﻿Rotation of Slightly Elastic Bodies. 33 



typical element when at rest and when in motion and the 

 original length of the rod is therefore 



x \'p co' 2 /\ = sin (d? y/p w 2 /\) / cos (a vW*>7\)- 



Neglecting the cube and higher powers of l/\, we may 

 replace this by the simpler form, 



as = a'o + 'i'opo(o 2 (3a 2 — tf 2 )/6\ 



to the order l/\. To the same order, 



T = p co 2 a 2 [1— x 2 la 2 ] . 



The greatest extension is ^p a) 2 a 2 /\, and this occurs at 

 the end about which the bar is rotating. The tension is also 

 greatest at this point and takes there the value p co 2 a 2 . 



As an example of the actual magnitudes of the quantities 

 in practical cases we may take a twenty feet steel bar, which, 

 when rotating about one end two hundred and fifty or three 

 hundred times a minute, increases in length about a tenth of. 

 an inch. 



dotation of an Infinite Elastic Circular Cylinder 

 about its Axis. 



Passing now to a simple problem in three dimensions, we 

 take the case of an infinite elastic cylinder of circular section 

 rotating about its axis. We may consider one of the circular 

 sections of the cylinder and use polar coordinates. At any 

 point (r, 0) let r l\ and T 2 be the transverse and radial ten- 

 sions per unit length, and T 3 the axial tension. We shall 

 consider the motion of the element of volume which when 

 at rest is bounded by the surfaces (z,z + 8z), (r, r + 6>), 

 (0, + 80). By the symmetry of the cylinder, the element 

 when in motion will continue to be bounded by the surfaces 

 (0, + S0) : and since the cylinder is of infinite length, 

 the element will continue to be bounded by the surfaces 

 (z, z + hz). Let p represent the radial dimension, so that 

 p — r is the radial extension at any point. Let a be radius 

 of the cylinder and a its density, when at rest ; let &> be the 

 angular velocity of the cylinder about its axis, and X and /x 

 the elastic constants for the material of which the cylinder 

 is made. 



The element of volume which we are considering is a 

 parallelepiped of sides dp, pd0, and dz. The forces on our 

 element of volume consist of (1) transverse tensions each 



Phil. Mag. S. 6. Vol. 44. No. 259. July 1922. D 



