A vessel of 5000 m\(^3\) displacement floating in seawater has a rectangular tank of length 6 m and width 10 m. The tank is half-filled with seawater as shown in the figure. The virtual reduction in metacentric height due to free surface effect is ______ m (answer in one decimal place).
Show Hint
Use FSC = i/V with the tank's breadth cubed in the moment of inertia, since the tank liquid matches the seawater density.
Step 1: Set up the free surface correction formula.
A slack tank with a free liquid surface causes a virtual rise in the centre of gravity, which is treated as a loss in metacentric height given by \( FSC = \dfrac{\rho_{liquid}}{\rho_{sw}} \times \dfrac{i}{V} \).
Here \(i\) is the second moment of area of the tank free surface about its own fore-and-aft centreline axis, and \(V\) is the vessel displacement volume.
Step 2: Read the tank dimensions from the figure.
The profile view shows the fore-and-aft length of the free surface as \(l = 6\) m, and the sectional view shows the athwartship breadth as \(b = 10\) m.
The free surface tilts about the fore-and-aft axis, so the moment of inertia uses the breadth cubed: \( i = \dfrac{l \, b^3}{12} = \dfrac{6 \times 10^3}{12} = 500 \ \text{m}^4 \).
Step 3: Apply the density ratio.
The tank is filled with seawater, the same liquid the vessel floats in, so \( \rho_{liquid} = \rho_{sw} \) and the density ratio is exactly 1.
Final Answer:
Divide the free surface inertia by the displacement volume.
\[ FSC = \frac{500}{5000} = 0.1 \ \text{m} \]
\[ \boxed{FSC = 0.1 \ \text{m}} \]
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