# Rotational Motion and Angular Momentum

## Kinematics of Rotational Motion


### Learning Objectives

By the end of this section, you will be able to:
1. Observe the kinematics of rotational motion.
2. Derive rotational kinematic equations.
3. Evaluate problem solving strategies for rotational kinematics.
Just by using our intuition, we can begin to see how rotational quantities like , , and  are related to one another. For example, if a motorcycle wheel has a large angular acceleration for a fairly long time, it ends up spinning rapidly and rotates through many revolutions. In more technical terms, if the wheel’s angular acceleration  is large for a long period of time , then the final angular velocity  and angle of rotation  are large. The wheel’s rotational motion is exactly analogous to the fact that the motorcycle’s large translational acceleration produces a large final velocity, and the distance traveled will also be large.
Kinematics is the description of motion. The kinematics of rotational motion describes the relationships among rotation angle, angular velocity, angular acceleration, and time. Let us start by finding an equation relating , , and . To determine this equation, we recall a familiar kinematic equation for translational, or straight-line, motion:
Note that in rotational motion , and we shall use the symbol  for tangential or linear acceleration from now on. As in linear kinematics, we assume  is constant, which means that angular acceleration  is also a constant, because . Now, let us substitute  and  into the linear equation above:
The radius  cancels in the equation, yielding
where  is the initial angular velocity. This last equation is a kinematic relationship among , , and  —that is, it describes their relationship without reference to forces or masses that may affect rotation. It is also precisely analogous in form to its translational counterpart. 
Starting with the four kinematic equations we developed in One-Dimensional Kinematics, we can derive the following four rotational kinematic equations (presented together with their translational counterparts):
In these equations, the subscript 0 denotes initial values (, , and  are initial values), and the average angular velocity  and average velocity  are defined as follows:
The equations given above in  can be used to solve any rotational or translational kinematics problem in which  and  are constant.
There is translational motion even for something spinning in place, as the following example illustrates.  shows a fly on the edge of a rotating microwave oven plate. The example below calculates the total distance it travels.

### Section Summary

1. Kinematics is the description of motion.
2. The kinematics of rotational motion describes the relationships among rotation angle, angular velocity, angular acceleration, and time.
3. Starting with the four kinematic equations we developed in the One-Dimensional Kinematics, we can derive the four rotational kinematic equations (presented together with their translational counterparts) seen in .
4. In these equations, the subscript 0 denotes initial values ( and  are initial values), and the average angular velocity  and average velocity  are defined as follows:
    

### Problems & Exercises
