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

18

ROTATION ABOUT A FIXED AXIS

10. The phrase "moment of inertia" means in the first
instance "the importance of the inertia"; and its significance
will be easily grasped if we consider the obvious importance
in the motion of some rigid bodies of the configuration or
arrangement of the particles which form them.
A golfer might conceivably possess a "putter" of the same
total mass as his "driver "; but the difference in their usefulness
for conveying force to the ball is clear at once. This is of
course due to the different configuration of particles in the two
clubs. Similarly, a cricketer "takes the long handle" when
hard hitting is his primary object.
Or again, suppose we have two wheels, of the same total
mass, rotating about fixed axes with the same angular velocity,
the only difference in them being that one has most of its mass
concentrated near the centre, the other at the rim. It will be
found that the one with the heavy rim requires a much greater
force to stop its rotation than the one with a heavy centre and
light rim.
The devices for finding the value of different moments of
inertia, either without or with the aid of the Calculus are
discussed fully in various text books. The following two
elementary examples will illustrate the method when the
Calculus is employed.
11. To find the moment of
inertia of a thin rod of mass
M and length 1, about an axis
which passes through one end __ 9
of the rod and is perpen- A' o A
dicular to its length.
Let OA be the rod of which
PQ is a small element dx, at
a distance OP (=x) from the Fo. 7.
axis, Fig. 7.
Let u =mass per unit length of the rod. Then the moment
of inertia of the whole rod
= ZIA dx. X2
= J, dx. x2
13
I3
=M12
12.. k2=-, I being the length of the rod.