Centre of gravity of the spherical cap

Question

How to calculate centre of gravity of the spherical cap, with known mass, R and h ?

Answer 1

For a solid sphere, the mass of the section is proportional to $r^2$, i.e. to $R^2-z^2$ (assuming horizontal sections).

Hence the mass

$$\propto\int_{R-h}^R(R^2-z^2)dz=Rh^2-\frac{h^3}3$$ and the moment around an horizontal axis by the sphere center,

$$\propto\int_{R-h}^Rz(R^2-z^2)dz=h^2R^2-Rh^3+\frac{h^4}4.$$

The distance of the gravity center to the sphere center is

$$R\frac{1-\dfrac hR+\dfrac{h^2}{4R^2}}{1-\dfrac h{3R}}.$$

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For small $h$,

$$R-\frac23h$$ is a good approximation.

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If the sphere is hollow, you replace $R^2-z^2$ by $\sqrt{R^2-z^2}$, and WLOG $R=1$,

$$\frac23\frac{\left(2h-h^2\right)^{3/2}}{\dfrac\pi2-\sqrt{2h-h^2}(1-h)-\arcsin(2h-h^2)}$$