Last Updated on September 10, 2026 by Maged kamel
How to determine y-bar for a right-angle case 1?
For more information about the difference between case-1 and case-2, please refer to post-2.
Using a horizontal strip to get the y-bar for a right-angle case 1.
We will start by using a horizontal strip to determine the y-bar value, or the Cg vertical distance to the y-axis.
We have X and Y axes, respectively, and the base of the Triangle with a dimension of b.
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, where m is a slope equal to -h/b *x, and the intersection with the y-axis =h. The Area of the strip is the Product of (dy*x).
That’s why the AC equation is Y =-( h/b) *x+h. our horizontal strip breadth=x and the width=dy.
First, it is good to examine the equation of the inclined line BC by substituting the value of x=0, which is point C, and checking that the corresponding y value=h, when using the equation y=-(h/b)*x+h).
We have Y=h when x=0.
For the second point, point B, we have x=b, based on the line equation, the corresponding y value is zero.
Integrate the horizontal strip to find the Area of the right-angle Triangle.
The Area of the Triangle is the sum of all the tiny horizontal strips, which we can express by integrating over the strip from the start, y=0, to the end, y=h, considering the strip moving vertically.
Since the strip width is x and its height is 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 estimate the Area dA as x*dy. Since integration is in the vertical direction, we will omit the x Expression by substituting its value in terms of y.
The value of x derived from the line equation can be set equal to (h-y)/h*(b), where y is the height of the strip from the base of the Triangle.
Proceed with the integration in the vertical direction from y=0 to y=h; we get the final Area = 0.50*b*h, which is the known formula for the Area of a right-angle Triangle: half the base times the height, where the base equals b, and the height equals h.
Integrate the horizontal strip to get the first-moment Area and y-bar for the right Triangle case -1.
The Expression of the first Moment of the strip about the x-axis can be written as dA*y-strip, where the strip Area is dA and the y distance for the 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 the total Area and the vertical CG distance from the X-axis, which in our case is b*h^2/6, where b is the Triangle base and h is the height. The vertical distance between the CG and the X-axis, which is designated as the Y bar value, will be obtained by simply dividing the first Moment of Area by the Area. For a right-angle Triangle, y-bar = h/3, or one-third of the opposite side height.
The data for the X bar and Y bar for the right-angle Triangle are shown in the next slide image.
You can view or download the PDF for this post and the following post.
For a good external reference, please refer to the following link.
The next post is X-bar for a right-angle case 1 using a vertical strip.