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Defining Rotation and its Variables

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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.