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

ROTATION ABOUT A FIXED AXIS

19

From the definition of moment of inertia as being the value
of 1mr2, it is clear that the moment of inertia for any body
is the sum of the moments of inertia of its component parts,
and, therefore, if 0 were the middle point of a rod A'A, the
moment of inertia about the central perpendicular axis would
M'12
be 2 - (M' being the mass of a length I which is in this
case half the rod), Mh2
or -,
3
M being the whole mass and h half the length of the rod.
To find the moment of inertia of a solid disc of mass M,
radius a, and thickness b, about its central perpendicular
axis.
Let us consider first a thin circular section ABC of the
disc (Fig. 8), the whole mass of which is dM, and let / represent the mass of this section per unit area. Divide the section
further into thin rings, one of which is represented in the
figure of radius r and thickness dr.
^ A A..... C (\B
FIG. 8.
The mass of this ring is u27rr dr.
Every particle of the ring is at a distance r from the axis;.. the moment of inertia of the ring is
/,27rr dr. r2, or 2/A7rr3dr;.. the moment of inertia of the thin section ABC
= 2/Ar3dr
4 [ 0
7ra4
2
dM2
= 2 a,

where dM is the mass of the thin circular section.