94 Problem of Uniform Rotation. 



Case I. — When the velocity of any point on the disk is 

 small compared with the velocity of light, we have 



A. * 

 C0S <P = 7 2 <k ■>;., • 



*( i+! ST 



The conditions of this case will be satisfied if as is small 

 compared with c. 

 Thus we may write 



cos4>=l-3/2°J, 



c 



whence 



2 c . 6 

 s = — ;=■ - sin £ (iv.) 



This indicates that the form of the vertical section in this 

 case is a curve of the cycloid family, an epicycloid. 



Case II. — When the velocity of a point on the outer part 

 of the disk approaches the velocity of light, since we have 



56* 



and v = ya } 



we pet 



2*2 



or.s 



» 



sea 



v = v^ 





From this we see that for all values of s which differ from 

 zero by any finite quantity, v = c when a is infinite. Thus 

 no point on the disk can be made to move with a velocity 

 greater than that of light, which is exactly what w r ould be 

 expected from relativity principles. 



Further, from the equation 



1 



( 1+ "^y 



we see that, when a becomes very large, cos <j> is small, and 

 is also sensibly independent of 5, unless s is very small. 

 Hence, when the angular velocity becomes very large, the 

 disk approaches the form of a right circular cone of small 

 angle, except near the centre of the disk. 



When w is infinite, all points at a finite distance from the 

 centre of the disk are at zero distance from the axis of 



