We were discussing “Mass moment of inertia” and “Area
moment of inertia” in our previous posts. We have also seen mass moment of inertia for the rectangular section about a line passing through the center of gravity.

Today we will see here the method to determine the
mass moment of inertia for the rectangular section about its base line with the
help of this post.

###
*Let
us first determine here the mass moment of inertia for the rectangular section
about a line passing through the base of the given rectangular section*

*Let us first determine here the mass moment of inertia for the rectangular section about a line passing through the base of the given rectangular section*

Let us consider one rectangular section
ABCD as displayed in following figure. Let us assume that one line is passing
through the base of the rectangular section and let us consider this line as
line CD and we will determine the mass moment of inertia for the rectangular
section about this line CD.

B = Width of the rectangular section ABCD

D = Depth of the rectangular section ABCD

T= Uniform thickness of the rectangular section ABCD

V= Volume of the rectangular section ABCD

V = B x T x D

M= Mass of the rectangular section ABCD

M = ρBDT

(I

_{m})_{CD}= Mass moment of inertia of the rectangular section about its base line i.e. CD

*Now we will determine the value or expression for the mass moment of inertia of the rectangular section about a line passing through the base of the*

*rectangular section*

Let us consider one rectangular elementary strip
with depth dY and at a distance Y from the base line CD as displayed in above
figure.

Area of rectangular elementary strip, dA = dY.B

Mass of the rectangular elementary strip, dm = ρ x T
x dA

Mass of the rectangular elementary strip, dm = ρ x T
x dY. B

Mass of the rectangular elementary strip, dm = ρBT.
dY

Mass moment of inertia of the elementary strip about
the base line = dm.Y

^{2}
Mass moment of inertia of the elementary strip about
the base line = ρBT. Y

^{2}dY
Now we will determine the mass moment of inertia of
entire rectangular section about its base line i.e. CD. And it could be easily
done by integrating the above equation between limit 0 to D.

Therefore, mass moment of inertia of the
rectangular section about the line CD will be determined as displayed here in
following figure.

(I

_{m})_{CD}= ρBT.D^{3}/3
(I

_{m})_{CD}= ρBTD.D^{2}/3
(I

_{m})_{CD}= ρV.D^{2}/3###
**(I**_{m}) _{CD} = M.D^{2}/3

_{m})

_{CD}= M.D

^{2}/3

Do you have any suggestions? 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.
Determination of the mass moment of inertia of the hollow rectangular section in
the category of strength of material.

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