MOMENTS AND CENTROIDS 53 
If the axis passes through the centre of the rod instead of through 
WE 
one end, it follows that I= 17 and i? = 13° 
2) Rectangle or parallelogram (Fig. 55) base of length a, and 
( 
altitude >, about an axis X,X,, coinciding with the base 
Consider an element of width dx parallel to the axis, and at a 
Xi 
| 
a 
2 
; 
Ss Le 
x 
Fig. 56. 
Fia. 55. 
distance z from it. The area of this element is adz, its moment of 
inertia is az*dx, and the total moment of inertia 
a es ab b? 
I,= [jeotae= = af’ ede = 3 and k}= 7" 
If the axis passes through the centre of gravity of the rectangle or 
parallelogram and is parallel to the base, then it follows that 
abs 3 8 
={9? and k? = 73° 
(3) Triangle (Fig. 56) base of length a, and altitude }, about an 
axis X,X, coinciding with the base. 
Consider an element of width dz parallel to the axis, and at a 
a(b — w)da 
distance « from it. The area of this element is its moment 
ae , and the total moment of inertia 
of inertia is 
_(? a(b -x)a*dx _a “(F- r)-% 
Fi faeces ‘(ate - dz) =9( oa 
0 
Tf the axis XX passes aan the centre of gravity G of the triangle 
and is parallel to the base, then by Theorem IT., Art. 68, p. 51, 
=1+00(2) _ abs 
1=1+ Jar . Therefore [= 36 
If the axis X,X, passes through the vertex of the triangle and is 
' parallel to the base, then 
2b\?_ al®  2ab® ab 
= i = : 
T,=1+ ja (F) = 36 +—— = ise 5 
