Prof. Laithwaite's Gyroscope Experiments Part VIII

More experiments with zero precessional starting velocity

I repeated the cuspidal nutation experiment performed previously, for other whole numbers of nutations over 180º, with results tabulated below:—

Nutations       Wheel      Time        Minimum       Maximum        Integral of
in 180º of        rot.          reqd. to    precessional   precessional    centrifugal
precessional    speed    precess      velocity            velocity             force resolved
motion            [rad/s]    180º [s]    [rad/s]            [rad/s]               in y-dirn. [N-s]

      3                  227.4       1.898            0                  3.522                 -0.0548
      4                  271.9       2.215             0                  2.95                   -0.00067
      5                  308.7      2.486             0                  2.588                -0.0239
      6                  340.7      2.724             0                  2.36                     0.0036
  

In all these cases, there is essentially zero centrifugal force in the y-direction. To help explain why we should expect this, let's consider a simple "inertial propulsion" problem:—

An inertial propulsion problem. The mass has to
reverse direction at the locations shown dashed.

In the above image, a mass (red) is attached to one end of an arm (black line). The other end of the arm has a central pivot to a vehicle. The mass undergoes 180º of revolution, and so the centrifugal force it exerts at the pivot drives the vehicle forward. The problem comes in starting and stopping the mass's motion at the start and finish of its half-revolution. Various ways of doing that could be proposed, e.g. using springs attached to the vehicle, or by delivering/recollecting torque at the pivot-end of the arm etc. However, there is no way of using this idea to achieve any net unreacted force on the vehicle, without disobeying Newton's laws of motion, especially the third one.

Since the precessional speed of the cuspidal-nutating gyroscope discussed above is indeed brought to zero at times; in particular when it coincides with the ±x-axis, then it too cannot be expected to deliver any net force in the y-direction over its half-revolution of precession. If it did, it would have to disobey Newton's laws just as much as the simple mass in the inertial propulsion example would.

Maximum precessional starting velocity

Next, I looked at a case where the cuspidal-nutating gyroscope starts and finishes its 180º of precession with maximum, rather than minimum velocity:—

The four-cusp case modelled above now occurs for a wheel speed of 290 rad/s. Its precession starts at 2.75 rad/s, and it completes 180º of precession in 2.22 seconds. The integrated value of its Fy graph is 98.797 N-s. This essentially balances the -99.0015 N-s found for a 30kg ("dead") mass completing the other (backward) 180º of revolution at the same radius and rotational speed of 2.75 rad/s. (Small discrepancies in all these results are only because of my trial-and-error initial "tuning" of the models). Once again, this modelling shows only orthodox behaviour.