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Let us imagine that actual area A is converted to a long narrow strip of area A. Refer Fig. 7.2.

We introduce rectangular coordinate x, y.

Refer Fig. 7.3(a)

The moment of inertia of the long narrow strip (parallel to x axis) about

x axis = K2x A

Ix = K2X A

Refer Fig. 7.3(b},

The moment of inertia of long narrow strip (parallel to y axis) about y axis

=KY2A

=K2Y A

Refer Fig. 7.3(c),

Similarly ,                                       IX = IX +A dx 2

IY =IY +A dY2

IZ =IZ +A d2

K2 =K2 +d2

Note that the axes (x’, y’) passes through the centroid of the area A where IX , Iy and are the moment of inertia of A about the centroidal axes (x’, y’, z’).

Equation (7.2) is the mathematical expression of perpendicular axis theorem which states that the moment of inertia of an area about an axis at origin normal to x, y plane is the sum of moments of inertia about the corresponding x and y axis.

Equations (7.5) are the mathematical expressions of parallel axis theorem which states that if the moment of inertia of a plane area about an axis in the plane of area through the C.G of The moment of inertia of narrow circular ring of area A (shaded portion) about 0 is

= K2Z A

IZ = K2Z A

In summary ,

We known ,                                     IZ = IX +IY

K2Z A =K2X A +K2Y A

K2Z = K2X +K2Y