Finding the Centroid of the surface of a hemisphere

[Darragh]

Hi gang, I was given this question, but I can't quite seem to get it all the way...
1.Determine the coordinates of the centroid of the surface of a
hemisphere, the radius of which is r, with respect to its base.

Homework Equations

A=2[itex]\pi r^{2}[/itex]
A: Surface Area of the Hemisphere.

Coordinates of the centroid: ([itex]\overline{X}, \overline{Y}, \overline{Z}[/itex])

[itex]\overline{Y}A = \int ydA[/itex]

I set up the coordinate system so [itex]\overline{X}[/itex] = 0 and [itex]\overline{Z}[/itex] = 0

The Attempt at a Solution

I tried finding the circular elements of the hemisphere

r[itex]_{el}[/itex]=[itex]\sqrt{r^{2}-y^{2}}[/itex]
r[itex]_{el}[/itex]: Radius of the element

dA=2[itex]\pi\sqrt{r^{2}-y^{2}}[/itex]dy

Then I used [itex]\overline{Y}( 2\pi r^2) = \int^{r}_{0} y 2 \pi \sqrt{r^2-y^2} dy[/itex]
but I'm pretty sure this is wrong, any ideas or pointers? Thanks in advance.

 

[rude man]

There's a sneaky, roundabout way to this:

1. Compute the work needed to assemble the hemisphere from elements of mass dm. The hemisphere when finished sits on its flat base. An element of mass dm is attracted to the Earth center by a force g*dm, and work to put it in place = gy*dm where y is height above ground for the element dm.

2. Then the centroid must be at height h such that Mgh = total work needed to assemble the hemisphere, where M = mass of entire hemisphere.

Even if you don't want to submit the answer this way, you can double-check your result by this relatively easy integration.