Metacentric height from transverse load shift on a ship A load of magnitude w is moved transversely by a distance d across the deck of a ship of displacement W, causing it to heel by an angle θ (small). Determine the metacentric height (GM) from these quantities.

Introduction / Context:Transverse stability of ships is characterized by the metacentric height GM. A convenient onboard test uses a known weight shift across the deck to induce a small heel angle; from the observed angle, GM can be computed.

Given Data / Assumptions:

• Ship displacement (weight) = W.
• Movable load = w shifted transversely by distance d.
• Heel angle θ is small (tan θ ≈ θ in radians), ensuring metacentric theory validity.

Concept / Approach:Equate the heeling moment produced by the load shift to the righting moment provided by GM. Heeling moment = w * d. Righting moment for small angles = W * GM * tan θ. Setting them equal yields GM = (w * d) / (W * tan θ).

Step-by-Step Solution:

Verification / Alternative check:Dimensional check: GM has dimensions of length; (w/W) is dimensionless; d/ tan θ has length → consistent.

Why Other Options Are Wrong:Options (a), (c), and (d) invert factors improperly, giving wrong dimensions or scaling.

Common Pitfalls:Using degrees directly inside tan without conversion when computing numerically; applying the formula at large heel angles where metacentric theory is invalid.

Final Answer:GM = (w * d) / (W * tan θ)