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).
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Use FSC = i/V with the tank's breadth cubed in the moment of inertia, since the tank liquid matches the seawater density.
Free surface effect happens because a partly filled tank lets liquid slosh sideways whenever the ship heels, and that shifting liquid behaves like a virtual rise of the vessel's centre of gravity instead of adding real stability. The size of this virtual rise depends only on the tank's own free surface shape and the ship's total displacement volume, not on how deep the tank is or how full it is.
From the figure, the tank's free surface is a rectangle of fore-aft length 6 m and athwartship breadth 10 m. For a rectangular free surface, the relevant second moment of area about the centreline is \( i = l b^3 / 12 \), which uses the breadth raised to the third power because that is the direction the liquid actually swings across. Substituting the values gives \( i = 6 \times 1000 / 12 = 500 \ \text{m}^4 \).
The general free surface correction is \( FSC = (\rho_{liquid}/\rho_{sw}) \times i/V \). Since the tank holds seawater and the vessel also floats in seawater, the density ratio cancels to 1, leaving \( FSC = i/V = 500/5000 \). This works out to 0.1 m, so the metacentric height is effectively reduced by 0.1 m by this free surface.
