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

36

REPRESENTATION OF ANGULAR MOMENTUM

Hence the total angular velocity of the body about OA, i.e.
about the line fixed in space with which OA happens at the
instant to be coinciding,
= 6, + "o2 cos a
= W if a ==90~.
Thus, if we compound two angular velocities about axes at
right angles to each other, each component is a total velocitynot a velocity relative to some moving plane or frame.
So too with three rotations about axes mutually at right
angles.
Conversely, if we resolve a single rotation into three components
mutually at right angles, each component is a total velocity,
and each is independent of the other two.
The importance of the above will be seen more clearly in
Chapter VI.
29. Representation of angular momentum. If we have a
wheel rotating about its axle H'OH with angular momentum
a,, say, then using the same conventions
H E as when representing angular velocity, we
E,. can take the line OH to represent this
> K angular momentum. Therefore, angular
momentum is a vector quantity, and hence,a;-2 it follows that angular momentum, like
angular velocity, obeys the parallelogram law.
- I h For instance: if OA (Fig. 16) be taken to
represent the component angular momentum Il1w of a body about OA, OB to represent the component I2o2 about OB, then
the resultant angular momentum Iw of the
parallelogram, and is represented by 00,
1H. el and W2 being defined in the same way
FIG. 1i. as in Art. 24.
30. It should be noted that the resultant axis of total angular
momentum of a body is not necessarily the same as the resultant
axis of total angular velocity: for Ilol, I2oW2 are not proportional to w,, W2 respectively, unless I1=12.
31. Referring now to Fig. 18, we see that if OE is a line
making an angle a with the axle OH, then, OH being taken to
represent Iw about the axle, OH cos a, i.e. OK, represents the
resolute about OE, where HKO is a right angle.
We might also have taken the area, A say, of the wheel's
disc to represent Win whic case A.cosa represents the
res iLe abutLthe axis OE. In other words the resolute would
be represented by the area of the projection of the disc as seen
from a point E on OE. This method of estimating the resolute
about any axis of the angular momentum of a rotating wheel
is sometimes convenient.