224 



PRACTICAL MATHEMATICS 



121. Let P be a point on a lamina about which there is a small 

 elementary area a. (Fig. 57.) 



Let G be the centroid of the lamina, and GX, GY, be two axes 

 at right angles to one another, drawn in the plane of the figure. 

 The co-ordinates of P with reference to the axes GX and GY 

 are x and y. 



Let O be any point in the lamina, and the axes OX and OY be 

 drawn parallel to the axes GX and GY respectively. 



The moment of inertia of the elementary area a about the 

 axis GY is ax 2 , then the moment of inertia of the whole area 

 about that axis is *Lax 2 , 



or 



IGY - 



Yl 



FIG. 57. 



Also the moment of inertia of the elementary area a about the 

 axis OY is a(x + I) 2 where I is the distance between the two 

 parallel axes GY and OY. 



Hence I OY = 'La(x + I) 2 



= 'Lax 2 + ZXalx + Lai 2 



= Lax 2 + 2lLax + l 2 La 



= I GY +21A.X+ I 2 A 



= I GY +AJ 2 (1) 



Since Ax = as the axis GY passes through the centroid. 

 Similarly if ra is the distance between the axes OX and GX, 



then I ox = I GX + Aw 2 (2) 



Let GZ and OZ be axes drawn perpendicular to the plane of 

 the lamina through the points G and O respectively. 



