Rigid Body Motion - Dynamics and Relativity - page 227

the vertical in a co-ordinate system that has its origin at the pivot and in which e 3
is parallel to the instantaneous direction of the axle. If we consider only the angular
momentum of the flywheel then L is parallel to e 3 . R represents the position vector
of the centre-of-mass, and so the gyroscope experiences a gravitational torque about
the pivot given by:

τ = R × m g . (10.96)

The torque is directed into the page in Figure 10.18. In the inertial lab frame this
torque must produce an instantaneous change in the angular momentum which is
also into the page. This is achieved by rotating the direction of the e 3 axis and
hence also the direction of L . The instantaneous change in L is represented in the
horizontal plane in Figure 10.19. Using Eq. (4.18) we have

τδt ≈ δL (10.97)

d L

df

L sin q

Figure 10.19 The changing horizontal component of L induced by the gravitational torque
on the gyroscope.

and

L sin θδφ ≈ δL, (10.98)

from which we obtain

τ
L sin θ .

ω p ≡ φ = (10.99)

As long as θ doesn't change, τ is constant and we obtain a solution in the lab frame
with ω p also constant, which represents the uniform precession of the gyroscope
axle about the vertical.
The above treatment is straightforward and gets us quickly to a result that tells
us how the gyroscope is able to precess, i.e. L rotates at exactly the correct rate so
as to compensate for the gravitational torque about the pivot, so there is no torque
'left over' to cause the gyroscope to topple. However, the simple approach is
unsatisfying in a couple of ways. Firstly it assumes that all the angular momentum
of the system is generated by the rotation of the flywheel about the e 3 axis, and
ignores the angular momentum associated with the rotation of the whole system
about the vertical direction. Secondly, it assumes that the gyroscope executes its
precession at a fixed angle to the vertical and therefore says nothing about the
possibility of θ changing with time, as would happen if we were not able to release
the gyroscope at the correct angle. We will address these deficiencies in turn: the
first in the lab frame by including the angular momentum associated with the
precession; the second by using Euler's equations in a rotating frame of reference.