Centroid of area drawn over a hemisphere
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?
1 Answer
$\begingroup$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: $$ z_{centroid}= {\int_\Omega R^2 z\sin\theta\, d\theta d\phi \over \int_\Omega R^2\sin\theta\, d\theta d\phi}= {\int_\Omega R^2 R\cos\theta\sin\theta\, d\theta d\phi \over \int_\Omega R^2\sin\theta\, d\theta d\phi}= {RB\over A}. $$
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