no one could have anticipated in 2001.

Also, to be blunt, this is Hollywood: it just looks cooler to have the ship tumbling like that, and it allows for the dramatic scene when the Soviet cosmonaut and Curnow sail straight for the “motionless” center of Discovery (as if Curnow would ever be allowed into space if he was prone to panic attacks like that).

12

I saw the movie last week, but obviously wasn’t paying close enough attention, because I missed the part where they explained that the new rotation was caused by the conversion of the rotation of the centrifuge.

However, doesn’t it seem that, even if that could have caused the change in direction, the new rate of rotation was way too fast? Wouldn’t the new rate be proportional to the the mass of the structure that was set rotating? In other words, since the mass of the centrifuge must have been a tiny fraction of the mass of the entire craft, wouldn’t converting its rotation to a new moment create a very slow end-over-end rotation? Much slower than we see in the movie.

It also seemed highly unlikely to me that any natural causes could have set Discovery in such a stable flat spin as we see. I can’t imagine how the structure of the thin middle section could possibly be rigid enough to remain intact as the ship starts to take on its new spin. Why wouldn’t it “crack the whip,” or snap apart in any of a thousand ways? The forces needed to keep the center rod perfectly straight while the two big “rocks” at the end start spinning around their common center of gravity seem impossibly complex and unlikely to occur unaided.

Finally, consider this: in its normal flight mode, the structure of Discovery would always be in compression along its length, as the thrusters in the rear pushed the mass in front. But when it’s rotating end-over-end, the mid section is in tension and holding together the huge masses of the crew compartment at the front and the thrusters at the rear. I think it’s safe to say that this is a situation that Discovery’s designers never anticipated, and probably wouldn’t have designed the ship to handle.

So I say it’s next to impossible that any natural phenomena could have gotten the Discovery to rotate that way, that fast, that stably, without breaking the ship into pieces.

What do you say?

13

The degree of “compression” due to thrust would be minimal; Discovery’s engines were ion thrusters, capable of constant thrust but only at very low sustained levels. However, one would assume that large disposable boosters were used to break Earth orbit (or that Discovery’s engines are capable of short periods of high thrust as demonstrated in 2010) and so one would built the engine/reactor pod to habitat pod structure to withstand not only direct compression but also to resist bending, local buckling, and vibration loads. In this case, making it also capable of withstanding the tensile loads from the endo-rotation displayed in the film is pretty much a trivial exercise. Given that structural weight is not a huge limitation, one would want to design a certain level of robustness into the structure.

Chronos has already discussed the dynamics of rotation so I have nothing to add beyond reinforcing the notion that a structure rotating about one of the two primary axes will not spontaneously start rotating about another axis; however, any amount of tidal interference or resonance from the bodies of the Jovian system may have imparted additional modifying torques to cause Discovery to tumble whichever way. It’s hard to tell from the film which way the centrifuge rotates, though I think I always assumed it was ‘flat’ with respect to the bridge, i.e. in the yaw direction (based upon the apparent orientation of the tunnel from the bridge).

Stranger