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
= b2 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 xG yG
where, obviously
x G = b/2
y G = d /2
I x y (0) = bd × (b/2) × ( d/2)
= b2 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 xG′ yG′
where with respect to B,
xG′ = b/2 ; yG′ = ( - d/2)
∴ I x y (B) = (bd ) (b/2)(-d/2)= - b2 d 2/4