Angular Velocities: wC/A = wC/B + wB/A
Angular velocity is a vector that describes the rotational motion of one rigid body relative to another rigid body. The magnitude of the angular velocity vector is equal to the rate at which the one body is rotating relative to the other. The direction of the angular velocity vector is the direction of the axis of rotation of the one body relative to the other. Unfortunately for rigid bodies traveling merrily through space, it is often very difficult to identify or visualize the angular velocity. Say you become involved in an altercation and someone tosses a chair at you. It is clear that the chair is rotating as it approaches you but to identify at any time its axis of rotation and rate of rotation is a daunting challenge! One suspects that both the rate of rotation and axis of rotation are changing as the chair approaches but more than that - who can say?
Presuming we have avoided the chair, the formula given above can rescue us. Let us first define the terms:
wC/A = Angular velocity vector of a rigid body named C relative to a rigid body named A. Note that in discussing the motion of rigid bodies we can equally well discuss the motion of a coordinate system attached to the body. If the body rotates, then so does the coordinate system attached to it. Thus we can read the formula as the angular velocity of a coordinate system C relative to a coordinate system A. In many problems the reference coordinate system A will be a system attached to the ground (a fixed or inertial reference system). The magnitude of the vector at any instant will be the rate at which system C is rotating relative to system A at that instant. The direction of the vector at any instant will be the direction of the axis about which system C is rotating relative to system A at that instant.
wC/B = Angular velocity vector of that body (or coordinate system) C relative to some other intermediate rigid body (or coordinate system) B.
wB/A = Angular velocity vector of the intermediate rigid body (or coordinate system B) relative to the rigid body named A.
Thus the formula states that the angular velocity of system C relative to system A is equal to the sum of two angular velocities, that of system C relative to system B plus that of system B relative to system A.
Great, but how does this help us describe the complex rotational motion of a rigid body? One can often introduce intermediate coordinate systems such that the overall angular velocity can be decomposed into a sum of relative angular velocities. Hopefully for each of these relative angular velocities both the rate of rotation and the axis of rotation will be "obvious".
What in the world do we mean by introducing "intermediate" coordinate
systems? And how can the rates of rotation and axes of rotation be
"obvious"? This is best illuminated by considering an example.
Spherical coordinates are often used to describe the location of a particle.
Spherical coordinates are defined in the text in Figure 2/16 on page 81.
Formulas for the velocity and acceleration in spherical coordinates are
defined in Equations 2/18 and 2/19 on page 83. As we shall see these
equations can be developed based on the concept of angular velocity combined
with some knowledge about how to differentiate vectors. For now we
will restrict ourselves to a study of the coordinate system involved in
spherical coordinates. Click here to enter
the mystical world of spherical coordinates.
Copyright (1998) by Nels Madsen. Last Updated:
February 28, 1998