| dc.description |
A RECENT communication<sup>1</sup> discusses the uniform rotation of a rigid circular disk as a problem in special relativity. Phipps is in error in supposing that there is more than one way to measure the peripheral velocity in a Lorentz frame in which the centre of the disk is at rest. He will see that the velocities v and rÏ are equal (if it is not at once obvious from the kinematics in S) by drawing a Minkowski diagram showing the approximately parallel world lines of adjacent markers. These move with velocity v in S, and are at rest in S<sup>*</sup>. S<sup>*</sup> ascribes a spatial displacement Î x<sup>*</sup> to the pair of events consisting of successive transits of a point fixed in S. Since the two markers are at rest in S<sup>*</sup> Î x<sup>*</sup> may instead be determined from a pair of events in their histories simultaneous in S<sup>*</sup>. But then Î x<sup>*</sup> is just the rest displacement, namely, 2Ï r/nâ 1â v<sup>2</sup>/c<sup>2</sup>. This reflects the well-known fact that in a rigid rotating (but non-holonomic) co-ordinate system the ratio circumference : radius is given by 2Ï : â 1â v<sup>2</sup>/c<sup>2</sup>. Substituting the above value for Î x<sup>*</sup> into the Lorentz formula, we find that the â round-tripâ velocity rÏ and the relative velocity v are in fact equal. |
