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

GYROSCOPIC RESISTANCE

45

As a further illustration of the gyroscopic couple, it will
be noticed that if we try to turn the axle of a wheel which
is not rotating we meet with no resistance beyond that of the
inertia of the body, i.e. w=0, and the gyroscopic couple IcwQ
becomes zero. So in the Introductory Chapter, when the
gyroscope (Fig. xxII.) is prevented by the clamp at Z from
turning about the vertical, i.e. when Q = 0, the gyroscopic couple
is again zero and the body offers no resistance to being turned
about YY', except that of inertia. Hence we see that a rotating
wheel will ofer no gyroscopic resistance unless it is free to
precess.
Again, when we apply the equation Mga=Iw2Q, in Art. 35,
to obtain the equation of steady motion of the gyroscope, we
are expressing the fact that the gyroscopic couple I-wQ is
balancing the couple Mga due to gravity.
[12.] What corresponds to the gyroscopic couple in uniform circular
motion?
If we neglect the inclination to the vertical of a wheel as
it rolls round a curved path, the expression for the gyroscopic
couple simplifies. For if v is the rate at which the wheel is
travelling, r its radius, R the radius of the curve,
v v
(=-, 12==,
M2V2
and Io2-5,
k being the radius of gyration.
The investigation when the inclination of the wheel to the
vertical is taken into account, will be found in Chapter VI.
FURTHER EXAMPLES.
1. A metal disc of radius 1 ft. and mass 2 lbs., is made to roll uniformly
round a curve of radius 12 ft. The gyroscopic couple is 5 ft.-lbs.
Neglecting the inclination of the disc to the vertical, find the rate of
rolling.
2. An engine on six wheels, each of radius r, is rounding a curve,
radius R. If V is the speed of the train, iM the mass of each wheel,
k the radius of gyration, and 2h the width of the gauge, show that the
gyroscopic couple due to the wheels is
6M1k2R V,2, -.(R2- 2)'
Owing to their rotation, which wheels tend to come off the rails?
3. If the wheel of the gyroscope in Fig. xxv. is a solid disc spinning
with velocity o, and precessing with velocity 12, find its radius if 2a=the
length of the axle.
4. The propeller shaft of a Torpedo-Boat Destroyer is a cylinder of
radius a ft., and weight M lbs. When the shaft is revolving with an
angular velocity o the speed of the ship is V ft./sec., and the radius of