. The London, Edinburgh and Dublin philosophical magazine and journal of science. hB, any length ab, at a distance r from 0, will contract fromr . 80 to r . S6a / 1 2 wnen the disk is rotating, so that ab is moving with linear velocity v. As Ehrenfest pointed out, the Oa will have no tendencyto change, and if this condition is to be fulfilled it is im-possible for the disk to remain in the plane form. It mustassume a cup-like form, whose horizontal sections will, fromsymmetry, be circles, and whose shape is such that ab has / v2contracted to ab\/ 1 2, while Oa is unaltered. * Communicated by t
. The London, Edinburgh and Dublin philosophical magazine and journal of science. hB, any length ab, at a distance r from 0, will contract fromr . 80 to r . S6a / 1 2 wnen the disk is rotating, so that ab is moving with linear velocity v. As Ehrenfest pointed out, the Oa will have no tendencyto change, and if this condition is to be fulfilled it is im-possible for the disk to remain in the plane form. It mustassume a cup-like form, whose horizontal sections will, fromsymmetry, be circles, and whose shape is such that ab has / v2contracted to ab\/ 1 2, while Oa is unaltered. * Communicated by the Authors, Problem of Uniform Rotation. 93 If, therefore, AOA represent the vertical section of thefinal form of the disk containing the axis of rotation OX,. we shall have Oa measured along the arc equal to 9% while / v*~aB measured perpendicular to OB will be r\f 1 -%. In this way both the conditions demanded by the relativityprinciple will be satisfied. Writing Oa = s and aB = y, according to the usualnotation, we have y /c2 — v2 /c2 - ?/20>2 o) being the angular velocity of the disk. y2c2 = GO Differentiating, and arranging terms, we have dy or (c3 + o>y>^=S(C*-yV) (c2 + (o2s2)y cos 0 = s(o2 — ;y2&)2). a;.) Substituting in (ii.) the value of y from (i.) we have,taking the positive root of the equation, (J + ,JJ\J^C0SJ,- = (<3- 0,W V whence (c2 + &)V)3/2cosj) = c3. .... (iii.) This gives the intrinsic equation of a section of the diskwhen rotating with angular velocity w, and contains noapproximations. 94 Problem of Uniform Rotation. Case I.—When the velocity of any point on the disk issmall compared with the velocity of light, we have A. *C0S <P = 7 2 ;., • *(i+!ST The conditions of this case
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