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

96

EXAMPLES

gravity and the point of suspension, C being the moment of inertia about
the axis of figure. Show that when motion is steady,
Q2 sin 4(Cw - Al2 cos ) cos 0 = Mga sin ( - 0),
12(l sin 0+a sin O) =g tan 0.
If another gyrostat were spun on the head of this one, which way would
the spin have to be in order to maintain steady motion?
8. A heavy sphere is held in contact with a rough circular wire which
is fixed in a horizontal plane, and a horizontal impulse is then applied to the
sphere, causing it to roll round steadily. If c is the radius of the ring and b
that of the sphere, and if a is the constant inclination to the vertical of the
radius through the point of contact, &2 the angular velocity of the point
of contact, show that the magnitude of the impulse is such as would impart
to the sphere, if it were free, a velocity
72 (c-b sin a),
and that the relation between f2 and a is given by
722(c - b sin a)= 5g tan a.
NOTE. The point of contact of the sphere with the wire is (instantaneously) a fixed point.
9. A homogeneous sphere' of radius a is loaded at a point on its surface
by a particle whose mass is -th of its own; if it move steadily on a smooth
horizontal plane, the diameter through the particle making a constant angle
a with the vertical, and the sphere rotating about it with uniform angular
velocity o, prove that o2 must be not less than
5g cos a(2n + 7)/an (n + 1)
if the particle is at the upper extremity of the diameter; and show that the
particle will revolve round the vertical in one or other of two periods whose
sum is 4:rnao/5g.
NOTE. The moment of inertia for the whole body about an axis through
its centre of gravity G, perpendicular to the diameter through G, is
+1 2~+ a72, where 11 is the mass of the sphere.
10. A sphere is rotating within a spherical concentric light shell of radius
a, placed on a rough horizontal plane, about an axis through the common
centre. Show that if the centre of the sphere describes a circle of radius r
with uniform velocity v, while spinning with velocity o, then the inclination
of the axis of rotation is given by
k2 sin a (v cos a - reo) = var,
where k is the radius of gyration of the sphere.
11. A homogeneous right circular cone, radius of base a, height A,
spinning about its axis with velocity o, maintains a constant azimuthal
motion i2 with its axis inclined to the vertical at an angle 3 and its base in
contact with a rough horizontal plane. Show that the necessary condition is
a2w2 sin /3 - 20 {a2 + 2} 2 sin,3 cos /3
10 ~ I ~L- ~111 P 0 4J
=g (Ca cos 3 +asin )
r being the radius of the circle described by the centre of gravity of the
cole.
Additional problems on steady motion will be found among
the Miscellaneous Examples at the end of Chapter IX.