FIEST AND SECOND MOMENTS 



bh 

 2 



27 

 h 



as shown in Fig. 22, the base of each wedge being , its height -, 

 and its volume 



= base x altitude = . 



Since the centroid of a triangular wedge, like that of a triangle, is 

 at a distance of f its altitude from the vertex, 



2 h h 



Therefore 



For any plane area the /'s with respect to two parallel axes are 

 related as follows: 



Let 00 denote an axis 

 through the centroid of the 



figure, A A any parallel axis, _o f \ / o 



and d their distance apart 

 (Fig. 23). Also let I denote 

 the / of the figure with re- 

 spect to the axis 00, and I A 

 with respect to the axis AA. 

 Then, from the definition of /, 



FIG. 23 



But since 00 is a centroidal axis, ^yAA = for this axis. There- 

 fore, since y*kA = / , the above expression becomes 



(23) 



I A = I 



From this relation it is evident that the / for a centroidal axis is 

 less than for any parallel axis. 



As an application of this formula, find the / for a rectangle with 



7 70 7 



respect to its base. From what precedes, JT = . Also, d= ~ and 



1 "_ L 



A = bh. Hence the / for a rectangle with respect to its base is 



bh 3 . ,, M\ 2 6tf 

 2/ = 3 ' 



