As if the posts in this thread weren't long and nerdy enough....
I experimented with a new LEGO "rig" (below) designed to match the kinematics of the Laithwaite "apparatus" in question. The rig is easily reconfigured to test some of the ideas we've been throwing around.
Premise: If Laithwaite's gyro were really able to lose weight just by spinning, as he claimed, then any device with similar kinematics, sufficient spin angular momentum, and good enough bearings should be able to reproduce his result
in exactly the way he claimed. I believe this latest rig is such a device.
Didn't have time to post a video, so I'll describe the rig and its various configurations before moving on to the important findings. To quote Ben Franklin, "I'd have written a shorter letter if I'd had more time."
Rig components and improvements: The
upright is the light gray vertical support at the center of the rig. It pivots on its dark gray base about a vertical axis coinciding with the gyro's
precession axis. The horizontal bars coming out from the bottom of the upright are for precession manipulation. A
beam with a gyro at one end and an adjustable counterweight at the other rests on a
fulcrum at the top of the upright. A lockable
hinge connects the gyro to the beam. The
gyro proper includes the spinning rotor (seen here with a white cap on its outboard end), the rotor axle, and a very short arm connecting the axle to the hinge. All of these components find counterparts in Laithwaite's apparatus.
The new gyro rotor has a much higher axial moment of inertia than the one in my last video, re-embedded below:
https://www.youtube.com/watch?v=zv0kQTCctHARelease speed is down from ~4,000 to ~3,000 RPM, but the new rig brings much more spin angular momentum to bear on the matter. To amplify the gyroscopic effects of interest, I made the beam's moments of inertia about the precession axis and fulcrum as small as possible. Friction at my cantilevered rotor axle is higher now but still tolerable. As before, friction is negligible everywhere else.
Main goal: To be clear, the Laithwaite demo in question starts at 6:04 in the video below. His apparatus started in a static state (
State 1, 6:41) with the beam balanced for a gyro dangling
vertically -- or so he said. It ended in a dynamic state (
State 2, e.g., 7:07) that he offered as proof that the gyro lost weight due to spin alone. If true, my experments suggest that State 2 would indeed be a plausible outcome. But some 40 years later, modern physics
still forbids rotor weight loss of any kind under Laithwaite's stated conditions, and my results are entirely consistent with that as well.
Correction: Though Laithwaite didn't mention it and may not have realized it, his gyro was dangling
a few degrees shy of vertical when he balanced his beam for State 1. That subtlety turns out to be the key to explaining (i) how his apparatus got from State 1 to State 2 without his intervention in between, and (ii) why I was initially unable to reproduce his demo.
https://www.youtube.com/watch?v=ezJAmT19xwoIf we say (i) that the beam is "up" when the hinge is above the fulcrum, and (ii) that the gyro is "up" when its free end is above the plane of the beam, then State 2 represents a
stable {beam up, gyro down} configuration with the gyro spinning and precessing at the same time. Laithwaite's claim that spin reduces the gyro's weight rests squarely on the "stable" and "beam up" parts of State 2. (Under Laithwaite's stated conditions, the "gyro down" part is to be expected regardless.)
My main goal, then, was to see if the rig could go from State 1 to State 2 without my help.
It could not.Correction: It could, but only by doing what Laithwaite actually did in setting up Stage 1, and not what he said or implied he was doing.
Main findings: (i) State 1 and State 2 are far apart with regard to static equilibrium. (ii) To get the rig from a true State 1 to a
transient State 2 with the hinge unlocked, I had to manipulate the precession rate carefully en route. When I stopped, the State 2 configuration fell apart almost instantly. (iii) The only way to get the rig to a
stable State 2 was to start,
not with State 1, but with the beam balanced for State 2 in advance.
If Laithwaite did any of these things to take his apparatus from State 1 to State 2, he violated his stated conditions. I can't catch him at it in the video, but I'm convinced that he cheated. Think illusionist.Correction: Should have said, "(iii) The only way to get the rig to a
stable State 2 was to start, not with Laithwaite's claimed State 1, but with his actual State 1, which slightly biased his apparatus toward State 2, perhaps inadvertently."
Centered fulcrum: Laithwaite put his fulcrum in the midplane of his beam, just as I did here. As ta0 pointed out, this
centered fulcrum leaves a properly balanced beam with an unlocked hinge in a metastable state at all beam angles. Centering the rig's fulcrum allows beam angles up to ±60° from the horizontal when the gyro doesn't interfere. Lathwaite's beam, on the other hand, reached hard stops against his upright at ±45°. Yes, his beam was pegged at +45° in State 2, but that doesn't explain my inability to reproduce his demo.
Eccentric fulcrum: The rig can also accommodate an
eccentric fulcrum lying just above the plane of the beam, as in Iacopo's experiment. The resulting self-leveling tendency was strong in Iacopo's case and weak in mine.
Finding: The rig's behavior is consistent with Iacopo's earlier contention that a centered fulcrum and a
slightly eccentric one yield very similar results.
Hinge angle: We'll say that the hinge angle
h between the beam and the gyro's spin axis is negative when the gyro's down in the sense defined above. The rig above is in State 1 with the unlocked hinge dangling at
h = -90°. Importantly, Laithwaite's apparatus entered State 2 at
h ~ -40°.
The rig's hinge can be locked at 0°, ±20°, ±40°, and ±90°. Unlocking it allows
h to range freely between -120° and +120° when the upright doesn't interfere. Laithwaite's ostensibly unlocked hinge couldn't quite reach ±90°
, but that restriction doesn't explain my inability to reproduce his demo.
Correction: Turns out that this subtle restriction on Laithwaite's apparatus
does explain why I couldn't reproduce his demo the first time around.
