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What Is Metacentric Height?

Joe Williams
Joe Williams

Metacentric height is the distance between a floating body’s metacenter and its center of gravity. It is a measure of a floating body’s stability such that a ship with a large metacentric height is more stable than a ship with a smaller one. A ship that has a large metacentric height also has a shorter rolling, which is less desirable for passenger ships. A passenger ship should therefore have a metacentric height large enough to be stable, but not so large that it is uncomfortable for passengers.

A floating body such as a ship behaves like a pendulum when it rolls back and forth in the water. A ship has a natural frequency that determines the speed with which it rolls in the water. This frequency depends on the mass at the end of the pendulum’s swing arm such that a greater mass causes a slower roll rate. A ship’s swing arm is the imaginary line between the center of gravity and metacenter of the ship, so the metacentric height is the length of this swing arm.

A ship with cargo has a greater metacentric height than one without.
A ship with cargo has a greater metacentric height than one without.

The ship’s metacenter is equal to its inertia resistance divided by its displacement. The ship’s inertia resistance is a measure of the degree to which the ship resists overturning at its waterline. Its displacement is the volume of the ship that is below the water line. A ship that is wide and shallow has a high metacenter, as does a ship that is narrow and deep. Ships that are narrow and shallow or wide and deep have a low metacenter.

A ship that has a high metacentric height is difficult to overturn but rolls quickly. A ship with a low metacentric height overturns easily but rolls slowly. The ideal passenger ship must strike a balance between these two extremes in behavior. The addition of ballast in the hold of the ship lowers the ship's center of gravity and therefore increases its height. This means that a ship that is laden with cargo will be more stable in the water.

When a ship heels, or tilts to one side, its center of buoyancy moves to the side. The metacenter is the point at which a vertical line through the heeled ship’s center of buoyancy intersects with the vertical line through the upright ship’s center of buoyancy. It can be treated as a stationary point for ships that are only slightly heeled, and it must be calculated for ships that are greatly heeled.

Discussion Comments


It's kind of scary that there is a correlation between how unsafe a ship is and how comfortable it is for passengers. I'd rather be on a ship that was very safe but mildly uncomfortable.

I guess that's why immigrants from a hundred years ago always described the journey as being so difficult. They probably had a lot more sea sickness because the ship was built for safety, rather than for the comfort of the passengers.


@KoiwiGal - I think in some cases the captains would deliberately overload the ships to make them more likely to capsize. Either the ship would make the journey safely, in which case they get a pay day from that, or the ship would capsize, in which case the owner of the ship could still collect a lot of money in insurance.

I think it was concern over the fact that these captains were basically scamming the insurance companies that lead to the change in the law, rather than concern over the lives lost.

I imagine that ship design is much better these days as well, but it's still possible for ships to go down if the crew acts in an irresponsible way.


Apparently a long time ago it was standard to completely ignore these measurements and just load ships up as much as possible, in order to make the most money. They would go to sea completely unbalanced and would often capsize.

Eventually they made it a law that ships had to be floating above a certain line on the hull of the ship or they wouldn't be allowed to leave port. I imagine the line was worked out with regard to the metacentric height.

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    • A ship with cargo has a greater metacentric height than one without.
      By: benjaminlion
      A ship with cargo has a greater metacentric height than one without.