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

MISCELLANEOUS EXAMPLES

131

4. A heterogeneous sphere has its centre of gravity G at a distance
c from the centre 0 of the sphere, and the radius of gyration about any line
through G is R. The sphere is placed on a fixed smooth horizontal plane,
spinning with angular velocity & about a radius inclined at an angle a to
the vertical; and at the instant of release the axis OG is in the vertical plane
containing the axis of rotation and makes an angle f3 with that radius.
Show that G describes a horizontal straight line if
R2Q2 sin a sin 3 =gc. sin2 a + 3.
5. A shell in the form of a prolate spheroid whose centre of gravity is at
its centre contains a symmetrical gyrostat, which rotates with angular
velocity o about its axis, and whose centre and axis coincide with those of
the spheroid. Show that in the steady motion of the spheroid on a perfectly
rough horizontal plane when its centre describes a circle of radius c with
angular velocity f2, the inclination a of the axis to the vertical is given by
{Mbc(a cot a+b) - Ab cos a+ C(a sin a + c)} f2 + C'bo2 - Mgb(a- b cot a)=O,
where M is the mass of the shell and the gyrostat, A the moment of inertia
of the shell and the gyrostat together about a line through their centre
perpendicular to their axis; C, C' those of the shell and gyrostat respectively
about the axis, a the distance measured parallel to the axis of the point of
contact of the shell and plane from the centre, and b its distance from the
axis. (Camb. Math. Tripos.)
6. A sphere whose diameter is equal to the difference of the radii of two
spherical shells is placed between them, while each shell is made to rotate
with uniform angular velocity about some fixed axis through their common
centre, though the angular velocity and axis need not be the same in each
case. Show that the centre of the sphere will describe a circle uniformly,
provided there is no slipping.
7. A rough inclined plane is driven round a vertical axis with constant
angular velocity, and a sphere is placed on it. Show that, if the plane is
inclined to the vertical at an angle greater than about 18~, steady motion
is possible with the centre of the sphere describing a circle in space.
8. A rough horizontal disc can turn about an axis perpendicular to its
plane, and a right circular cone, vertical angle a, rests on the disc with
its vertex at the axis. If the disc be made to rotate with angular velocity
2, show that the cone takes up an amount of kinetic energy equal to
/ (cos2a sin2 a)
C and A being respectively the moments of inertia of the cone about its
axis and a line perpendicular to the axis through the vertex.
9. A uniform solid right circular cone of vertical angle 2/3 is placed on a
rough inclined plane of slope a, so that the generator in contact is horizontal.
Prove that the cone will always be in contact with the plane, and will
oscillate through two right angles, if
I+ 3 sin218
tan a< ~3 —/
9 sin ]g cos f3'
10. A homogeneous right circular cone, of which the C.G. is at the
distance h from the vertex and the semi-vertical angle is a, rolls on a
rough inclined plane, starting from rest when it touches the plane along
a horizontal line. Prove that, when the generator in contact makes an
angle / with the horizontal lines in the plane,
_f2 = 2mgh sin2 a sec a sin y sin Vk,
I2