M092 · Fluid mechanics

Buoyancy & Metacentric Stability

Can a floating hull still overturn? Yes. Floating only needs the weight to equal the buoyancy. Staying upright needs more: as the hull heels, buoyancy and weight must form a torque that pushes it back. Whether they do depends on the shape of the underwater volume and how high the ballast sits, and a hull that rights itself at small angles can have that torque reverse at large ones.

Move a ballast mass inside a sealed box hull or a flared prism hull. Compare the exact displaced geometry, the restoring curve at finite heel and the initial stability. Then infer the initial metacentric height from small-heel moments you record.

Total mass $\text{Ready}$

Displaced volume $\text{Ready}$

Weight $\text{Ready}$

Buoyant force $\text{Ready}$

Signed righting arm $\text{Ready}$

Signed restoring moment $\text{Ready}$

Initial metacentric height $\text{Ready}$

Mass centre transverse position $\text{Ready}$

Mass over fully submerged displacement $\text{Ready}$

Initial height from raw moments $\text{Ready}$

Small-heel fit residual $\text{Ready}$

Recorded stations $\text{Ready}$

Force or volume balance residual $\text{Ready}$

Model state $\text{Ready}$

Floating stability and shallow-water flow

Where should the ballast go?

Hull deck breadth

B

4.00\,m

Hull depth

D

2.00\,m

Prism length

L

8.00\,m

Keel-to-deck breadth ratio

f

1.00\,

Mean sealed hull density

\rho_H

350.00\,kg/m^3

Liquid density

\rho

1000.00\,kg/m^3

Ballast mass

m_b

2500.00\,kg

Ballast transverse fraction

\xi_b

0.00\,

Ballast height fraction

\zeta_b

0.25\,

Clockwise heel

\theta

12.00\,{}^\circ

Moment perturbation gain

\epsilon

0.00\,

The hull is a sealed prism of uniform density, with a movable point ballast inside its cross-section. Sideways position is positive to the right, height is measured up from the keel, and positive heel is clockwise. At each heel angle you set, the Lab solves for the floating depth; it does not simulate rolling. Orange marks the centre of mass and blue marks the centre of the displaced volume. The blue submerged mesh shows the displaced volume, not water inside the sealed hull.

The upright metacentric height describes stability only for small tilts from upright. The full curve comes from cutting the hull section at the waterline, and it can change sign at large heel. Off-centre ballast creates a moment even when the hull is upright; a positive initial height guarantees neither upright equilibrium nor safety at larger angles. A hull that cannot displace enough water shows a fully submerged candidate, and acquisition is blocked.

An acquisition sets seventeen heel angles within four degrees. A cubic fit to the raw moments, with an offset term, estimates the initial slope; the mass and gravity are known separately. Because the slope comes from finite angles, it carries a truncation error that depends on the hull shape. The perturbations are repeatable and bounded by the selected gain in newton metres. No confidence interval or external calibration is claimed.

The hull is a watertight prism in still water. The heel angle is imposed, and the hull sits at the depth where its weight and buoyancy balance. Rolling motion, flooding and trim are not modeled, and nothing here certifies ship safety.

Start with the essentials

Focus question: Does floating mean stable?
One-sentence intuition: Compare the full geometric model, its small-angle approximation and the separately recorded data.

Core mathematical model

Vertical force balance

\rho g\nabla=mg

To float in equilibrium, the hull must displace enough water. A sealed hull that is overloaded cannot reach this balance.

Centre of the displaced volume

\begin{aligned}\nabla&=LA_s\\\boldsymbol r_B&=\frac{1}{A_s}\int_{A_s}\boldsymbol r\,dA\end{aligned}

Cut the tilted cross-section at the waterline, then integrate the submerged area and its centroid.

Restoring moment at any heel angle

\begin{aligned}GZ&=x_B-x_G\\M_R&=mg\,GZ\end{aligned}

For a floating hull, a positive moment resists a clockwise heel. A negative arm makes the heel worse.

Initial metacentric height

\begin{aligned}BM&=\frac{I_{\mathrm{wp}}}{\nabla}\\GM&=KB+BM-KG\end{aligned}

The second moment of the waterplane is taken about its centroid. This height describes only small angles and does not replace the curve for finite heel.

Initial slope measured from raw data

\begin{aligned}\frac{M_i}{mg}&\simeq c+a\theta_i+b\theta_i^2+d\theta_i^3\\\widehat{GM}&=a\end{aligned}

Angles are in radians. Fit a cubic with a free offset over four degrees, using the raw moments and the separately known mass.

Common difficulties

A reference curve is not a measurement

Typical misconceptionThe reference curve on screen is a measured result.

Better mental modelOnly the sensor channels you record count as observations, and changing the setup clears them. The perturbations are deterministic, so they give no statistical certification.

Run the experiment

01
Move weight down
Compare Low ballast with High ballast, unstable.
What to observe: Both hulls float, but near upright their restoring curves slope in opposite directions.
02
Look beyond the initial slope
Drag the gold heel selector out to large angles.
What to observe: The curve from the full geometry moves away from the small-angle sine approximation.
03
Move the ballast sideways
Select Off-centre ballast and look at zero heel.
What to observe: The initial stability can be positive while the upright hull still feels a nonzero moment.
04
Measure first, then infer
Acquire the small-heel sweep, add perturbations and export the CSV.
What to observe: The slope estimate comes from the raw moments. Clear them and the estimate disappears.