(i.e. magnitude) and the axis of rotation (i.e. direction). In Figure 1.1, the speed of rotation is 
(1.4)
where the right handed set of unit vectors,
Figure 1.2 Even 2D problems are 3D.
Using this definition of the angular velocity, the motion of the tip of vector
which, by the rules of the vector cross product, has magnitude,
and a direction that, according to the right hand rule2 used for cross products, is perpendicular to both
Combining these two terms to get
(1.5)
The time derivative of any vector,
It is important to understand that the angular velocity vector,
1.2 Performing Kinematic Analysis
Before proceeding with examples of kinematic analyses we state here the steps that are necessary in achieving a successful result. This first step in any dynamic analysis is vitally important. The goal is to derive expressions for the absolute velocities and accelerations of the centers of mass of the bodies making up the system being analyzed. In addition, expressions for the absolute angular velocities and angular accelerations of the bodies will be required. It is at this first step of the analysis that degrees of freedom are defined and constraints on relative motion between bodies are satisfied.
For this general description of kinematic analysis, we assume that we are analyzing a system that has multiple bodies connected to each other by joints and that we are attempting to derive an expression for the acceleration of the center of mass of a body that is not the first in the assembly.
The procedure is as follows.
1 Find a fixed point (i.e. one having no velocity or acceleration) in the system from which you can begin to write relative position vectors that will lead to the centers of mass of bodies in the system.
2 Define a position vector that goes from the fixed point, through the first body, to the next joint in the system. This is the position of the joint relative to the fixed point.
3 Determine how many degrees of freedom, both translational and rotational, are required to define the motion of the relative position vector just defined. The degrees of freedom must be chosen to satisfy the constraints imposed by the joint that connects this body to ground.
4 Define a coordinate system in which the relative position vector will be written and determine the angular velocity of the coordinate system.
5 Repeat the previous three steps as you go from joint to joint in the system, always being careful to satisfy the joint constraints by defining appropriate degrees of freedom.
6 When the desired body is reached, define a final relative position vector from the joint to the center of mass.
7 The sum of all the relative position vectors will be the absolute position of the center of mass and the derivatives of the sum of vectors will yield the absolute velocity and acceleration of the center of mass.
1.3 Two Dimensional Motion with Constant Length
Figure 1.3 shows a rigid rod of length,
