An elementary treatment of the theory of spinning tops and gyroscopic motion, by Harold Crabtree.

78

PRACTICAL APPLICATIONS

say, to a gust of wind or a shift of load) to the resulting angular
precessional displacement, relative to the car, of the axis of
the gyrostat; for the larger the impulse required to produce
a given displacement, the larger the resistance which the car
offers to being upset.
Now the arrangement of the car is such, that after some time
t the car has no angular momentum about the line of rail;
hence the
impulse of applied couple= impulse of erecting couple
= Iwl2 dt
= IWO)
=I0,
2 being the precessional velocity and 0 the angle of precession.
Hence, the efficiency = Iw.
If we now increase the linear dimensions n-times, the
upsetting couple produced by a corresponding lateral displacement of load, say, will be n4-times as great: for the displacement
is n-times greater and the load is n3-times greater.
Hence, for equal efficiency we must have Iw n4-times greater.
Now I is n5-times greater; for I=Mk2 where M is n3-times
and k is n-times as great. Consequently, if w is n-times less,
we shall have equal efficiency.
For further information concerning the Mono-railway, the
reader is referred to Professor Perry's article in Nature,
March 12, 1908.