Defining Rotation and
its Variables
We begin our study of rotational motion by
defining exactly what is meant by rotation, and establishing a new set of
variables to describe rotational motion. From there we will revisit kinematics to generate
equations for the motion of rotating bodies.
Definition of Rotation
We all know generally what it means if an object
is rotating. Instead of translating, moving in a straight line, the object
moves about an axis in a circle. Frequently, this axis is part of the object
that is rotating. Consider a bicycle wheel. When the wheel is spinning, the
axis of rotation is simply a line going through the center of the wheel and
perpendicular to the plane of the wheel.
In translational motion, we were able to
characterize objects as point particles moving in a straight line. With
rotational motion, however, we cannot treat objects as particles. If we had
treated the bicycle wheel as a particle, with center of mass at its center
point, we would observe no rotation: the center of mass would simply be at
rest. Thus in rotational motion, much more than in translational motion, we consider
objects not as particles, but as rigid bodies. We must take
into account not only the position, speed and acceleration of a body, but also
its shape. We can thus formalize our definition of rotational motion as such:
A rigid body moves in
rotational motion if every point of the body moves in a circular path with a
common axis.
This definition clearly applies to a bicycle
wheel, due to its circular symmetry. But what about objects without a circular
shape? Can they move in rotational motion? We shall show that they can by a
figure:
Figure
%: An arbitrarily shaped object rotating about a fixed axis
The figure shows an
object with no circular symmetry, rotating 90 o about
a fixed point A. Clearly all points on the object move about a fixed axis (the origin
of the figure), but do they all move in a circular path? The figure shows the
path of an arbitrary point P on the object. As it is rotated 90 o it
does move in a circular path. Thus any rigid body rotating about a fixed axis
exhibits rotational motion, as the path of all points on the body are circular.
Now that we have a clear definition of exactly
what rotational motion is, we can define variables that describe rotational
motion.
Rotational Variables
It is possible, and beneficial, to establish variables
describing rotational motion that parallel those we derived for translational
motion. With a set of similar variables, we can use the same kinematic
equations we used with translational motion to explain rotational motion.
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