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Thursday, 9 March 2017

MASS MOMENT OF INERTIA OF A RECTANGULAR PLATE ABOUT A LINE PASSING THROUGH THE CENTRE OF GRAVITY

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 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

IXX = Area moment of inertia of the rectangular section about the X-X axis
(Im) 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.Y2
Mass moment of inertia of the rectangular elementary strip about the X-X axis = ρBT. Y2dY

 
Now 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
(Im) XX = ρBT.D3/12
(Im) XX = ρBTD.D2/12
(Im) XX = ρV.D2/12

(Im) XX = M.D2/12

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

(Im) YY = M.B2/12

Do you have any suggestions? Please write in comment box

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

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