Imagine a scenario where we are talking about a driving shaft placed on a truck.
I am doing kinematic analysis on the subject and faced the following paradox:
The input shaft rotating speed is constant ( let's name it $\omega$ ), which implies a constant magnitude linear velocity for a point rotating @ a fixed radius.
Also, this implies zero angular acceleration. This means that the linear acceleration is zero as well.
Now, from my understanding of vector math:
in order for the magnitude of the derivative of a vector to be zero, the derivative of each component has to be zero, i.e
$\ |a| == 0 => \ a_x == \ a_y == \ a_z == 0 $
The problem is that with my analysis, the velocity components of v are resembling sin ofrms of functions of time, thus their time derivative is by no means zero.
I use principal rotations and coordinate transformations matrices to move from one coordinate frame to another. The one frame i am looking at, has a fixed axis ( the same one as the World reference ) w.r.t which it is rotating.