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

CENTRE OF GRAVITY INITIALLY AT REST

101

102. To determine the least velocity at which a top will
spin in a vertical position. In this case we have from equation
(iii) Art. 98, W(I -cos0)
A sin20
CW
A(1 +cos 0)'
If now we suppose 0 so small that 02 may be neglected, we
see that c
4 = 2A, a constant.
Again, from equation (ii) Art. 98, the value of 4, when 0 is
stationary, is given by,2 2Mga(1 - cos 0)
A sin2 0
2Mga
A (1 +cos 0)
- A'
neglecting squares of small quantities.
Hence, in order that the top may return to the vertical
position after a small displacement, it is necessary that
02W2 Mga
4A2> A
or C2W2> 4MgaA.
When w2 is less than this value, the disturbed motion does not,
as the disturbance is indefinitely diminished, tend to return to
the vertical motion; on the other hand, it does not necessarily
depart very far from it. Both limits of oscillation in agreement
with C2 might be near the vertical.
The above conditions show that it is very difficult to spin a
top for which C is small in comparison with A; for instance
it is almost impossible to spin a pencil on its point.
Example. Taking the top of the previous article we must
have 4MgaA
4.386.2.4.642
252
8.64
(0> 25 4/193
25
> 284-5 rad./sec..the number of revolutions a second must be
> 45 approx.,
which is practically the same number as in the preceding case.