Breif description of Post 4-inertia.

4- Easy steps for X bar for a right angle case-2.

How to determine x bar for a right angle case-2?

For more information about the difference between case-1 and case-2, please refer to post-2.

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.

Using a horizontal strip to get x bar for a right angle case 2.

X bar for a right angle case-2

We will start by using a horizontal strip to get the value of the X bar for a right angle case-2 or the CG horizontal distance to the y-axis.
We have X and Y axes respectively and the base of the triangle.

We have line AB with the length of b, the rise of the triangle is=h, and the inclined portion AC, equation: y =m*x, m which is a slope is equal to+h/b *x and the intersection with y-axis =0. The following four steps are shown in the previous slide image.

Perform integration for the horizontal strip to get the area of the right-angle triangle.

The area of the triangle is the summation of all the tiny horizontal strips, which can be expressed by using the integration for the strip from the start which is y=0 to the end which is y=h, considering moving the strip in the vertical direction.
Since the strip width is (b-x) and its height=dy, we are going to use the relation by y and x as derived from the equation of line BC.

We will estimate the area dA as the product of x*dy, since integration is in the vertical direction, we will omit x expression by substituting its value in terms of y. the x value=(b*y/h).

The area of a right angle case 2 using a horizontal strip.

Proceed with the integration we will get the final area=0.50*b*h, which is a known formula for the area of a right-angle triangle, that is the product of half base* height.

Perform integration for the horizontal strip to get the first-moment area about the Y-axis.

The expression of the dA*x-strip will be represented by the first moment of area about the y-axis, where x-strip is the horizontal distance from the Cg of the strip to the y-axis.

Derive the expression for the first moment of area for a right angle case 2 by using a horizontal strip.

The expression of dA*x-strip is shown in the next slide image and integration will be carried out in the vertical direction from y=0 to y=h.

We notice that x strip from the cg of the strip to y-axis=x+(b-x)*0.50=0.50(b+x).

The final A*x bar represents the product of total area * the horizontal CG distance from the y-axis will be found as in our case=b^2*h/3, where b is the triangle base while h is the height.

X bar for a right angle case-2 final step.

Derive the expression for the first moment of area for a right angle case 2 by using a horizontal strip.

Continue the estimation of the integration of the product of A *x bar the full details are shown in the next slide image.

The final value of the X bar for a right angle case 2.

X bar value will be obtained by simply dividing the first moment of area /Area, we will get x bar for a right angle=2*b/3 or two-third of the base width.

Using a vertical strip to get x bar for a right angle case-2.

Another approach by use of a vertical strip to get the value of the X bar or the CG horizontal distance to the y-axis.
We have X and Y axes respectively and the base of the triangle.

The area of a right angle case 2 using a vertical strip.

We have line AB with the length of b, the rise of the triangle is=h, and the inclined portion AC, equation: y =mx+C, m is a slope is equal to +h/b *x and the intersection with y-axis =0.

The width of strip =dx and its height=y. The area of that strip is dA=y*dx, since we are integrating it into the x-direction we will omit the expression of y, by substituting its value in terms of x. The procedure is shown in the next image picture. The area=0.50*b*h, which is the same result obtained earlier by using the horizontal strip.

Perform integration for the vertical strip to get the first-moment area about the Y-axis.

The expression of the dA*x-strip will be represented by the first moment of area about the y-axis, where the x-strip is the horizontal distance from the Cg of the strip to the y-axis. The expression of dA*x-strip is shown in the next slide image and integration will be carried out in the vertical direction from x=0 to x=b.

The final A*x bar represents the product of total area * the horizontal CG distance from the y-axis will be found as in our case=b^2*h/3, where b is the triangle base while h is the height.

X bar for a right angle case-2 final step.

The final value of the X bar for a right angle case 2.

X bar value will be obtained by simply dividing the first moment of area /Area, we will get x bar for a right angle=b/3 or Two-third of the base width.

This is a link for the pdf file used in the illustration of this post.

This is a link for a good external reference here is The Engineering Toolbox.
This is the link to the next post is y bar for a right angle case-2.

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