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

132

MISCELLANEOUS EXAMPLES

and that the point of action of the resultant normal reaction between the
plane and the cone is at a distance
Ahcos a+ 3 sin a tan y sin ~}
from the vertex; where I is the moment of inertia about a generator, and
y is the inclination of the plane to the horizontal. (Coll. Exam.)
11. A solid uniform prolate spheroid whose axes are 2a, 2b, 2b, spins
steadily on a smooth horizontal table. It has angular velocity n about
its axis of figure, that axis has angular velocity o about the vertical, and
h is the constant height of the centre above the table. Show that
5(a2 + b2)(h2- b2) g
n b4 A
and that, if n has its least value,
(o25(a2- b). g. (Camb. Math. Tripos.)
a2 + b2 h
12. A uniform solid sphere of radius c rolls under gravity in contact with
a perfectly rough elliptic wire of semi-axes a and b, whose plane is horizontal:
the centre of the sphere moving in a vertical plane through the major axis
of the ellipse. Prove that if X be the angular velocity of the sphere when
its centre is at a height z above the major axis,
02= 2gb6(h- z)
b6(5Z2 + 2C2) + 5 (z2 + C2 - b2) (a2 - b2)3
the value of z when o=0 being h, and c > b.
13. A circular disc has a thin rod pushed through its centre perpendicular
to its plane, the length of the rod being equal to the radius of the disc.
Prove that the system cannot spin with the rod vertical, unless the velocity
of a point on the circumference of the disc is greater than the velocity
acquired by a body after falling from rest vertically through a height
ten times the radius of the disc. (Coll. Exam.)
14. A wheel with 4n spokes arranged symmetrically rolls with its axis
horizontal on a perfectly rough horizontal plane. If the wheel and spokes
be made of a very fine heavy wire, prove that the condition for stability is
T2> 3 22n+r
4 4n + 3 —
where a is the radius of the wheel and V its velocity. (Coll. Exam.)
15. Two light rods OP, PQ, each of length 2a, are smoothly jointed at P,
and are the axes of equal gyrostats whose centres of mass are at the middle
points of the rods. The gyrostats spin with equal angular velocities n in
such directions that both would spin in the same way if OPQ were a straight
line. 0 is fixed and Q slides above 0 on a smooth vertical rod OZ. If M is
the mass of each gyrostat, A and C its principal moments of inertia, and a
mass m is suspended from Q, show that steady motion is possible with a
precession l2, in the same sense as the resolved part of any angular velocity
n along OZ, provided that k - lies between unity and zero, where
k = Cn 2(M+ m)ga
k=(A +Ma2) and = (A +.Mfa2)'
Show further that the motion is always stable. (Camb. Math. Tripos.)
16. Prove that the least velocity v with which a thin circular disc
(radius a) must be started in order to roll steadily on a rough horizontal
plane in a straight line, or very nearly in a straight line, is given by
v2> -ga; and that the period of a small oscillation is
27r A(A+Ma2) 2
27 C C2(C+ Ma2) - MgaAJ