Prof. Laithwaite's Gyroscope Experiments Part IX

Negative precessional starting velocity

To conclude these investigations I had a quick look at two more cases where the gyroscope starts and finishes its 180º of precession with negative velocity in the direction of precession.

1. Looping Nutation

The looping nutation case shown in this video occurs for a wheel speed of 244.7 rad/s. Its precession is started at -2.0 rad/s, (i.e. it starts off in the negative-y-direction) and it completes 180º of precession, including four loops of nutation, in 2.23 seconds. Even though its net precession is still over the positive-y-direction as before, it now exerts a net negative force in the y-direction. The integrated value of its Fy graph is -72.01 N-s. This essentially balances out the 71.9 N-s found for a 30kg "dead" mass completing another forward 180º of revolution at the same radius and rotational speed of -2.0 rad/s. The combined locus for this is shown below:—

Locus for a gyroscope wheel, "live" and undergoing looping nutation over 180º,
then acting as a "dead" mass for the remaining 180º of a full cycle.
Both half-cycles are in the forward (positive-y) direction, but there is no net force.

It's obvious enough that force from the half-loops at the start and finish of the cycle cancel out the forward force generated over the remainder of the cycle, as would occur similarly for the simpler locus shown below:—

2. Gyro wheel locus on a cylinder rather than a sphere

Examination of the force graph for the negative-force looping nutation case raised the question of what would happen if the radial position of the gyro wheel from the vertical z-axis could be kept constant throughout the experiment. Would it still deliver a balancing-out net negative force?  Would it still be able to precess at all?

In the model shown above a rod (purple, of 1 gram mass) is added, with a 3-d.o.f. rotational joint to the wheel's center, and a z-direction translational joint to the central axle. This forces the wheel to keep a constant radius from the z-axis, while still transferring its weight via its shaft to the central pivot as before. The original wheel joint is modified to allow translation as well as rotation on the shaft. (Note that the model is valid as is, although I haven't bothered about avoiding superposition of object images. In a real physical model, gimbals etc would be required to avoid such clashes).

Provided this model is given a somewhat higher wheel speed than before, it can still undergo looping nutation. For a wheel speed now 410 rad/s and a starting precession of -2.0 rad/s as before, it completes 180º of precession in 2.565 seconds, with six rather than four loops. The integrated value of its Fy graph is still negative. It has the same "balancing-out" value of -72.0 N-s as before, again matching (i.e. equal and opposite to) the Fy graph for a "dead" mass completing the remaining 180º at 2 rad/s.

Once again these two results agree with the "obediance to Newton's laws re inertial propulsion" argument given previously.