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 x_G  y_G

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 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)= - b² d ²/4