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

80

PRODUCTS OF INERTIA

Let x, y, z be the co-ordinates of any particle P of the body,
referred to these axes.
Then the angular momentum of the particle P is:
About OZ, nmwr.r or mr2w.
About OX, -mwr cos 0. z
=- - Zr. -. z =3-nmz. X.
r
About Y, - ncr sin 0. z
= - mr. -. z = - m. y.
r
Z
LT
/ A/
L - 7- x:- ' T
Y N
FIG. 37.
Hence it is seen that, owing to the rotation of the body
about OZ, there is a contribution of angular momentum about
OX equal to - Enowz. x, and about 0 Y equal to - mowz. y,
in addition to Emr2W, or I1w, about OZ.
The expressions Emzx, Emzy are called products of inertia,
but we shall be able to neglect these in dealing with solids of
revolution, as will be seen in Art. 79.
78. It should be noticed that although the rotation w of
the body about OZ involves in general angular momentum
about both OX and 0 Y, yet it involves no component angular
velocity of the body about OX and OY; for the resolute of o
about OX or OY is w cos 90~, which is zero.
This can be further illustrated by considering two points
A, B moving in parallel lines with the same velocity so that
AB is perpendicular always to the lines of motion. If the
velocity at any instant is v, though the angular velocity of
B about A is zero, the angular momentum of B about A is
mvc, where c is the distance AB.
79. Particular case of a solid of revolution. If the body
is a solid of revolution whose axle is the axis of rotation OZ,
as, for instance, an ordinary top spinning vertically on its toe,
it is clear (Fig. 38) that for every particle P revolving about
OZ in direction (XY) and possessing a definite linear velocity