Metacentric Height (Ship Stability)

Equivalent forms

BM = \frac{I}{V}

GM = KB + BM - KG

\text{restoring torque} = \rho g V \cdot GM \cdot \sin\theta

Stability of a thousand-tonne ship reduces to whether one height is positive.

Symbol	Name	SI unit	Dimension	Range
$I$	Waterplane Moment Second moment of area of the waterline plane about the roll axis	m⁴	L⁴	1 – 5000
$V$	Displaced Volume Submerged volume of the hull	m³	L³	1 – 5000
$\overline{BG}$	Centre Offset Height of centre of gravity above centre of buoyancy	m	L	-2 – 5
$GM$	Metacentric Heightoutput Height of metacentre above centre of gravity	m	L	-2 – 10

Unit systems

Symbol	SI	CGS	Imperial
$I$	m⁴	cm⁴	ft⁴
$V$	m³	cm³	ft³
$\overline{BG}$	m	cm	ft
$GM$	m	cm	ft

Where it holds

Small-angle (initial) stability for a freely floating rigid body in still water;

BM = I/V

holds for heel angles up

to \sim 10^{\circ}

before the waterplane shape changes nonlinearly and large-angle GZ curves take over.

Dimensional analysis

$[I/V] = L^{4}/L^{3} = L$ ; $[BG] = L$ ; $GM = L - L = L \checkmark$ (a length)

Discovery

Pierre Bouguer · 1746

In his 'Traité du navire' Bouguer introduced the metacentre to put ship stability on a rigorous footing, replacing shipwrights' rules of thumb. Euler developed parallel results; GM remains the single most quoted number in naval-architecture stability books.

Try this

Why does a top-heavy-looking ferry stay upright while a slim canoe flips? One length decides it.

A barge has waterplane second moment of area I = 200 m⁴ and displaces a submerged volume V = 500 m³, with its centre of gravity 0.3 m above the centre of buoyancy. Find the metacentric height GM.

Research status: stable

Real-world applications

Ship and ferry design and loading limits
Inclining-experiment certification
Offshore platform and pontoon stability
Iceberg and floating-body orientation

Common misconceptions

A wider beam is always safer — wider raises BM (good) but can pair with a high G; both matter
GM measures how far it can roll — it sets the stiffness/righting rate, not the ultimate range of stability (that is the GZ curve)
A very large GM is ideal — too large gives a brutally stiff, fast roll that is uncomfortable and can shift cargo

Experimental verification

The 'inclining experiment' — moving a known weight across the deck and measuring the resulting heel — directly yields GM and is a mandatory step in ship certification; results match the I/V formula within survey tolerance.

Derivation

When the hull heels by a small angle, the emerged and immersed wedges shift the centre of buoyancy.

Integrating that shift shows the metacentre rises a distance

BM = I/V

above the centre of buoyancy, where I is the waterplane second moment.

The metacentric height above G is

GM = BM - BG = I/V - BG

.

Limiting cases

GM > 0⟶ Stable, returns uprightMetacentre above G gives a restoring torque — the ship rights itself after a roll.

GM = 0

⟶ NeutralNo net righting moment; the vessel lolls at any small angle.

GM < 0⟶ CapsizeMetacentre below G makes any heel self-amplifying — the boat rolls over.

What if…

What if cargo is stacked higher, raising G?

BG grows, GM shrinks toward zero and then negative — the classic mechanism behind top-heavy capsizes.

What if the hull is made beamier?

Waterplane I scales with beam cubed, so BM and GM jump — why rafts and catamarans are so hard to tip.

1

GM of a barge

Given ·

I:: 200
V:: 500
\overline{BG}:: 0.3

Find · GM

Steps

Metacentric radius: $BM = I/V = 200/500 = 0.4\,\mathrm{m}$
$GM = BM - BG = 0.4 - 0.3$
$GM = 0.1\,\mathrm{m}$ — positive but marginal; small extra top weight would capsize it

Result ·

BM = I/V = 200/500 = 0.4\,\mathrm{m}

;

GM = BM - BG = 0.4 - 0.3 = 0.1\,\mathrm{m}

> 0, so the barge is (just) stable.

Archimedes' Principle→Hydrostatic Pressure→Pascal's Principle→