Stability of floating bodies: If I is the second moment of area of the waterline plane about the longitudinal axis and V is the displaced volume, the height of the metacentre above the centre of buoyancy (BM) equals:

Introduction / Context:

Initial stability of floating vessels is assessed using the metacentric height GM. A key geometric quantity is BM, the distance between the metacentre M and the centre of buoyancy B, determined solely by the waterplane geometry and displaced volume.

Given Data / Assumptions:

• Small heel angles (initial stability).
• Homogeneous fluid with unit weight rho * g.
• Longitudinal axis consideration (roll stability).

Concept / Approach:

For small angles, the shift of the line of action of buoyancy produces a righting moment proportional to the waterplane second moment and inversely proportional to the displaced volume. The canonical relation is BM = I / V. Then overall metacentric height GM = KB + BM − KG, where KB and KG are geometric positions of B and the centre of gravity, respectively.

Step-by-Step Solution:

Verification / Alternative check:

Dimensional check: I has units of m^4, V has m^3, so BM has m—consistent for a length measure.

Why Other Options Are Wrong:

• V/I inverts the relationship.
• IV or I/(rhog) have incorrect dimensions.
• sqrt(I/V) has wrong functional form and dimensions.

Common Pitfalls:

• Confusing waterplane second moment of area I (m^4) with the hull mass moment of inertia (kg·m^2).

Final Answer:

BM = I / V