﻿On the Rotation of Slightly Elastic Bodies. 31 



novelty attaches to some of the earlier results, it seems 

 desirable to include them. 



The simplest problem of! this nature is, of course, that of 

 the thin circular hoop rotating about its centre. When such 

 a hoop of radius a and density p is spun round its centre 

 with constant angular velocity w the value of T, the tension 

 per unit length in the Loop, is well known. For an element 

 ds of its length has an acceleration aco 2 inwards, and the 

 resultant of the tensions at its ends is Tds/a per unit area 

 inwards. Hence the equation of motion is 



Tds/a = paco 2 , 



giving T = pa 2 co 2 . 



If, however, the hoop is slightly elastic, and X the value of 

 Young's modulus for the material of which the hoop is 

 made, and r the radius of the hoop when in motion, the 

 equation of motion of the stretched element ds becomes 



T/V = ?'a> 2 . pair. 



Applying Hooke's law to the stretched element, we have, 



T = X(r-a)/a. 

 Hence eliminating T, 



prato 2 = X(r — a) /a. 



In practice X is always large, and if we may neglect 1/X and 

 higher powers of 1/X the appropriate value of r/a, which 

 differs from unity by a quantity of order 1/X, is 1 + pa 2 co 2 /X. 

 The value of the tension to the same order is pa 2 co 2 . 



The effect of a rotation is therefore to increase the radius a 

 of the hoop to a(l +//,), where fj, = pa 2 co 2 l\, a number depending 

 on the density, the elasticity, and the radius of the hoop, and 

 on the rate at which it is rotating. 



As regards the practical order or magnitude of pa 2 (o 2 /X 

 the extension per unit length, we may take a steel wire for 

 which X is about 2*12 xlO 12 dynes per square centimetre, 

 and p is about 7*5. In order that Hooke's law may hold, the 

 extension per unit length must not exceed 10~ 3 , roughly 

 speaking. If the velocity of a point on the rim is in the 

 neighbourhood of 1*9 x 10 4 cm. per second — which is ap- 

 proximately the case in a twenty-foot flywheel making two 

 hundred and fifty revolutions a minute — we find that the 

 extension per unit length is about 7*9 XlO -4 , which comes 

 within the limits of applicability of Hookers law, and that 



