Brief decription for  post 5-round shapes-area and Cg for a circular segment.

5-How to find the area and Cg for a circular segment?

Area and Cg for a circular segment.

Reference handbook 10.00 value for the area and Cg for a circular segment.

There is a list of the common round shapes area and CG value. Our fifth case is the case of a circular segment.

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.

The circular segment is a portion of a circular sector where a cut by a chord is made. Thus the circular sector can be treated as a circular sector from which two triangles are cut. For a circular sector bounded by an angle of 2θ, each triangle has a base of a cosθ and a height of a*sinθ.

In order to determine the area and Cg for a circular segment, we will deduct the area of the two triangles from the circular sector area. The details pf the both circular sector and the two triangles are shown in the next slide image.

Area and Cg for a circular segment.

The external axis Y is passing by the left point of the circular segment which has a radius value of a. The x-axis is Bisecting the angle of the circular sector.

Area and Cg for a circular segment.

The first area A1 is the area of the circular sector, from the previous post we have estimated it as equal to a^2*θ, where a is the radius and θ is half of the enclosed angle of 2*θ.

The second area A2 is the area of the two triangles, the two areas are equal to 0.5*(2a*sinθ)*a *cosθ=a^2 *sinθ*cosθ.

From trigonometry we have sin(2θ)=2*sinθ*coθ, we can readjust the expression for the area of the two triangles to be equal to a^2*0.5*sin(2θ).

The area of the circular sector can be estimated as A1-A2, which is equal to a^2*(θ-0.5*sin(2θ).

Area of the circular segment.

The steps followed to the area for the circular segment are shown in the next slide image.

Cg for a circular segment-first moment of area about the Y-axis.

The Cg for a circular segment about Y-axis can be obtained by using the expression (A1*Xcg1-A2*Xcg2)/(A1-A2)

For the first moment of area for a circular sector, we will multiply the area which is a^2*θ by the distance of its Cg to the Y-axis, which is Xcg1, which is equal to (2/3)*a^2*sinθ.

For the first moment of area for the two triangles, we will multiply the area which is 2*0.5*sin(2θ) by the distance of its Cg to the Y-axis, which is Xcg2, which is equal to (2/3)*a*cosθ.

Step 1 to estimate the first moment of area of a circular segment.

The calculations of the first moment of area for the circular sector and for the two triangles are shown in the next slide.

For more simplification of the expression for the first moment of area for the circular segment about the Y-axis, we will utilize the expression for sin^2θ=1-cos^2θ.

The value of the first moment of area about the Y-axis.

The final expression for the first moment of area about the Y-axis is shown in the next slide image.

The value of xcg or the circular segment.

For the expression of Xcg or the distance from Cg of the circular segment, we will divide the first moment of the area by the value of the area of the circular segment. The Xcg is shown in the next slide image.

Cg for a circular segment-first moment of area about the X-axis.

For the area and Cg for a circular segment about X-axis. We can use the expression (A1*Ycg1-A2*Ycg2)/(A1-A2)

For the first moment of area for a circular sector, we will multiply the area which is a^2*θ by the distance of its Cg to the X-axis, which is zero.

For the first moment of area for the two triangles, we will multiply the area which is 2*0.5*sin(2θ) by the distance of its Cg to the Y-axis, which is zero.

The value of yCg for the circular segment.

The final value of the first moment of area about the x-axis is zero since the area is not equal to zero, this indicates that the Cg is located at the X-axis, and has a zero vertical value.

We have completed the subject of the area and Cg for a circular segment.

The next post will be on how to estimate the area and Cg for an ellipse.

This is a link to a very useful site: Engineering statics open and interactive.

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