39-] DETERMINATION OF CENTROIDS. ! 9 



C x * C y * C x * C v * 



M-x\ I pxdxdy, M-y = \ \ pydxdy; 



y^j *Jy^ */*i \Jv\ 



and if the mass be homogeneous, i.e. p = const., since then the 

 first integration can at once be effected : 



* 



or similar expressions for y as independent variable. 



In polar co-ordinates, tKe element of area is rdrdd, and we 

 have x=r cos 6, y r sin 6 ; hence 



cos OdrdO, S *y = sm OdrdB; 



or, performing the first integration, 



Vcos&/0, 5-7=4- 



3 



It will be noticed that these last formulae express also that 

 the infinitesimal sector | r^dQ is taken as element, the centroid 

 of this element having the co-ordinates f rcosO, 



39. As a somewhat more complicated example let us consider 

 a circular disc of radius a, in which the density varies directly 

 as the distance from the centre (Fig. 9). Let a circle described 

 upon a radius as diameter be cut out of this disc ; it is required 

 to find the centroid of the remainder. 



Let O be the centre of the disc of radius a, C that of the 

 disc of radius \a\ G l the centroid of the latter, G the required 

 centroid; and put OG 1 =^ 1> OG=x. Then if M l be the mass 



