Problem of Uniform Rotation. 



93 



If, therefore, A'OA represent the vertical section of the 

 final 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 relativity 

 principle will be satisfied. 



Writing Oa = s and aB = y, according to the usual 

 notation, we have 



y 



/c 2 — v 2 /c 2 - ?/ 2 0> 2 



o) being the angular velocity of the disk. 



y 2 c 2 = 



GO 



Differentiating, and arranging terms, we have 



dy 



or 



( c 3 + o>y>^= S ( C *-yV) 



(c 2 + (o 2 s 2 )y cos = s(o 2 — ;y 2 &) 2 ). 



a;.) 



Substituting in (ii.) the value of y from (i.) we have, 

 taking the positive root of the equation, 



(J + ,JJ\J ^ C0S J , - = ( < 3- 0,W V 



whence 



(c 2 + &)V) 3 / 2 cosj) = c 3 . .... (iii.) 



This gives the intrinsic equation of a section of the disk 

 when rotating with angular velocity w, and contains no 

 approximations. 



