Find out the product of inertia of a rectangular area:
Find out the product of inertia of a rectangular area b × d with respect to its sides as illustrated in Figure
Solution
Let sides OA and OB is the reference axes of area for the needed product of inertia.
∴ I _{x,y (0)} = ∫ dA xy
= b^{2} d ^{2} /4
If G is the centroid of the area, then I _{xy} (G) = 0 by virtue of symmetry.
On the other hand,
I _{x,y (0) }= I _{xy (G )} + A x_{G} y_{G}
where, obviously
x _{G} = b/2
y _{G} = d /2
I _{x y (0) } = bd × (b/2) × ( d/2)
= b^{2} d ^{2}/ 4
Though, product of inertia w. r. t. axes BC and Y is negative as illustrated below.
I _{x y (B)} = I _{xy (G )} + A x_{G}′ y_{G′}
where with respect to B,
x_{G}′ = b/2 ; y_{G}′ = ( - d/2)
∴ I x y (B) = (bd ) (b/2)(-d/2)= - b^{2} d ^{2}/4