X bar for a right angle-case-1-using vertical strip.
For more information about the difference between case-1 and case-2, please refer to post-2.
Using a vertical strip to get x bar for a right angle case-1.
For another approach to get the X bar for a right angle-case-1 is by using 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. 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 =h.
The area of the strip is the product of (y*dx), which is the area of the hatched strip.
The width of the strip =dx and its height=y.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 value of y can be expressed as equal to (-h*X/b)+h. The procedure is shown in the next image picture. The area=0.50bh, 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.
to get the X bar for a right angle-case-1, using a vertical strip. Start using a vertical strip for which, 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 horizontal direction from x=0 to x=b.
The detailed process of integration can be found in the next slide image. 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/6, where b is the triangle base while h is the height. The area=0.50bh, which is the same result obtained earlier by using the horizontal strip.
X bar for a right angle final step.
X bar for a right angle-case-1. The value of X bar value will be obtained by simply dividing the first moment of area /Area. the first moment of Area can be found as equal to (b^2*h/6). We will get an x bar for a right angle=b/3 or one-third of the base width. The next slide image shows the value of the X-bar.
This is the link to view or download the pdf used for the illustration of this post.