Metacentric height (GM) is the vertical distance between a floating body’s centre of gravity (G) and its metacentre (M). It is the single most important measure of a ship’s initial transverse (rolling) stability: when GM is positive, the vessel is stable and rights itself after a small tilt; when GM is negative, it capsizes. Numerically, the transverse metacentric height is found from GM = KM − KG, where both distances are measured from the keel (K).
What is the metacentre and where does GM come from?
When a ship floats upright, its weight acts downward through the centre of gravity (G) and the buoyancy force acts upward through the centre of buoyancy (B), the centroid of the displaced water volume. The two are in line. When the ship heels by a small angle, the underwater shape changes, so B shifts sideways to a new position. The vertical line of buoyancy through the new B intersects the original centreline at a point called the metacentre (M). For small angles of heel, M stays effectively fixed, which is why GM is a reliable stability constant.
The distances are built up from the keel:
- KB — height of the centre of buoyancy above the keel.
- BM — the metacentric radius, from B up to M.
- KM = KB + BM — height of the metacentre above the keel.
- KG — height of the centre of gravity above the keel.
Combining these gives the working definition: GM = KM − KG = (KB + BM) − KG.
What is the formula for metacentric height?
The metacentric radius BM is calculated from the geometry of the waterline plane:
BM = I / V
where:
- I = second moment of area (moment of inertia) of the waterplane about the longitudinal centreline axis — this axis governs transverse, i.e. side-to-side rolling, stability.
- V = volume of water displaced by the vessel.
Because I depends on the cube of the waterplane breadth, a wider hull produces a much larger BM and therefore greater transverse stability. The full transverse metacentric height is then:
GM = KB + (I / V) − KG
When is a floating body stable?
Stability depends entirely on whether the metacentre lies above or below the centre of gravity:
| Condition | Position of M and G | Behaviour |
|---|---|---|
| GM > 0 | M is above G | Stable equilibrium — a righting moment returns the body upright |
| GM < 0 | M is below G | Unstable — an overturning moment increases the heel; capsizes |
| GM = 0 | M coincides with G | Neutral equilibrium — the body stays in the tilted position |
A larger positive GM gives a stiff, quick-rolling vessel with a strong righting effect, while a small positive GM gives a tender ship that rolls slowly and comfortably but with less reserve stability. Naval architects therefore design GM within a balanced range rather than simply maximising it.
How is metacentric height measured experimentally?
GM is determined practically using the inclining experiment. A known weight is shifted a measured distance across the deck, causing the vessel to heel through a small angle, which is read from a plumb line or pendulum. The metacentric height follows from:
GM = (w × x) / (W × tanθ)
where:
- w = the movable (shifting) weight,
- x = horizontal distance the weight is moved,
- W = total weight (displacement) of the floating body including w,
- θ = resulting angle of heel.
In the laboratory, this principle is reproduced with a metacentric height apparatus: a floating pontoon or model ship fitted with a transverse sliding mass and an angle indicator. Students move the mass in known increments, record the tilt angle, and compute GM, comparing the measured value against the theoretical GM = KM − KG. The experiment makes the abstract idea of the metacentre directly observable and reinforces the relationship between weight distribution, geometry, and stability.
Why metacentric height matters
Understanding GM is fundamental wherever bodies float or must resist overturning:
- Ship and boat design — ensuring adequate righting stability and acceptable roll behaviour in service.
- Loading and cargo planning — positioning cargo and ballast to keep KG low and GM safely positive.
- Offshore and marine structures — pontoons, barges, floating docks, and platforms.
- Naval architecture and ocean engineering education — a core fluid-mechanics concept for buoyancy and flotation.
Uses and applications in the laboratory
The metacentric height experiment is a standard part of fluid mechanics and hydraulics courses in engineering colleges and universities. It is used to study buoyancy, flotation, equilibrium of floating bodies, and the influence of centre-of-gravity position on stability — preparing students for marine, civil, mechanical, and ocean-engineering applications.
Scientico India is an ISO 9001:2015 and CE certified manufacturer and exporter of engineering and science laboratory equipment, supplying institutions across 60+ countries since 1993. Explore our Fluid Mechanics Lab Equipment range, including metacentric height apparatus for teaching and research.
Frequently Asked Questions
What is metacentric height in simple terms?
Metacentric height (GM) is the vertical distance between a floating body’s centre of gravity (G) and its metacentre (M). It measures how stable a ship is: a positive GM means the vessel rights itself after tilting, while a negative GM means it will capsize.
What is the formula for metacentric height?
The transverse metacentric height is GM = KM – KG, where KM = KB + BM and BM = I/V. Here I is the second moment of area of the waterplane about the longitudinal centreline axis and V is the displaced volume. Experimentally, GM = (w x x) / (W x tan theta) from the inclining experiment.
When is a floating body stable based on metacentric height?
A floating body is in stable equilibrium when GM is positive (the metacentre M lies above the centre of gravity G), unstable when GM is negative (M below G), and in neutral equilibrium when GM equals zero (M coincides with G).
Why is metacentric height important for ships?
Metacentric height determines a ship’s initial transverse stability and roll behaviour. A larger GM gives a stiff, fast-rolling vessel with strong righting moments, while a small positive GM gives a tender, comfortable roll with less reserve stability, so it guides hull design, cargo loading, and ballasting.
How is metacentric height measured in the lab?
It is measured using the inclining experiment. A known weight w is moved a horizontal distance x across a floating model, the heel angle theta is read from an indicator, and GM is calculated as (w x x) / (W x tan theta), where W is the total displacement. Lab apparatus uses a pontoon with a sliding mass and angle scale.
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