Art. 72. 



102 MOMENTS OF INERTIA. 



For a triangle, Fig. 74, 



or one-third as much as for a rectangle of the same base and 

 height. 



For a circle, Fig. 75, 



r+r rl 



= I v 2 xdv = 4: I r 2 sm 

 J -r JO 



f 

 7 = 4r 4 / 



JO 



r cos<J>d<f> 



d <> = 



The moment of inertia is four times that of one quadrant. 

 In like manner / for a rectangle is twice that for half of it, and 

 it follows that / for a rectangle about one edge is-^ bh 3 in which 

 h is twice the height, that is 



h = 2hi and 1= 



It is often necessary to find moments of inertia for axes 

 parallel to the neutral axes. In Figs. 72 and 73, for example, 

 the neutral axis A T JV of the combination is the one about which 

 the moments of inertia of the elements of area (rectangles and 

 angles) is desired. Equation (11) is applicable; thus in Fig. 67, 



idA+d 2 I dA 



/- 

 - 



/+vi r+vi r-H 



(v+d)*dA = -- I v 2 dA+2d I vd 

 -V2 J V2 J V2 



r+vi r+vi r+vi 



Sinre / v 2 dA = /, / vdA = and / dA = A 



J V2 J V2 J V2 



[See equations (11) and (13)]. 



l^^I+Ad 2 (16) 



This equation is of much practical importance and the stu- 

 dent should be familiar with its meaning. Equation (16) means 

 that the moment of inertia of any area, about an axis parallel to 

 the neutral axis, is equal to the moment of inertia of the area 

 about its neutral axis, plus its area times the square of the dis- 

 tance between the axes. 



