Question 65

The Correct Answer is A The correction to the height of the metacenter above the keel, $\text{KG}$, due to transverse free surface effects is commonly denoted as the **Free Surface Effect (FSE)** or **Free Surface Correction (FSC)**, which is subtracted from the Metacentric Height ($\text{GM}$) or added to $\text{KG}$ to find the corrected metacentric height $\text{GM}'$. This correction is calculated by the formula: $$\text{FSE} = \frac{\sum (i_t \rho_l)}{\Delta}$$ where: * $i_t$ is the transverse moment of inertia of the free surface area of a tank. * $\rho_l$ is the density of the liquid in the tank. * $\Delta$ is the vessel's displacement (weight). When using moments (units of weight $\times$ length$^2$), the formula simplifies to: $$\text{FSE} = \frac{\sum \text{Transverse Free Surface Moments}}{\text{Displacement}}$$ The term "Transverse Free Surface Moments" refers to the product $(i_t \rho_l)$, which represents the moment that affects the vessel's stability due to the liquid sloshing transversely. Therefore, the correction to $\text{KG}$ (or $\text{KM}$, or $\text{GM}$) is found by dividing the vessel's displacement ($\Delta$) into the **sum of the transverse free surface moments of the vessel** ($\sum (i_t \rho_l)$). *** **Explanation of Incorrect Options:** **B) transverse baseline of the vessel:** The baseline is a reference line (usually the bottom of the keel) and has no direct mathematical relationship to the calculation of the free surface correction. **C) sum of the vertical moments of the vessel:** Vertical moments ($\text{W} \times \text{KG}$) are used to calculate the vessel's center of gravity ($\text{G}$), but they are not used as the numerator in the free surface correction formula, which specifically requires moments related to the sloshing liquid's inertia. **D) transverse free surface correction for the vessel:** The quotient (the division result) *is* the free surface correction (FSC). Therefore, you cannot find the correction by dividing the displacement into the correction itself. The displacement is divided into the *moments* that cause the correction.