Angular velocity
97
relative to the centre of mass resulting from a rotation through an infinitesimal
angle do (see (9.1)): dr = dR + do xr. Dividing this equation by the time
dt during which the displacement occurs, and putting
dr/dt = V,
dR/dt =
do/dt = La
(31.1)
we obtain the relation
V = V+Sxr.
(31.2)
Z
X3
P
X2
r
o
R
X1
Y
X
FIG. 35
The vector V is the velocity of the centre of mass of the body, and is also
the translational velocity of the body. The vector S is called the angular
velocity of the rotation of the body; its direction, like that of do, is along the
axis of rotation. Thus the velocity V of any point in the body relative to the
fixed system of co-ordinates can be expressed in terms of the translational
velocity of the body and its angular velocity of rotation.
It should be emphasised that, in deriving formula (31.2), no use has been
made of the fact that the origin is located at the centre of mass. The advan-
tages of this choice of origin will become evident when we come to calculate
the energy of the moving body.
Let us now assume that the system of co-ordinates fixed in the body is
such that its origin is not at the centre of mass O, but at some point O' at
a distance a from O. Let the velocity of O' be V', and the angular velocity
of the new system of co-ordinates be S'. We again consider some point P
in the body, and denote by r' its radius vector with respect to O'. Then
= r'+a, and substitution in (31.2) gives V = V+2xa+2xr'. The
definition of V' and S' shows that V = Hence it follows that
(31.3)
The second of these equations is very important. We see that the angular
velocity of rotation, at any instant, of a system of co-ordinates fixed in
the body is independent of the particular system chosen. All such systems
t
To avoid any misunderstanding, it should be noted that this way of expressing the angular
velocity is somewhat arbitrary: the vector so exists only for an infinitesimal rotation, and not
for all finite rotations.
4*
98
Motion of a Rigid Body
