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Area Moments of Inertia in Rectangular and Polar Coordinates

Let us consider a thin lamina of area A as shown in Fig. 7.1. We take a differential area dA which is at a distance x and y from y and x axis

Moment of inertia of d A about x axis = dIx = y2 dA. . .

Total moment of inertia of A aboutx axis

= Ix = ʃ  dIx = ʃ y2 dA

 

Moment of inertia of dA about y axis= dIy = x2 dA :.

Total moment of inertia of A about y axis

= Iy = ʃ dIY =ʃ x2 dA

 

The moment of inertia of d A about 0 (z-axis)

= dIz = r2 Da

 

Total moment of inertia of A about 0 (z-axis)

Iz = ʃ dIZ = ʃ r2 da

 

In summary ,   I x = ʃ y2 dA

Iy = ʃ x2 dA

Iz = ʃ r2 dA

Iz = ʃ r2 dA

= ʃ (x2 +y2 )Da

= ʃ ( x2 dA + ʃ y2 dA

= I= +Iy

Iz = Ix +Iy

 

S.I. unit of area moment of inertia

= L4 where L= length unit (= m)

= m4