Buoyancy and Stability

Why it matters

Understanding buoyancy and stability is essential for designing ships, submarines, and other floating structures. It ensures that these structures can float and remain stable in water, preventing capsizing and ensuring safety.

Key ideas

• Buoyancy: The upward force exerted by a fluid on a submerged or floating object. It is equal to the weight of the fluid displaced by the object.
• Archimedes' Principle: States that the buoyant force on an object is equal to the weight of the fluid displaced by the object.
• Stability: Refers to the ability of a floating body to return to its original position after being tilted. Stability depends on the center of gravity and the center of buoyancy.
• Metacenter: For an infinitesimal heel about a specified axis, the metacenter is the intersection of the new buoyancy-force line with the original vertical through the centre of buoyancy. Buoyancy acts through the centroid of displaced fluid, B; M is a geometric stability construction.

Formulas

• Buoyant Force: F_b = ρ_f · V_d · g
  • F_b: Buoyant force (N)
  • ρ_f: Density of the fluid (kg/m³)
  • V_d: Volume of displaced fluid (m³)
  • g: Acceleration due to gravity (9.81 m/s²)
• Stability Condition: GM = BM - BG
  • GM: Metacentric height (m)
  • BM: Distance between the center of buoyancy and the metacenter (m)
  • BG: Distance between the center of buoyancy and the center of gravity (m)

For small heel of a floating body, BM = I_waterplane/V_displaced, where I_waterplane is the waterplane second moment about the heel axis. A signed expression is GM = KB + BM - KG. GM > 0 gives initial restoring stability, GM < 0 instability, and GM = 0 neutral first-order stability. Large-angle stability needs the full righting-arm curve. A completely submerged body has a different criterion: with fixed B and G, B above G gives restoring stability.

Worked example

Given: A wooden block with a volume of 0.5 m³ is floating in water. The density of the block is 600 kg/m³. Calculate the buoyant force.

1. Calculate the weight of the block.

   • Formula: W = ρ_b · V · g
   • ρ_b = 600 kg/m³, V = 0.5 m³, g = 9.81 m/s²
   • W = 600 · 0.5 · 9.81 = 2943 N

2. Since the block is floating, the buoyant force equals the weight of the block.

   • Buoyant Force, F_b = 2943 N. With water density 1000 kg/m³ the displaced volume is 0.3 m³, not the entire block volume. The geometry is not supplied, so metacentric stability cannot be calculated.

Common mistakes

• Confusing the density of the object with the density of the fluid.
• Forgetting that the buoyant force equals the weight of the displaced fluid, not the object.
• Miscalculating the metacentric height, leading to incorrect stability analysis.

For GATE ME

Questions often involve calculating the buoyant force, determining stability conditions, and applying Archimedes' Principle. Practice problems on floating bodies, submerged bodies, and stability analysis.

Quick check

1. What is the principle that determines the buoyant force on an object?
2. How does the metacentric height affect stability?
3. What happens to the buoyant force if the density of the fluid increases?

Answers: 1. Archimedes' Principle 2. Positive GM gives initial restoring stability; a larger positive GM increases initial restoring moment for the same weight and small angle. 3. It increases at fixed displaced volume. A freely floating body of unchanged weight instead displaces less volume and retains buoyant force equal to its weight.