When an empty tank below the waterline is bilged (flooded), the water that enters is at the same level as the sea and, being below the waterline, the added weight of water is exactly balanced by the added buoyancy of the extra volume displaced (the ship sinks slightly). The net effect is that the centre of gravity of the added water is low (in the tank, near the bottom of the ship), while the added buoyancy acts at the centre of buoyancy of the flooded volume, which is also low. Because the added weight is low, the overall centre of gravity KG is lowered, and because the added buoyancy is low, the centre of buoyancy KB is also lowered but by less than the weight effect. The result is that GM increases (the ship becomes stiffer). Additionally, once the tank is full there is no free surface, so there is no free-surface loss of GM. Hence bilging an empty low tank increases GM and improves stability (though it increases displacement and draft).

Ship displacement 10,000 t, GM = 0.5 m, period of roll in still water T = 20 s. A mass of 50 t is discharged from a position 14 m above the centre of gravity.

Radius of gyration: T = 2 pi k / sqrt(g GM), so k = T sqrt(g GM)/(2 pi) = 20 x sqrt(9.81 x 0.5)/(2 pi) = 20 x 2.2146/6.283 = 7.05 m.

Discharging a weight above the CG lowers the CG: the CG falls by GG = w x h/(Delta - w) = 50 x 14/(10000 - 50) = 700/9950 = 0.0704 m. So the new GM = 0.5 + 0.0704 = 0.5704 m (KM unchanged).

New period T' = 2 pi k / sqrt(g GM') = 6.283 x 7.05 / sqrt(9.81 x 0.5704) = 44.28 / sqrt(5.595) = 44.28/2.365 = 18.72 s.

Answer: the new period of roll is about 18.7 s (the ship rolls faster because GM increased).