We were
discussing “Area moment of inertia” and “Radius of gyration”,” and also we have seen the “Basic principle of
complementary shear stresses” with the help of our previous posts.

Now we are
going further to start our discussion to understand the theorem of
perpendicular axis with the help of this post.

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**Let us first see here ****theorem of perpendicular axis**

According
to the theorem of perpendicular axis, if I

_{XX}and I_{YY}are the moments of inertia of an irregular lamina about tow mutually perpendicular axis X-X and Y-Y respectively in the plane of lamina, then there will be a moment of inertia I_{ZZ}of lamina around the Z-Z axis which will be perpendicular to the plane of lamina and passing through the intersection of X-X and Y-Y axis.
Moment of
inertia I

_{ZZ}of lamina around the Z-Z axis will be given by following equation as per the theorem of perpendicular axis.
I

_{ZZ}= I_{XX}+ I_{YY}
Let us see
the following figure which indicates one irregular lamina in the plane of X-Y
and area of the lamina is A. Let us assume one small elemental area dA in the
plane of X-Y.

Where,

X=
Distance of the C.G of the small elemental area dA from OY axis

Y= Distance
of the C.G of the small elemental area dA from OX axis

R= Distance
of the C.G of the small elemental area dA from OZ axis

We can
easily note it here that distance of the C.G of the small elemental area dA from
OZ axis could be written as mentioned here.

R

^{2}= X^{2}+Y^{2}
Moment of
inertia of small elemental area dA about the OX axis = dA.Y

^{2}
Hence, moment
of inertia of entire area A about the OX axis, I

_{XX}will be determined as mentioned here.
I

_{XX}=Ʃ dA.Y^{2}
Moment of
inertia of small elemental area dA about the OY axis = dA.X

^{2}
Hence, moment
of inertia of entire area A about the OY axis, I

_{YY}will be determined as mentioned here.
I

_{YY}=Ʃ dA.X^{2}
Moment of
inertia of small elemental area dA about the OZ axis = dA.R

^{2}Hence, moment of inertia of entire area A about the OZ axis, I

_{ZZ}will be determined as mentioned here.

I

_{ZZ}=Ʃ dA. R^{2}
I

_{ZZ}=Ʃ dA. [X^{2}+ Y^{2}]
I

_{ZZ}=Ʃ dA.X^{2}+ Ʃ dA.Y^{2}
I

_{ZZ}= I_{YY}+I_{XX}###
**I**_{ZZ} = I_{XX} + I_{YY}

_{ZZ}= I

_{XX}+ I

_{YY}

Therefore we
can brief here the theorem of perpendicular axis as moment of inertia of an irregular
lamina about an axis normal or perpendicular to it will be equal to the sum of the
moment of inertia about any two mutually perpendicular axis in the plane of the
lamina.

Do you have any suggestions or any amendment required in this post? Please write in comment box.

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**Reference:**

Strength
of material, By R. K. Bansal

Image Courtesy: Google

We will see another important topic i.e. Theorem of parallel axis, in the
category of strength of material, in our next post.