Solving the "Paradox" in Galilean Relativity

[bigerst]

seems to be a "paradox" in galilean relativity

Hello

I'm having a little bit of trouble with so-called rest frames. I will distinguish two cases.

consider frame S, and a particle moving along the x-axis at speed v.

Case I: consider the rest frame S' traveling along with x at speed v, so in S' x remains at rest in the origin. this is typically the "rest frame" in relativity

Case II: this is what I consider to be a bit of paradoxical nature, consider the set of all frames such that
a)the particle is moving at v in the x direction
b)the particle is at the origin of this frame
c)this frame is at rest in S

naturally at any given time there is only 1 frame that satisfies these conditions, interestingly, if we let time go forward and mark out the origin of these frames with a red dot, the red dot travels at precisely v and in fact at any time it overlaps with the particle.

of course there is nothing too paradoxical about the argument, not one single frame is moving. relative velocity is still v. first am I correct in all of the above reasoning?

well here comes the part that bugs me, and it comes from Taylor's classical mechanics book on rotational dynamics. it states (if i interpreted correctly) that in a rotating body's "body frame" in which the axis are defined by the body's principal axis of rotation, there can be still a non-zero angular velocity. if that is true then i think it is evoking the "Rest frame" scenario of case II. however, it then uses the transformation properties of vectors of a truly self rotating frame, which is analogous to Case 1. So i don't get it, what am i missing here?

thanks

 

[mfb]

I don't understand where the problem is.
For a rotating body, there is no inertial frame where the whole body is at rest, as different parts of the body move with different velocity in all inertial frames (relative velocities are independent of the frames).
This is different from non-rotating objects.