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

90

BLUNT PEG

Now ML=sum of horizontal projections of ON and NL
= ON sin a -NL cos a
= Cw sin a - A2 sin a cos a (Art. 82).
Hence (Cw sin a-AQ sin a cos a) 2 = Mga sin a,
or CA[2- AQ2 cos a = Mga.
93. Steady motion of a solid of revolution spinning on a
rough horizontal plane. In Fig. 49 let GC be the axle of
the solid, a the angle it makes with vertical, P the point of
contact at any moment with the plane; let GE=c, PL perpendicular to AG=-y, and C and A be the principal moments
of inertia through G.
z
G
A x
y rv,~X-'X' 7S-n
P F
My
FIG. 49.
Here, since there is steady motion, cw is constant, and therefore
there is no component of F perpendicular to the azimuthal
plane, the reaction R being along the normal PE.
Since G describes a horizontal circle uniformly, if r is its
radius, we have F= M22r.
Also, since G does not move vertically,
R =3 g.
Considering the angular momentum about GX the horizontal
through G, as being made to rotate with velocity Q about the
vertical through G, we have (rotations right-handed):
angular momentum rotated
= Cwe sin a- AfQ sin a cos a about GX,
velocity of rotation = about GZ,
torque producing rotation
= Rc sin a - F (c cos a + y cosec a) about YG.