A spherical shell of radius a is just filled with liquid of density $\rho$  and the system rotates with uniform angular velocity $\omega$  about vertical diameter.

 
 
$$\begin{gathered} P_B-P_A=\rho g(z+a) \\ {P}_C-P_B=\frac{\rho {\omega }^2{r}^2}{2} \\ {P}_C-P_A=\rho g(z+a)+\frac{\rho {\omega }^2{r}^2}{2} \\ \Delta P=\rho g(z+a)+\frac{\rho {\omega }^2}{2}(a^2-z^2) \\ for \Delta p to be max, \end{gathered}$$

$$\begin{gathered} \frac{d\left(\Delta P\right)}{d_z}=\rho g\left(1\right)+\frac{\rho {\omega }^2}{2}(-2z)=0 \\ z=\frac{g}{{\omega }^2} \\ dA=2\pi r\left(ad\theta \right) \\ P=\rho g(z+a)+\frac{\rho {\omega }^2}{2}(a^2-z^2) \\ dF= PQA \\ {F}_{net}=\int dF\cos \theta =\int [\rho g(z+a)+\frac{\rho {\omega }^2}{2}(a^2-z^2)]2\pi r\left(ad\theta \right)\cos \theta \\ =\int [\rho g(a\sin \theta +a)+\frac{\rho {\omega }^2}{2}\left(a^2{\cos }^2\theta \right)]2\pi r\left(ad\theta \right)\cos \theta \\ =\int \rho ga\left(2\pi r\cos \theta ad\theta \right)+\int \rho ga\sin \theta \left(2\pi r\cos \theta ad\theta \right)+\int \frac{\rho {\omega }^2{a}^2}{2}\left(2\pi ad\theta \right){\cos }^3\theta \end{gathered}$$

$=\rho g{a}^3\left(2\right)\underset{0}{\overset{\pi }{\int }}{\cos }^2\theta d\theta +\rho g{a}^3\left(2\pi \right)\underset{0}{\overset{\pi }{\int }}{\cos }^2\theta \sin \theta d\theta +\frac{\rho {\omega }^2{a}^4}{2}\left(2\pi \right)\underset{0}{\overset{\pi }{\int }}{\cos }^4\theta d\theta$

$$\begin{gathered} =\frac{5}{3}\rho g\pi a^3+\frac{1}{4}\rho \pi {\omega }^2{a}^4 \\ {\left(F_{net}\right)}_{upper}=\frac{1}{3}\rho g\pi a^3+\frac{1}{4}\rho \pi {\omega }^2{a}^4 \end{gathered}$$