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

126

THE SLEEPING TOP

137. To determine the constants K, k, and 8 in terms of 00
and o.
From equation (v) we have
= - 2X2o0o2 sin (2X2t + 28$);
32
4-+ X22(x- K002)2 = X22C200 4
Comparing this with equation (iv) with which it must be
identical we get, by equating the constant terms,
(K 2_ 2) 22 = 2X 2,..................... (vi)
showing that K is always numerically > k.
Writing 1 = 2
2
we have K 2 -k2 = 2,.......................(vii)
and from (v) K +k cos 2S= 1,........................(viii)
while k sin 2 = - a say......................(ix)
These three equations determine K, k, and 2S.
From (viii) and (ix) k2=(1-K)2 +a2;.. using (vii) K2 -(1-K)2- a2 = 2;
K l. + 2+ y2
2..
Also tan 28 = 2a
1 - a2 -- y2
and C2= (1- a )+a2.
From the last two equations we see that ko may be either
positive or negative, and also that 8 can have two values. If kc
is taken as positive, we see from equation (ix) that 8 must have
that value which makes sin 28 of the same sign as a. Hence
there is no ambiguity.
From equation (v) we see that the greatest and least values
of (0) are K+ k and K-k, and from (vi) the latter limit
is always positive; hence 0 never changes sign but varies
between al and a2, where ap, a are o0,K-k and o0,.K+k
respectively.
138. Particular case when K'=k'. It should be remarked
that this condition does not necessarily involve K=k, since 00
might be zero, which proves to be the case.