Rotational Kinetics

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.

Angular Displacement

The most important constraint placed on us when developing these variables is that they have to be a property of the object: any point on the object must have the same value for the variable. We therefore cannot use our old variables, such as velocity, because some parts of a rotating disk move at different speeds than others, and a single number for velocity would not describe the motion of the entire disk. So what is a property of every point on a rotating object? Since every point rotates in a circle about a common axis, we can say that the angular displacement is the same for any point on a rotating object. That is, the angle which each point sweeps out in rotating is the same at any given time for any point on the object:

Figure %: Point P on an object sweeping out an angle μ
The figure shows point P, located a distance r from the axis of rotation, moving a distance s as it rotates. The angle swept out by the point, which is the same for any point on the object, is given by:

 μ =

Where s is the arc length shown in , r is the distance from the point to the axis of rotation, and μ is the measure of the angle. Note: Up to this point we have measured angles in degrees. We now introduce a new, more useful measurement called a radian. A radian is defined by the following relation:

 1 revolution = 2Π radians = 360 o

90 degrees is equivalent to Π/2 radians, 180 degrees is equivalent to Π radians, etc. By convention, we define the positive direction of rotation to be counterclockwise.

Angular Velocity

Angular displacement is an equivalent quantity to linear displacement. Indeed, by taking the linear displacement of a given particle on an object and dividing by the radius of that point, we derive angular displacement. The equivalency between linear and angular displacement leads us to a further realization: just as we define linear velocity from linear displacement, we similarly define angular velocity from angular displacement. If an object is displaced by an angle of Δμ during a time period of Δt , we define the average angular velocity as:

 =

And, using calculus, we define the instantaneous angular velocity as:

 σ =

Like angular displacement, angular velocity is identical for every point on a rotating object, and essentially describes the rate at which an object rotates.

Angular Acceleration

The rotational corollary of linear acceleration is angular acceleration, the rate of change of angular velocity. In the same manner as we derived the equations for average and instantaneous velocity, we define angular acceleration:

 = α =

These equations for angular displacement, velocity, and acceleration bear striking resemblance to our definitions of translational variables. To see this, simply substitute x every time you see μ , v every time you see σ , and a every time you see α . The yield are the translational equations for displacement, velocity, and acceleration. This similarity will allow us to easily derive kinematic equations for rotational motion

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