238 



PRACTICAL MATHEMATICS 



f R 



For the whole circle I oz = 2:r I a? dx 



Jo 



= ~AR 2 , where A is the area. 



For a circular ring, if R 2 and R x be the external and internal 

 radii respectively, I oz = 27t I a? dx 



^A(R 2 + R 2 ), where A is the area. 



FIG. 68. 



Fig. 68 shows the circle placed in isometric projection, OX and 

 OY being two axes drawn at right angles to one another in the 

 plane of the figure. Because the circle is symmetrical about any 

 diameter, the moment of inertia is the same for all diameters. 



Hence 



" * 



But 



Therefore 



+ 



1 OY 



= 2I OX or2l oy 

 1 



)Y - g L OZ 



