THE MOMENT OF INERTIA OF A CIRCLE 237 



Distance of centroid from axis OX 



_L _L_ ( 8 \ 3 ? f^ 2 



lox " 12 X A/2 X Wl/ + 2 X W2/ 



3 



2V2 









16 



= 2-25 



For the whole section I ox = 2 (2-25 + 0-628 } 



= 5-756 

 Let the axis OY be drawn perpendicular to OX. 



Then I OY + I x = JOL + ! O K 



and I OY = 7-264 - 5-756 



= 1-508 



Hence the principal moments of inertia are 5-756 and 1-508. 

 To draw the momental ellipse, 



3-25 



9 = 



5-756 

 3-25 



1-508 



=0-751 



= 1-468 



and these are the semi-minor and semi-major axes to be measured 

 along OX and OY respectively. 



127. The Circle. Consider an elementary ring of width Sa; 

 bounded by concentric circles of radii x and (x + $x) respectively. 



FIG. 67. 



Area of the ring = 2rcr &r. 



Moment of inertia of the ring about an axis OZ which is 

 perpendicular to the plane of the circle and passes through the 

 centre = 



