42 STRENGTH OF MATERIALS 



easily be measured by means of a planimeter. Also let N denote t In- 

 static moment of the original section with respect to the liiu* -I//, 

 where the static moment an area with respect to any axis - 

 defined by the integral 



N= CydF, 



in which y is the distance of an infinitesimal area dF from the - 

 axis. The static moment is thus equal to the area of the section 

 multiplied by the distance of its center of gravity from the p. 

 axis. Then 



N = CydF = Cyzdy = / Cafdy = IF'. 



But, from the above definition, 



N=cF, 



where c is the distance of the center of gravity of the original sec- 

 tion from the line AB. Therefore cF= IF' ; whence 



5. 



which determines the position of the center of gravity. 



To find the moment of inertia, make a second transformation by 

 constructing for each z 1 a value z", such that 



Then the points P' on the first transformed curve are transformed 

 into a series of points P" on another curve, shown by the broken 

 line in Fig. 26. Let the area of this second curve be denoted 1- 



Then, since z" = z' | , and z' = z '-j , we have " = z \ Consequent 1 y. 







/ = 



JV'rfy = //"', 



which gives the moment of inertia of the original section with respect 



to the line AB. 



