We were taught an theorem for finding the centroid of an arbitrary area having uniform mass density drawn over Hemisphere.
The theorem states that the centroid of and arbitrary area $A$ of uniform mass density drawn over Hemisphere is located at $\frac{B}{A}\cdot R$ distance from base of Hemisphere:
$$Y_{\text{centroid}} = \dfrac{B}{A}\times R$$
where
- $A$ is total area drawn over Sphere
- $B$ is the parallel projection of area $A$ over Base of Hemisphere
- $R$ is radius of Hemisphere.
I have verified this theorem for some symmetrical shapes drawn on Hemisphere.
My questions are:
Is there a name for this equation/theorem?
Is there a simple proof for this?
Area $A$ and its projection $B$ are given, in spherical coordinates, by: $$ A=\int_\Omega R^2\sin\theta\, d\theta d\phi, \quad B=\int_\Omega R^2\sin\theta\cos\theta\, d\theta d\phi, $$ where I took the base of the hemisphere in the $x-y$ plane, $\theta$ is the polar angle and $\Omega$ is the integration domain.
The height of the centroid is given then by its $z$ coordinates, so by definition: $$ z_{centroid}= {\int_\Omega R^2 z\sin\theta\, d\theta d\phi \over \int_\Omega R^2\sin\theta\, d\theta d\phi}= {\int_\Omega R^2 R\cos\theta\sin\theta\, d\theta d\phi \over \int_\Omega R^2\sin\theta\, d\theta d\phi}= {RB\over A}. $$