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4.3 Metacenter and Metacentric Height

3 min readLast Updated on July 19, 2024

Metacentric height is crucial for understanding the stability of floating objects. It's the distance between an object's center of gravity and its metacenter, which determines whether it'll right itself or capsize when tilted.

Calculating metacentric height involves factors like shape, weight distribution, and displaced fluid volume. A positive value means stability, while negative spells trouble. This concept is vital for designing and operating ships, boats, and other floating structures safely.

Metacenter and Metacentric Height

Metacenter concept and significance

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  • Metacenter represents the point where the vertical line through the center of buoyancy intersects the line of action of the buoyant force when a floating body is slightly tilted (ships, boats)
    • Acts as the pivot point about which the body rotates when disturbed from its equilibrium position
  • Relative position of the metacenter and center of gravity determines the stability of a floating body
    • Metacenter above the center of gravity indicates a stable body that will return to its original position when disturbed (sailboats)
    • Metacenter below the center of gravity results in an unstable body that will capsize when disturbed (capsized canoes)

Calculation of metacentric height

  • Metacentric height (GMGM) measures the distance between the center of gravity (GG) and the metacenter (MM)
    • Calculated using the formula: GM=BMBGGM = BM - BG
      • BMBM represents the distance between the center of buoyancy (BB) and the metacenter (MM)
      • BGBG represents the distance between the center of buoyancy (BB) and the center of gravity (GG)
  • Rectangular floating objects calculate BMBM using the formula: BM=IVBM = \frac{I}{V}
    • II represents the moment of inertia of the waterplane area about the axis of rotation (rectangular barges)
    • VV represents the volume of the displaced fluid
  • Cylindrical floating objects calculate BMBM using the formula: BM=πr44VBM = \frac{\pi r^4}{4V}
    • rr represents the radius of the cylinder (floating oil drums)
    • VV represents the volume of the displaced fluid

Metacentric height vs stability relationship

  • Metacentric height (GMGM) serves as a measure of the initial stability of a floating body
    • Positive GMGM indicates stable equilibrium where the body returns to its original position when disturbed (floating docks)
    • Negative GMGM indicates unstable equilibrium where the body capsizes when disturbed (top-heavy boats)
  • Magnitude of GMGM determines the degree of stability
    • Larger GMGM results in greater stability and a stronger restoring force when the body is tilted (wide, shallow-draft vessels)
    • Smaller GMGM results in less stability and a weaker restoring force when the body is tilted (narrow, deep-draft vessels)

Problem-solving with metacentric height

  1. Calculate the metacentric height (GMGM) using the appropriate formulas for the shape of the object to determine the stability of a floating body
    • Positive GMGM indicates a stable body
    • Negative GMGM indicates an unstable body
  2. Consider factors that can affect the stability of a floating body when solving problems:
    • Changes in weight distribution altering the location of the center of gravity (GG) (loading cargo)
    • Changes in the shape of the waterplane area affecting the moment of inertia (II) and the location of the metacenter (MM) (damaged hull)
    • Changes in the volume of displaced fluid affecting the location of the center of buoyancy (BB) (flooding compartments)
  3. Use the calculated GMGM value to determine the angle of heel at which the body will capsize, if applicable
    • Angle of vanishing stability represents the angle of heel at which the body will capsize
    • Calculated using the formula: θmax=tan1(GMBG)\theta_{max} = \tan^{-1}(\frac{GM}{BG}) (maximum safe heel angle)
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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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