Overview

In order to apply Newton laws to particles or bodies we must be able to determine their velocities and accelerations. In some cases these kinematics are described in stationary 3-dimensional Cartesian frames and applying the Newton laws is straightforward. However in other cases deriving the kinematics in a stationary frame is not easy. An example is a particle that moves on the surface of a horizontal platform that revolves about its vertical axis. A more pragmatic example is a top spinning on a horizontal surface.

To simplify the analysis we shall study the motion of the frame and the motion of the body and the particle separately. We suppose that a frame rotates about some axis. As it rotates we take two consecutive snap shots of this frame one at times t and t+ ? t and we denote the frames at these two instances as S and S'. In this scenario it is assumed that the two frames share the same origin. In frame S, the unit vectors along the ( x, y, z) axes are ( i, j, k) respectively. These unit vectors at time t+ ? t (in frame S') become ( i', j', k'). We shall consider three special rotations: the first about the x-axis, the second about the y-axis and the third about the z-axis. Referring to Fig. E.1, we suppose that in the time interval [