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3.2 Hydrostatic Forces on Submerged Surfaces

3 min readLast Updated on July 19, 2024

Hydrostatic forces and buoyancy are key concepts in fluid mechanics. They explain how fluids exert pressure on surfaces and objects. Understanding these principles is crucial for designing structures that interact with fluids, from dams to ships.

Calculating hydrostatic forces on various surfaces and applying Archimedes' principle are essential skills. These concepts help engineers predict how objects will behave when submerged or floating, ensuring safety and efficiency in fluid-related applications.

Hydrostatic Forces and Buoyancy

Hydrostatic force on plane surfaces

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  • Hydrostatic force (FhF_h) on a submerged plane surface calculated by multiplying pressure at the centroid (pcp_c) and the surface area (AA)
    • Formula: Fh=pcAF_h = p_c A
    • Pressure at the centroid depends on the depth of the centroid (hch_c) and the specific weight of the fluid (γ\gamma)
      • Formula: pc=γhcp_c = \gamma h_c
      • Specific weight is the weight per unit volume of the fluid (water, oil)
  • Center of pressure (ypy_p) is the point where the resultant hydrostatic force acts on the surface
    • For a vertical surface, center of pressure is calculated using the formula: yp=IcAhc+hcy_p = \frac{I_c}{A h_c} + h_c
      • IcI_c is the second moment of area about the centroid (measure of the surface's resistance to bending)
    • For an inclined surface, center of pressure is calculated using the formula: yp=IcsinθAhc+hcy_p = \frac{I_c \sin \theta}{A h_c} + h_c
      • θ\theta is the angle between the surface and the horizontal plane (measured in degrees or radians)
      • Inclined surfaces include ramps, sloped walls of tanks

Hydrostatic force on curved surfaces

  • Hydrostatic force on a submerged curved surface calculated by integrating the pressure distribution over the surface area
    • Formula: Fh=pdAF_h = \int p dA
    • Integration accounts for the varying pressure along the curved surface (spherical tank, cylindrical pipe)
  • Horizontal component of the hydrostatic force on a curved surface equals the hydrostatic force on the vertical projection of the curved surface
    • Vertical projection is the shadow cast by the curved surface on a vertical plane
  • Vertical component of the hydrostatic force on a curved surface equals the weight of the fluid volume above the curved surface
    • Fluid volume above the surface contributes to the downward force
  • Center of pressure for a curved surface determined by taking moments of the pressure distribution about a reference point
    • Reference point can be any convenient location (bottom of the surface, free surface of the fluid)

Buoyancy and Archimedes' principle

  • Buoyancy is the upward force exerted by a fluid on an object immersed in it
    • Buoyant force counteracts the weight of the object (ship, submarine)
  • Archimedes' principle states that the buoyant force acting on an object equals the weight of the fluid displaced by the object
    • Formula: Fb=ρgVdF_b = \rho g V_d
      • FbF_b is the buoyant force (measured in newtons)
      • ρ\rho is the density of the fluid (measured in kg/m³)
      • gg is the acceleration due to gravity (9.81 m/s²)
      • VdV_d is the volume of fluid displaced by the object (measured in m³)
    • Principle applies to both fully and partially submerged objects (iceberg, floating log)

Buoyant force and object stability

  • For a floating object, the buoyant force equals the weight of the object
    • Formula: Fb=WF_b = W
    • At equilibrium, the object displaces a volume of fluid equal to its own weight
  • For a submerged object, the buoyant force equals the weight of the fluid displaced
    • If the buoyant force is greater than the object's weight, the object will rise (helium balloon)
    • If the buoyant force is less than the object's weight, the object will sink (anchor)
  • Stability of a floating object depends on the relative positions of the center of gravity (G) and the center of buoyancy (B)
    • The metacenter (M) is the point where the line of action of the buoyant force intersects the object's centerline when the object is tilted
      1. If M is above G, the object is stable (self-righting)
      2. If M is below G, the object is unstable (capsizing)
    • The metacentric height (GM) is the distance between G and M
      • A positive GM indicates stable equilibrium (boat)
      • A negative GM indicates unstable equilibrium (tipped buoy)
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© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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