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

ROTATION ABOUT A FIXED AXIS

23

17. Summary of preceding results.
Linear Motion. Rotational Motion.
Inertia or mass = M. Moment of inertia = Mik2 I.
Moment of Momentum
Momentum = Jy1. or -= In.
Angular Momentum J
Force = Mf. Moment of force = Ia.
Impulse = M(V - v). Moment of impulse = I(l - o).
Kinetic energy = Mv2. Kinetic energy = IJW2.
Work done by constant force P moving Work done by constant couple K in
its point of application through dis- turning body through angle 0
tance s =Ps= M(v2-v2). = KO= I(W12 - 2).
18. Table of moments of inertia for some simple solids
of common occurrence. Most of the following results (the
exceptions are noted by an asterisk) are included in a rule
enunciated originally by the late Dr. Routh, and known as
"Routh's Rule."
"The moment of inertia of a solid body about an axis of
sum of squares of perpendicular semi-axes
symmetry = mass x> 3 4 5
the denominator being 3, 4, or 5 according as the body is
rectangular, elliptical, or ellipsoidal."
It should be remarked that these perpendicular axes are
axes of symmetry, and that a circle and sphere are special
cases of an ellipse and ellipsoid respectively. The results can
be obtained by simple integration similar to that indicated in
the earlier part of the chapter. Except where otherwise stated
the axes are taken? through the centre of gravity.
(1) Thin straight rod-about an axis perpendicular to
its length:
* Through one end, M-, where I is the length.
'3
Mh2
Through the centre of gravity, -h-, where h isi the
half-length.
(2) Thin rectangular lamina, sides 2a, 2b —about an axis:
Perpendicular to plane, M. C' b
3
Parallel to side 2a, 3b
(3) Rectangular parallelopiped sides 2a, 2b, 2c:
Perpendicular to 2b, 2c, M. +
3