﻿248 Prof. J. G, Gray on the 



rotated about in its own plane through a small angle 0, 

 so that A is brought to A' and B to B'. In effect the small 

 portion AA' of the arc is transferred from one end of the arc 

 to the other. The mass of this element is mrd, and it has 

 been translated (virtually) through the distance 2r sin a/2, 

 where a is the angle AOB. If G is the centroid of the arc, 

 we have obviously 



mrO 2r sin a/2 = mra OG 0, 



or 



0G = 



2r sin u/2 



As a second example we take the case of a sector of a 

 circle OADC (fig. 3). Let the sector be turned in its own 

 plane through a small angle 6 about an axis through 0, so 

 that A comes to A', B to B\ The effect of the rotation has 

 been (virtually) to transfer the triangle OAA' to OOC. 



Fig-. 3. 



The centroid G of the sector has moved parallel to gg' 

 through a distance OG 6. The mass of the sector is JrW, 

 and that of the triangle OAA' is ^r 2 6cr, where a is the mass 



of the sector per unit area. 



Since aa' = k t sin « , we have 



99 



-r 2 6<r - r sin - = - r 2 u<r OG 0, 



or 



OG = 



4 r sin a/2 



Again, let it be required to find the position of the centroid 

 of a segment of a circle ABC (fig. 4). The segment is 

 turned in its own plane, about 0, through a small angle 0. 

 A is thus brought to A f and to 0\ If the mass per unit 

 of area of the segment is cr, the mass of the triangle DAA' 



