Find out the product of inertia of a rectangular area, Mechanical Engineering

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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

551_Find out the product of inertia of a rectangular area.png

= 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


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