We were discussing “Mass moment of inertia” and “Area moment of inertia” in our
previous posts. Today we will see here the method to determine the mass moment
of inertia for the rectangular section 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 center of gravity of the rectangular section*

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

Let us consider one rectangular section ABCD as
displayed in following figure. Let us assume that center of gravity of the given
rectangular section is C.G and axis X-X passing through the center of gravity
of the rectangular section as displayed in following figure.

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

_{XX}= Area moment of inertia of the rectangular section about the X-X axis
(I

_{m})_{XX}= Mass moment of inertia of the rectangular section about the X-X axis

*Now we will determine the value or expression for the mass moment of inertia of the rectangular section about the X-X axis which is*

*passing through the center of gravity of the rectangular section*

Let us consider one rectangular elementary strip
with thickness dY and at a distance Y from the X-X axis 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 rectangular elementary
strip about the X-X axis = dm.Y

^{2}
Mass moment of inertia of the rectangular elementary
strip about the X-X axis = ρBT. Y

^{2}dYNow we will determine the mass moment of inertia of entire rectangular section about the X-X axis which is passing through the C.G of the rectangular section ABCD. And it could be easily done by integrating the above equation between limit (-D/2) to (D/2).

Therefore, mass moment of inertia of the rectangular
section about the X-X axis after calculation, we will have

(I

_{m})_{XX}= ρBT.D^{3}/12
(I

_{m})_{XX}= ρBTD.D^{2}/12
(I

_{m})_{XX}= ρV.D^{2}/12###
**(I**_{m})
_{XX} =
M.D^{2}/12

_{m})

_{XX}= M.D

^{2}/12

Similarly, we will determine the mass moment of
inertia of the rectangular section about the Y-Y axis passing through the C.G of
the rectangular section ABCD and we will have

###
**(I**_{m}) _{YY} = M.B^{2}/12

_{m})

_{YY}= M.B

^{2}/12

Do you have any suggestions? Please write in comment
box

###
**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 rectangular section about a line passing through the base of rectangular section in the category of
strength of material.

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