Brief content of post-5-inertia posts.

5- Easy approach for Product of inertia Ixy for a rectangle.

Product Of Inertia Ixy for a Rectangle at the CG.

Brief content of the video.

In Unit number 5, we’re going to talk about the product moment of inertia for a rectangular section at the CG. We want to prove that the product moment of inertia for rectangular section =0. 

  We draw a rectangle for which the Base is b and the height is h, and we introduce the external two axes X and Y. Then we come at the cg and we draw another two axes, which are X’ and y’.
We divide this rectangle into 4 areas, namely A1, A2, A3, A4, and for each area, we are going to locate the CG of every area. for example, A1 is the area the width = (b/2) the height is (h/2).

For area no. 2, the c g Lies to the right of the y’ axis by distance =b/4 and Abovethe’ axis by h/4.This is a part of the video with a closed caption in English.

A third method is presented in the following video to estimate the Ixy g as a third proof. Again, we have our rectangle with base b and height h, and we have external two axes x and y at the left edge left corner of the rectangle, and we have x CG and y CG. But this time, our strip will be small with width =dx and height =dy.

We will estimate the horizontal distance as X to y CG and y to the CG. This is a part of the video with a closed caption in English.

You can click on any picture to enlarge, then press the small arrow at the right to review all the other images as a slide show.

This is the second video to get the product of inertia value by integration.

The first method used to get the product of inertia Ixy for a Rectangle.

To get the Ixy at the CG for a rectangle, we consider the moment of inertia about the two axes passing by the CG.
1- The rectangle can be considered as composed of 4 equal areas, like quarters, each quarter =A/4; for the first quarter, the x distance is at the left of the y’ axis, so the x distance is =-b/4, while the y1 is =h/4 with a positive sign. 

   2- For the remaining three quarters, we will check the signs of x&y; for the second area, the xis+ve & y is +ve; for the third area, the xis-ve and y are also-ve, for the last quarter, the x-distance is -ve, and y distance are also-ve.

The product of inertia for a rectangle

3- Summing all the products of Ai *xi*yi we will get Ixy at the Cg =0. the next slide image shows in detail the first method of estimation.

The second method to get the product of inertia Ixy for a Rectangle.

The second method to get the value of the product of inertia for the external edge and also at the Cg is as follows:
1-introduce a strip of width dy and breadth=b.  
2- estimate the Ixy=∫h*dy*x/2*y from y=0 to y=h.            

The second method is to get the value of the product of inertia.

3-the value of integration will be Ixy=Ab*h/4. 4 the value of Ixyg=Ixy-A*(b/2)*(h/2)=0. the details are shown in the next slid image.

The third method to get the product of inertia.

The third method to get the value of the product of inertia for the external edge and also at the Cg is as follows:1-introduce a strip of width dy and breadth=dx. 

The third method is to get the value of the product of inertia for a rectangle.

 2-Ixyg=∬dx*dy*x*y, x, and y are the distances to the X-CG and Y-CG respectively, the integration will be done from x=-b/2 to x=b/2 and from y=-h/2 to y=h/2.   
3-the value of integration will be Ixyg=0.           

Ixy at the external axes.

The value of the product of inertia for a rectangle by double integration.

Ixy at the external axes does not=0, but we will integrate the area (dA*x*y) from x=0 to x=b and from y=0 to y=h, to get the product of inertia at the two axes at the external corner of the rectangle section.

The product of inertia from a table.

This is a list of the data for the rectangle starting from the first moment of area to the value of the product of inertia.

Polar moment of inertia.

The polar moment of inertia=Ix+Iy, if we want the polar moment of inertia value at the CG, we add both the moments of inertia at the CG, we will get J0G=b*h(b^2+h^2)/12.

The polar moment of inertia for the rectangular section


While at the external axes, The polar moment of inertia  j0=b*h/3*(h^2+b^2).

This is the PDF used for the illustration of this post.
For an external resource, engineering core courses, the moment of inertia.
This is a link to the post, for the Moment of inertia.

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