MASS MOMENT OF INERTIA OF RECTANGULAR SECTION ABOUT ITS BASE

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 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.Y²

Mass moment of inertia of the elementary strip about the base line = ρBT. Y²dY

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.D³/3

(I_m) _CD = ρBTD.D²/3

(I_m) _CD = ρV.D²/3

(I_m) _CD = M.D²/3

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 hollow rectangular section in the category of strength of material.

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