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Stability of a Ship with Metacentre
https://propertiesoffluids.wordpress.com/2015/05/04/stability-of-immersed-and-floating-bodies/
The point G on the diagram is the centre of mass of the ship where the weight W acts and point B is the centre of buoyancy where the buoyant force acts on the body. When a ship encounters waves, it may tilt to one side. However, this is not something serious as the ship will have a restoring moment or overturning moment.
When the ship tilts, the centre of buoyancy shifts. By drawing normal lines from centre of gravity and the new position of Buoyancy, an intersection point can be found. This point is labelled as M and is called the metacentre.
By looking at the diagram, it can be seen that when the metacentre is above the centre of gravity, the ship is stable and there is a restoring moment which stabilizes the ship. When the metacentre is below the centre of mass, there is an overturning moment and which makes the ship unstable.
We can determine whether if a ship is stable and unstable mathematically. But first, we need to identify the metacentric height. Metacentric height is the difference of the positions of M and G.
If MG is positive, then the ship is stable and if its negative, its unstable. Metacentric height can be found by the formula below:
Where GM is metacentric height, Io is the second moment of area, V is the submerged volume of the body and BG is the difference of the positions of centre of buoyancy and centre of gravity.
It should be noted that the second moment of inertia depends on the shape of the immersed body.
The second moment of inertia for different shapes can be found below from the table:
Second Moment of Inertia for Different Shapes
Why is Stability Important? Real Life Example!
The Vasa ship sitting in a museum after recovery (By Javier Kohen [CC BY-SA 3.0], from Wikimedia Commons)
The Vasa was a warship built in the 17th century by the order of a Swedish king. It was built to have two decks in order to one-up Danish ships at the time. Being essentially a showpiece of Swedish wealth and engineering, it was filled with expensive art and statues, which the ship was not prepared to carry. The ship's centre of gravity was higher than of the centre of buoyancy, and after a slight breeze, the ship listed to the port side and there was no restoring moment to keep the ship from toppling over. Shortly after, Vasa started to take water from its gun ports and sank to the bottom of the sea, and with it the king’s ego.
Figure 02: The effect of centre of gravity on stability (Source: Wista Tutor).
References:https://www.simscale.com/blog/2017/12/vasa-ship-sank/
https://www.eng.uwo.ca/people/esavory/STABILITY%20OF%20FLOATING%20BODIES.pdf
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