﻿32 Dr. Dorothy Wrinoh on the 



the actual increase in the radius is about a fifth of an inch. 

 In this case the tension is about 1*6 x 10 9 . 



It is further evident that </ n \/p is the largest velocity 

 if an extension of more than n per unit length is to be 

 avoided. When the elastic limit for the material is known, 

 this result can be used to give an upper limit to the velocity 

 it is safe to use if risk of deformation of the hoop is to be 

 avoided. 



We may now proceed to the problem of a thin rod rotating 

 about one end with uniform angular velocity. 



Thin Mod Rotating about One End. 



Let a be the unstretched length of the rod, co the angular 

 velocity of rotation about one end 0, p the density when it 

 is unstretched, and X the value of Young's modulus for the 

 material of which the rod is made. Let T be the tension 

 in any section in the rod during the motion. Let the 

 distances of the same particle at rest and in motion be 

 x and x. The density of the moving element dx is 

 podx^/dx and its acceleration towards is xco 2 . The equa- 

 tion of motion of the element is therefore, 



ST = — p Bxq . xco 2 , 



where, by applying Hooke's law to the element originally of 

 length dx and now of length dx, we have 



T = \(dx/dx -l). 



Hence, eliminating T, we obtain the equation, 



d 2 x/dx 2 = — p Q m 2 x / A . 



The solution must give the value x = a when x = a if a is 

 the length of the rod when in motion. Accordingly it is 



x = asm(x V p (o 2 / '\) / sin (a Q ^p Q oo 2 j\). 



We may determine the value of a by means of the condition 

 that the tension vanishes at the free end, which is given 

 indifferently by x = a or x = a . Thus, 



« \/p G) 2 /\ = tan a \^p co 2 /\. 

 The equation relating the two corresponding positions of a 



