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

GENERAL MOTION

117

The other two equations become
-AA sin 0-2A f cos 0+ wO = 0,..................(i)
A0+ Cwof sin 0-A 2 sin0 cos 0 = Mga sin 0,.......(ii)
which completely determine the motion.
Multiplying the first of these by sin 0 and integrating we get
AAfr sin20 + Cc cos 0 = D, a constant,
which is the equation for conservation of angular momentum.
Again, multiplying (i) by 2f sin 0, and (ii) by 20, subtracting
(i) from (ii) and integrating
Af2r sin20 + A 02 = E - 2Mga cos 0,
which is the equation of energy.
It will be noticed that of the three equations of motion that
about axis (1) gives the condition for constancy of angular
'momentum ^Bou'T tee -ertlHca^^,ta aout axis (2) gives the
o _the azimuhlboutaaxisne, and th ou 3 thae
consancy of tepn.f thb tnjabout its axle.
In a similar way the equations may be obtained for the
general motion of a solid of revolution spinning on a smooth
horizontal plane; but the shortest method is that of writing
down the conditions for conservation of angular momentum
and conservation-of energy.
129. The following is typical of the more advanced problems
to be found among the Miscellaneous Examples at the end of
Chapter IX.
A rough horizontal plane is made to rotate about a fixed
vertical axis with constant angular velocity, and the centre of
a sphere lying at rest at the point where the axis meets the plane
is set in motion with a given horizontal velocity. Show that
the path of the centre in space is a circle, described with
uniform velocity.
Let P be the point of contact at any instant of the sphere
with the plane, 0 the point where the vertical axis OZ meets the
plane. Taking OPX as axis (1), OZ as axis (3), and O Y perpendicular to these as axis (2), we have 0 =0, 02 =0, and 03-0,
where 0 is the angle OP makes with some horizontal line fixed
in space.
Let Fl, F2 be the frictional forces at the point P in directions
(1) and (2), fi,-f2 the accelerations of the centre of gravity G in
these directions, wo, w2, c,3 the angular velocities of the sphere,
m and a the mass and radius respectively.
Taking moments about axes through G parallel to (1) and (2),
we have
al- 2= maf2,.................. (i)
and 2mat?2a2 + 5mna20cO = -Fa= -maf,.............. (ii)