Below are the findings for various rig configurations. In each case, the rig started out in perfect static balance at the stated hinge angle.
Hinge unlocked, fulcrum eccentric, beam balanced for h = -90°, gyro spinning: This rig configuration matches Iacopo's, and the findings match his perfectly. See his report and video at
http://www.ta0.com/forum/index.php/topic,2420.msg47407.html#msg47407 for details.
Hinge unlocked, fulcrum centered, beam balanced for h = -90°, gyro spinning: A static rig in this configuration
is in a true State 1, but it never evolves to a stable State 2 with the gyro spinning and precessing. Forcing the precession to a lower rate always produces a {gyro down} state with the beam oscillating asymptotically about a near-horizontal orientation. Forcing it to a higher rate usually produces a {gyro up} state with similar beam oscillations but occasionally results in a very transient State 2 configuration. That configuration might last longer with a rotor with more angular momentum or less friction, but it seemed
inherently unstable to me.
Correction: When I later balanced my beam for -90° <
h < -85°, as Laithwaite did (but not as he said or implied), the rig evolved to stable State 2 on its own. And as I showed with my previous LEGO test rig, it did so without Laithwaite's claimed loss of rotor weight due to spin.
Hinge locked at and beam balanced for h = 0°, fulcrum either way, gyro spinning: Locking the hinge shifts the gyro's nutation axis from the hinge to the fulcrum. Forcing precession to a lower rate always produces a {beam down, gyro down} state. Forcing it to a higher rate always produces a {beam up, gyro up} state. The beam angle deflections just described hold for any locked hinge angle but are best seen at
h = 0°.
Iacopo: Forcing the precession to a higher rate is the simplest way to produce a steady {beam up} state -- but only when the hinge is locked. If you were to lock the hinge in your experiment, your beam should go up when you push the spinning wheel away in the precession direction. But the effect might not be noticeable, as your beam has a huge moment of inertia about its fulcrum.
Hinge locked at h = -40°, fulcrum either way, beam balanced for h = -90°, gyro not spinning: In State 1, Laithwaite's apparatus was purportedly balanced for
h = -90°. It entered State 2 at
h ~ -40°. It has been suggested that from a static perspective, State 2 was close enough to State 1 to allow a metastable {beam up} configuration. The rig disproves that hypothesis. When I balance the rig for
h = -90° and then lock the hinge at
h = -40° with no spin, the gyro end of the beam drops like a rock, as above. (This is to be expected, as cos(-90°) = 0, and cos(-40°) = 0.77.) Given the length and weight of Laithwaite's large metal gyro, his State 2 would have been even farther from the static equilibrium purportedly established in State 1.
Addendum: In State 1, Laithwaite's apparatus was purportedly balanced for
h = -90° but actually balanced for -90° <
h < -85°, as Iacopo and ta0 pointed out early on. Nonetheless, his State 2 was still far from the static equilibrium established in his actual State 1.
Hinge locked at and beam balanced for h = -40°, fulcrum either way, gyro not spinning: Look at how much I had to lengthen the counterweight lever arm to get the beam into static balance at
h = -40°.
I believe that conventional physics (as opposed to Laithwaite's version) tells us that the gyro in a state of constant precession will apply all its weight at the pivot point, regardless of its inclination. That is equivalent to hanging a weight from its pivot or to just adding a point mass to the arm at that location. So it can be treated just like a rigid arm free to rotate around the main pivot. Let's assume the right and left sides of this virtual arm are statically balanced around the main pivot. If this pivot or fulcrum is at the height of the center of mass, any angle of inclination will be semi-stable. If the pivot is higher than center of mass, the stable position will be with the center of mass directly below it and that would make the arm horizontal. If the center of mass is above the pivot point, the balance is unstable and the arm can fall to either side (lowering the center of mass).
On the video, the arm appears to pivot around a pin at the center of the arm, what would make any angle stable if Laithwaite correctly balanced it with the hanging non-spinning gyro. But when he lets go the gyro it is not yet precessing and it will initially tilt and fall. So for a fraction of a second it will apply less weight to the arm, until precession takes over.
Finding: Totally agree, ta0. My experiments are consistent with every word here.
So you expect the pivot to bend the way it does. If this would be enough to explain how the gyro ends up, I am not sure 
I'd expect the hinge to bend the way it did under Laithwaite's stated conditions, but not the fulcrum. Momentary loss of gyro weight during the gyro's free-fall right after release would certainly set the beam to oscillating, but it wouldn't result in a stable {beam up} state against gravity. My experiments bear that out. Were my findings artifacts of having less angular momentum and more rotor friction than Laithwaite did? Conceivable, but I don't think so.
Put another way, conventional physics would never allow his apparatus to evolve from a claimed State 1 with
h = -90° to a
stable {beam up} configuration on its own. A gyro weight loss between states would certainly do the trick, but what combination of geometry, motion, and the 4 known fundamental forces of nature could cause such a weight loss under Laithwaite's stated conditions? None. So did Laithewaite discover a new and unsuspected 5th force of nature? He apparently thought so, but would we still be talking about 4 known forces some 40 years later if he really had? After all, it wouldn't take a CERN to study his proposed 5th force, just a spinning desktop gyro in a suitable mount.
Since I trust mainstream modern experimental physics a lot more than I trust Laithwaite and even my own eyes in this matter,
I have to conclude that Laithwaite was basically a clever illusionist specializing in gyros. I'm as puzzled as you are as to why an accomplished engineer in his position would chose such a path, but it's a fascinating question.
Correction: I failed to recognize the significance of Laithwaite's actual State 1, as opposed to his claimed State 1, and he may have done so as well. Hence, he may not have been an intentional illusionist, but he was certainly a sloppy physicist.
