(8) [mu] = gMh sin [theta]/CK = An^2/KN, and [mu] = C'K/A sin [theta] = KM/A,
(9) KM.KN = A^2n^2,
so that K lies on a hyperbola with OC, OC' as asymptotes.
Constrained motion of the gyroscope.
4. Suppose the top or gyroscope, instead of moving freely about the
point O, is held in a ring or frame which is compelled to rotate about
the vertical axis OC with constant angular velocity [mu]; then if N
denotes the couple of reaction of the frame keeping the top from
falling, acting in the plane COC', equation (4) S 3 becomes modified
into
(1) gMh sin [theta] - N = [mu].CK = sin [theta] G'[mu] - A[mu]^2 cos [theta],
(2) N = sin [theta] (A[mu]^2 cos [theta] - G'[mu] + gMh)
= A sin[theta] cos [theta] ([mu] - [mu]1)([mu] - [mu]2);
and hence, as [mu] increases through [mu]2 and [mu]1, the sign of N
can be determined, positive or negative, according as the tendency of
the axis is to fall or rise.