The centroid of a hemispherical surface

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Homework Help Overview

The discussion revolves around finding the centroid of a hemispherical surface with a specified radius. Participants are exploring various methods, including Pappus's theorem and integration techniques, to determine the correct approach for calculating the centroid's coordinates.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to apply Pappus's theorem but questions the validity of their result compared to the textbook answer. Some participants clarify the distinction between the centroid of a semicircle and that of a hemispherical surface, suggesting that Pappus's theorem may not be applicable. Others propose using integration of horizontal shells to find the surface area and centroid, but there are concerns about the integration setup and calculations.

Discussion Status

The discussion is active, with participants providing insights and alternative methods. There is recognition of the need to adjust the approach to calculating the centroid, particularly regarding the integration of horizontal shells. While some participants have reported progress, there is still exploration of the correct setup and calculations needed to reach a valid solution.

Contextual Notes

Participants are working under the constraints of homework guidelines, which may limit the methods they can use. There is an ongoing examination of assumptions related to the geometry of the hemispherical surface and the application of theorems in this context.

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I have to find the centroid of a hemispherical surface with radius r.
I want to use Pappus's theorem.S=2*Pi*x*s
The surface S, is 2*Pi*r^2 and the length of the curve s, is Pi*r so the distance from the centroid ,which lies on the y-axis by symmetry, to the x-axis is S/(2*Pi*Pi*r) = r/Pi

This is not the answer by textbook gives for the y coordinate of the centroid. What am I doing wrong? Please help me.
 
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x is the centroid of the generating surface which is a semicircle. And the centroid of a semicircle (a 2D object) is different to that of a hemispherical surface (a 3D object). I'm afraid you can't use Pappus.
 
Last edited:
Try this instead - http://mathworld.wolfram.com/Hemisphere.html

I believe the hemispherical is a semi-circle rotated by pi, or a quarter circle rotated by 2pi about an axis corresponding to a radius and corresponding to either end of the arc.
 
I understand I can't use Pappus's theorem.
So I try to cut the hemispherical shell into horizontal shells up the y axis. Each shell should have the surface 2*Pi*R where R=sqrt(r^2-y^2)
So dS = 2*Pi*R dy
But there is something wrong here, When I integrate from 0 to r I don't get the right answer for the surface of the hemispherical surface.
Could you help me some more, please.
 
I got it right. I had to calculate the width of the shell.And then integrate.
 
How can I calculate the centroid of the hemisphere surface using only simple integrals and rectangular coordinates?
 

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