How to Calculate Metacentric Height

If you hear people tossing around terms like "metacentric height," you're either taking a class in naval architecture, or you're preparing for a U.S. Coast Guard licensing examination. Determining your vessel's metacentric height requires that you calculate the exact location of the center of gravity and the center of buoyancy, and their height above your boat's keel. Knowing the metacentric height of your vessel tells you how to distribute your ballast water and fuel at the fuel dock to keep your boat from leaning to one side.

Things You'll Need

• Spreadsheet

Instructions

1. Multiply the weight of every part of the vessel, including structural members, components and fittings, by its distance from the beam (the widest part) of the vessel. Use positive numbers for weight in the forward part of the vessel, negative numbers for weight at the rear.Total the individual results. Divide by the total weight of the vessel. A negative result is the distance that "G" lies behind the beam; a positive result is distance ahead of the beam. For example, on an 8-ton vessel, if a winch 12 feet ahead of the beam weighs 300 pounds, then 12 x 300 = 3,600. If a winch 3 feet behind the beam weights 220 pounds, then -220 x 3 = -60. Total these two (3,600 + -220, or 3,600 - 220 = 3,380) and divide the result, 3,380 by 16,000. Multiply the result, 0.2115 by 12 to determine the distance from the beam in inches: 2.535 inches.

2. Multiply the weight of every part of the vessel, including structural members, components and fittings, by its distance from the vertical center of the hull. Use negative numbers for weights above the vertical center and positive numbers for weights below the vertical center. Total the individual results. Divide by the total weight of the vessel. The result is the distance that the center of gravity lies above the keel. If the 300-pound anchor chain is 8 feet below the vertical center, then: 8 x 300 = 1,200. If a 50-pound portable fuel tank is 1 foot above the vertical center, then: -50 x 1 = -50. Add the results (1,150) and divide by 16,000: 1,150 / 16,000 = 0.718. Multiply the result by 12 for the distance above or below the vertical center of the hull: 0.718 x 12 = 8.625 inches

3. Make a simple scale drawing of the vessel from the side. Make a vertical line on the drawing that extends from the top of the vessel to the bottom of the vessel, to mark the location ahead of, or behind the beam to show the line on which the center of gravity lies (called the transverse metacenter). In the above example, the center of gravity would be 2.535 inches ahead of the beam and 8.625 inches below the vertical center of the hull. Measure the distance from the keel to the center of gravity. If the vertical center of the vessel in this example is located 10 feet above the keel, then the center of gravity would be 9.28 feet above the keel.

4. Extract your vessel's displacement tonnage from its registration or documentation. Multiply the length of your vessel by its width. Divide the displacement tonnage in pounds by the result. If your vessel is 25 feet long and 8 feet wide, then: 25 x 8=200. If the displacement tonnage is 8 tons (16,000 pounds) then: 16,000 / 200 = 8 feet. Since the center of buoyancy is at the center of mass of the water displaced, divide by 2 (one-half of the height of the water displaced) This means that your vessel's center of buoyancy ("B") lies 4 feet (48 inches) above the keel.

5. Subtract the height of the center of buoyancy (B) from the height of the center of gravity (G) to calculate the metacentric height (GM): 9.28 - 8.00 = 1.28 feet

Tips & Warnings

• Doing this work on a calculator can become confusing. A spreadsheet, like Excel, simplifies the process.

References

• "Merchant Ship Stability;" H. J. Purvey; Brown, Son and Ferguson, 1983; p. 48ff; p. 77ff