Hydrostatic Forces, Buoyancy and Stability
Total pressure and centre of pressure on plane and curved surfaces, buoyancy and Archimedes' principle, metacentric height and stability of floating bodies.
Dams, gates and tank walls must resist the force of the fluid behind them. Floating bodies must be stable. Both problems use the hydrostatic pressure law p = ρgh.
Force on a plane surface
For a plane surface of area A submerged in a liquid, with its centroid at depth h̄:
- Total pressure force F = ρ g A h̄ (pressure at the centroid × area).
- It acts at the centre of pressure, below the centroid: h_cp = h̄ + I_G sin²θ/(A h̄), where I_G is the second moment of area about the centroidal axis and θ the angle with the free surface (θ = 90° for a vertical surface).
For a vertical rectangle of height d with its top at the free surface: h̄ = d/2 and h_cp = 2d/3.
Force on a curved surface
- Horizontal component = force on the vertical projection of the surface.
- Vertical component = weight of the liquid directly above the surface (real or imaginary).
- Resultant = √(F_H² + F_V²).
Buoyancy
Archimedes' principle: a body in a fluid is pushed up by a force equal to the weight of the fluid it displaces, acting at the centroid of the displaced volume (the centre of buoyancy B).
A floating body displaces its own weight: fraction submerged = ρ_body/ρ_fluid.
Stability of floating bodies
- Metacentre M: the point about which the body seems to rock for small tilts.
- Metacentric height GM = BM − BG, with BM = I/V (I = second moment of the waterplane area about the tilt axis, V = displaced volume).
- GM > 0: stable; GM = 0: neutral; GM < 0: unstable.
- A submerged body (no free waterplane) is stable only if G is below B.
- Period of rolling: T = 2π k/√(g GM). Large GM rolls quickly and uncomfortably.
Worked example
A vertical gate 2 m wide and 3 m high has its top at the water surface. Find the force and its depth.
- F = 1000 × 9.81 × (2 × 3) × 1.5 = 88.29 kN.
- h_cp = 2 × 3/3 = 2 m below the surface.
Practice
A vertical rectangular gate 1 m wide and 4 m deep has its top at the water surface. What is the total force on it in kN? (g = 9.81 m/s², give 2 decimal places.)
Show answer and explanation
Answer: 78.48F = ρgAh̄ = 1000 × 9.81 × 4 × 2 = 78,480 N = 78.48 kN.
A wooden block of density 600 kg/m³ floats in water. What fraction of its volume is submerged?
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Answer: 0.6Fraction submerged = ρ_body/ρ_water = 0.6.
A rectangular barge 10 m long and 4 m wide floats with a draft of 1 m. What is BM in metres about its long axis? (Give 3 decimal places.)
Show answer and explanation
Answer: 1.333I = Lb³/12 = 10 × 64/12 = 53.33 m⁴; V = 40 m³; BM = I/V = 1.333 m.
A floating body is stable for small tilts when:
Show answer and explanation
Answer: AGM > 0 (M above G) gives a restoring couple. G may be above B and the body still be stable.