Find out the axial moment of inertia:
Find out the axial moment of inertia of a rectangular area of base b and height d around centroidal axis GX and the base B_{1}B_{2}.
Solution
With Reference to Figure, where centroidal axis GX divides the area at mid-depth, i.e. d/2.
Figure
For a thin strip, illustrated shaded, of width b and thickness (very small) dy, all of the points on it are at a constant distance y from axis GX,
∴ dA = bdy
Letting y as positively upward from centroidal axis GX: for elements below
GX, y shall be treated as negative.
∴ ∫ A dy = 0
=bd^{3} /12
Likewise, referring to y axis through centroid G,
I_{GY} = db3 /12
Moment of inertia around base B_{1} B_{2} may be computed either directly or by utilizing parallel axis theorem.
Direct approach is as. Referring to Figure,
I B_{1} B_{2} = ∫ dA × ( y′)^{2}
On the other hand, using theorem of parallel axis, we have
I B_{1} B_{2} = I_{GX } + A ( y_{1} )^{2}
Where y_{1} = d/2 = perpendicular distance between GX and B_{1}B_{2}.
= b d ^{3}/12 + b d (d/2)^{2} ?
= b d ^{3}/ 3