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Consider a rigid body executing pure rotational motion (i.e., rotational motion which
has no translational component). It is possible to define an axis of rotation
(which, for the sake of simplicity, is assumed to pass through the body)this
axis corresponds to the straightline
which is the locus of all points inside the body which remain stationary as the body rotates. A general point
located inside the body executes circular motion which is centred on the rotation axis, and orientated
in the plane perpendicular to this axis. In the following, we tacitly assume that the axis
of rotation remains fixed.
Figure 67:
Rigid body rotation.

Figure 67 shows a typical rigidly
rotating body. The axis of rotation is the line . A general point
lying within the body executes a circular orbit, centred on , in the plane perpendicular to
. Let the line be a radius of this orbit which links the axis of rotation to the
instantaneous position of at time . Obviously, this implies that is normal
to . Suppose that at time point has moved to , and the radius
has rotated through an angle . The instantaneous
angular velocity of the body is defined

(309) 
Note that if the body is indeed rotating rigidly, then the calculated value of should
be the same for all possible points lying within the body (except for those points lying
exactly on the axis of rotation, for which is illdefined).
The rotation speed of point is related to the angular velocity
of the body via

(310) 
where is the perpendicular distance from the axis of rotation to
point . Thus,
in a rigidly rotating body, the rotation speed increases linearly with (perpendicular) distance
from the axis of rotation.
It is helpful to introduce the angular acceleration of a rigidly rotating body:
this quantity is defined as the
time derivative of the angular velocity. Thus,

(311) 
where is the angular coordinate of some arbitrarily chosen point reference
within the body, measured
with respect to the rotation axis.
Note that angular velocities are conventionally measured in radians per second, whereas angular
accelerations are measured in radians per second squared.
For a body rotating with constant angular velocity, ,
the angular acceleration is zero, and the rotation angle increases linearly with time:

(312) 
where
. Likewise, for a body rotating with constant angular acceleration,
, the angular velocity increases linearly with time, so that

(313) 
and the rotation angle satisfies

(314) 
Here,
. Note that there is a clear analogy between the above
equations, and the equations of rectilinear motion at constant acceleration introduced in
Sect. 2.6rotation angle plays the role of displacement, angular velocity plays the role of (regular) velocity, and
angular acceleration plays the role of (regular) acceleration.
Next: Is rotation a vector?
Up: Rotational motion
Previous: Introduction
Richard Fitzpatrick
20060202