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- TL;DR Summary
- How do linear and circular motion compose in SR?

I have a pencil of Iron of length ##L## rotating about its center in a plane at constant angular velocity ##\omega##. The tip of the pencil in Newtonian mechanics has linear velocity ##\frac{\omega L}{2}##. It can exceed ##c##, of course.

Now let us complicate this. Assume the center of the pencil (thus the pencil as a whole) moves linearly along the Ox axis (while rotating at ##\omega##) at constant velocity ##c-\epsilon## in the Lab frame.

What is the linear speed of tip while in motion, calculated in the lab frame in terms of ## c, \epsilon, \omega, L##? Pretty sure it cannot exceed ##c##.

It is tempting to say that in the lab frame the radius of the disk generated by the rotating pencil shrinks to ##\gamma L/2##, but how would the linear velocity of the tip be? It cannot be simply by multiplication with ##\omega##, it would exceed ##c##.

Now let us complicate this. Assume the center of the pencil (thus the pencil as a whole) moves linearly along the Ox axis (while rotating at ##\omega##) at constant velocity ##c-\epsilon## in the Lab frame.

What is the linear speed of tip while in motion, calculated in the lab frame in terms of ## c, \epsilon, \omega, L##? Pretty sure it cannot exceed ##c##.

It is tempting to say that in the lab frame the radius of the disk generated by the rotating pencil shrinks to ##\gamma L/2##, but how would the linear velocity of the tip be? It cannot be simply by multiplication with ##\omega##, it would exceed ##c##.