Breif description of Post 5-inertia.

5-Y bar for a right angle case-2.

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

Our subject is to determine the y bar for a right angle case-2, but first, what are the differences between case-1 and 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 y bar for a right angle case-2.

We will start by using a horizontal strip to get the value of the y bar or the Cg vertical 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 =mx+C m which is a slope is equal to +h/b *x, and the intersection with y-axis =0.

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.

The steps to estimate the y bar value for a right angle case 2 with a horizontal strip.

Since the strip width is dy and it is at a distance y from the x-axis. 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 (b-x)*dy.

Since integration is in the vertical direction, we will omit the x expression by substituting it value in terms of y. the x value=b*y/h.

The area of a right-angle triangle 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 x-axis.

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

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

The final A*y bar represents the product of total area * the vertical CG distance from the y-axis will be found as in our case=b*h^2/6, where b is the triangle base while h is the height.

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

Y bar value will be obtained by simply dividing the first moment of area /Area.

We estimate the product of dA by y, it will be equal to (b-b/h*y)dy*y

The expression for the first moment of area of a right-angle triangle using a horizontal strip.

We will perform the integration from y=0 to y=h. Finally, the x bar value is =b*h^2/6

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

Another approach by use of a vertical strip to get the value of the y bar or the Cg vertical distance to the X-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 =mx+C m which is a slope is equal to +-h/b *x and the intersection with y-axis =0.

The steps to estimate the area for a right angle case-2 with a vertical strip.

The width of the strip =dx and its height=y. the strip area dA=y*dx, since we are integrating 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 X-axis.

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

The expression for the first moment of area of a right-angle triangle using a vertical strip.

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

The details of the integration are shown in the next slide image.

The expression for the first moment of area of a right-angle triangle using a vertical strip.

The final A*y bar for a right angle case-2 represents the product of total area * the vertical CG distance from the X-axis will be found as in our case=b*h^2/6, where b is the triangle base while h is the height.

Y bar for a right angle case-2 will be obtained by simply dividing the first moment of area /Area, we will get y bar for a right angle=h/3 or one-third of the opposite side height.

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

This is a link for a good external reference, please refer to the Centroid of an Area by Integration.

this is a link to the next post-Area and Cg of a triangle.

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