FIEST AND SECOND MOMENTS 



21 



of the centroid G from any side is one third the distance of the 

 opposite vertex from that side. 



15. Centroid of circular arc. For a circular arc CD (Fig. 13) the 

 centroid G must lie on the diameter OF bisecting the arc. Now 

 suppose the arc divided into small segments, and from the ends 

 of any segment PQ draw PR 

 parallel to the chord CD, and 

 QR perpendicular to this chord. 

 Since the moment of the entire 

 arc with respect to a line AB 

 drawn through perpendicular 

 to OF must be equal to the sum 

 of the moments of the small seg- 

 ments PQ with respect to this 

 line, the equation determining 

 the centroid is 



X Q arc CD = V PQ X x. FIG. 13 



But from the similarity of the triangles PQR and OQE we have 



PQ = OQ = r 

 PR QE x 



Therefore PQ - x = PR r, and consequently ] PQ - x = ]T PR r, 

 or, since the radius r is constant, 



2) PQ ' x = r^ PR = r - chord CD. 

 The position of the centroid is therefore given by 



chord 





(15) 



. radius. 



arc 



If the central angle COD is denoted by 2 a, then arc=2ra and 

 chord = 2 r sin a, and therefore the expression for the centroid may 

 be written 



For a semicircle 2 a = TT, and consequently 



