Analysis
To find the force exerted by the liquid on the lower half of the sphere, we can utilize the concept of Buoyancy and vertical force balance. The net buoyant force is the vector sum of the downward force on the upper surface and the upward force on the lower surface.
Derivation
Let $F_{bottom}$ be the upward force on the lower hemisphere and $F_{top}$ be the downward force on the upper hemisphere due to the liquid.
The Net Buoyant Force ($F_B$) is given by Archimedes’ principle:
$$F_B = F_{bottom} – F_{top} = \text{Weight of Displaced Liquid}$$ $$F_B = \rho \left( \frac{4}{3}\pi r^3 \right) g$$The downward force $F_{top}$ on the upper hemisphere is equal to the weight of the liquid column above it. Since the sphere is just submerged, the liquid above the curved upper surface is the volume of the enclosing cylinder minus the volume of the hemisphere itself:
$$V_{above} = V_{cylinder} – V_{hemisphere} = (\pi r^2 \cdot r) – \frac{2}{3}\pi r^3 = \frac{1}{3}\pi r^3$$ $$F_{top} = \rho \left( \frac{1}{3}\pi r^3 \right) g$$Now, substituting $F_{top}$ back into the force balance equation:
$$F_{bottom} = F_B + F_{top}$$ $$F_{bottom} = \frac{4}{3}\pi r^3 \rho g + \frac{1}{3}\pi r^3 \rho g$$