Hydrostatic Force

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Hydrostatic Force on a Submerged Surface Hydrostatic force is the force exerted by a fluid at rest on a submerged surface. It acts perpendicular to the surface. 1. Basic Concepts Pressure: $P = \rho g h$ $\rho$: density of the fluid (e.g., water $\approx 1000 \, \text{kg/m}^3$) $g$: acceleration due to gravity ($\approx 9.81 \, \text{m/s}^2$) $h$: depth of the fluid from the free surface to the point of interest Force (Constant Pressure): $F = P \cdot A$ $A$: area of the submerged surface For submerged objects, pressure varies with depth, so integration is often required. 2. Hydrostatic Force on a Horizontal Surface If a surface is horizontal, the depth $h$ is constant across the entire surface. The hydrostatic force is simply $F = \rho g h A$. The force acts downwards, perpendicular to the surface. 3. Hydrostatic Force on a Vertical or Inclined Plane Surface For a non-horizontal plane surface, the pressure varies with depth. We use the concept of the centroid. Total Force: $F = P_c A = \rho g h_c A$ $h_c$: depth of the centroid (center of gravity) of the submerged area from the free surface. $A$: total area of the submerged surface. Center of Pressure (CP): The point where the total hydrostatic force acts. It is generally below the centroid. Vertical distance to CP from free surface: $y_{cp} = \bar{y} + \frac{I_{xx,c}}{A\bar{y}}$ $\bar{y}$: distance from the free surface to the centroid along the inclined plane. If the surface is vertical, $\bar{y} = h_c$. $I_{xx,c}$: moment of inertia of the submerged area about its centroidal axis parallel to the free surface. $A$: area of the surface. Depth of CP from free surface: $h_{cp} = h_c + \frac{I_{xx,c} \sin^2 \theta}{A h_c}$ $\theta$: angle of inclination of the surface with the horizontal. For vertical surface, $\theta = 90^\circ$, $\sin^2 \theta = 1$. 4. Common Shapes and Centroidal Moments of Inertia ($I_{xx,c}$) Shape Area ($A$) Centroid ($h_c$ or $\bar{y}$) $I_{xx,c}$ (about centroidal axis) Rectangle (width $b$, height $d$) $bd$ $h_c = \text{depth to mid-height}$ $\frac{bd^3}{12}$ Triangle (base $b$, height $d$) $\frac{1}{2}bd$ $h_c = \text{depth to } d/3 \text{ from base}$ $\frac{bd^3}{36}$ Circle (radius $R$) $\pi R^2$ $h_c = \text{depth to center}$ $\frac{\pi R^4}{4}$ 5. Calculation Steps for Plane Surfaces Identify the fluid density ($\rho$) and gravity ($g$). Determine the shape and dimensions of the submerged surface. Locate the centroid of the submerged area. Calculate the depth of the centroid ($h_c$) from the free surface. Calculate the area ($A$) of the submerged surface. Calculate the total hydrostatic force $F = \rho g h_c A$. Calculate the moment of inertia ($I_{xx,c}$) of the area about its centroidal axis parallel to the free surface. Calculate the depth of the center of pressure ($h_{cp}$). 6. Hydrostatic Force on a Curved Surface For curved surfaces, the hydrostatic force is best calculated by resolving it into horizontal and vertical components. Horizontal Component ($F_H$): $F_H$ on a curved surface is equal to the hydrostatic force on the vertical projection of the curved surface onto a vertical plane. $F_H = \rho g h_c A_V$, where $A_V$ is the area of the vertical projection. $F_H$ acts at the center of pressure of the vertical projection. Vertical Component ($F_V$): $F_V$ is equal to the weight of the actual or imaginary fluid volume above the curved surface extending to the free surface. $F_V = \rho g V_{fluid}$, where $V_{fluid}$ is the volume of fluid above the curved surface. $F_V$ acts through the centroid of this fluid volume. Resultant Force ($F_R$): $F_R = \sqrt{F_H^2 + F_V^2}$ Angle of resultant force with the horizontal: $\alpha = \tan^{-1} \left( \frac{F_V}{F_H} \right)$ 7. Buoyancy (Archimedes' Principle) Buoyant Force ($F_B$): An upward force exerted by a fluid that opposes the weight of an immersed object. $F_B = \rho g V_{displaced}$ $\rho$: density of the fluid $g$: acceleration due to gravity $V_{displaced}$: volume of fluid displaced by the object. The buoyant force acts through the centroid of the displaced fluid volume (center of buoyancy). 8. Applications Design of dams, gates, tanks, and other hydraulic structures. Stability analysis of floating and submerged objects (ships, submarines). Measurement of fluid levels and pressures.