Figure 8.35 shows a semi-cylindrical massless gate (of width...

Figure $8.35$ shows a semi-cylindrical massless gate (of width $R$ ) pivoted at the point $O$ holding a stationary liquid of density $\rho .$ A horizontal force $F$ is applied at its lowest position to keep it stationary. The magnitude of the force is Fig. $8.35$ (A) $\frac{9}{2} \rho g R^{3}$ (B) $\frac{3}{2} \rho g R^{3}$ (C) $\rho g R^{3}$ (D) Zero

Figure $8.35$ shows a semi-cylindrical massless gate (of width $R$ ) pivoted at the point $O$ holding a stationary liquid of density $\rho .$ A horizontal force $F$ is applied at its lowest position to keep it stationary. The magnitude of the force is
Fig. $8.35$
(A) $\frac{9}{2} \rho g R^{3}$
(B) $\frac{3}{2} \rho g R^{3}$
(C) $\rho g R^{3}$
(D) Zero

Key Concepts

Torque Equilibrium

Torque equilibrium involves balancing the moments produced by various forces about a pivot or any chosen point. In hydrostatic problems, the moments due to the distributed pressure forces are countered by applied forces to maintain stability. This concept is critical when analyzing systems where forces are applied at specific points to prevent rotation or tipping.

Center of Pressure

The center of pressure is the point on a submerged surface where the total resultant hydrostatic force can be considered to act. It is determined by the distribution of pressure over the surface and usually does not coincide with the geometric centroid. Understanding its location is essential for analyzing moments and ensuring equilibrium in structures exposed to fluid pressure.

Hydrostatic Pressure Distribution

This concept describes how the pressure exerted by a fluid at rest increases linearly with depth due to the weight of the fluid above. It is crucial for understanding the forces acting on submerged surfaces because the pressure at any point on the surface depends solely on the depth below the free surface and the density of the liquid, making it fundamental to hydrostatic force calculations.

Integration of Pressure Over a Surface

When calculating the resultant force on a submerged body, it is necessary to integrate the pressure over the entire surface area. This process sums the small forces acting on differential elements of the surface, providing the total force and its point of application. This method is universally applicable in hydrostatics for bodies with arbitrary shapes.

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