M092 · Fluid mechanics

Buoyancy & Metacentric Stability

Move a ballast mass through a sealed box or flared prism. Compare exact displaced geometry, finite-heel restoring curves and initial stability; infer the initial metacentric height from acquired small-heel moments.

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-to-displacement capacity ratio $\text{Ready}$

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

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

Recorded stations $\text{Ready}$

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

Model state $\text{Ready}$

GEOMETRY / CONSERVATION / ACQUISITION

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 uniform-density prism; the movable point ballast lies inside its cross-section. Transverse position is positive to the right, height upward from the keel, and positive heel clockwise. Heave is solved at each imposed heel; this is not a roll simulation. Orange marks the mass centre; blue the displaced-volume centre. The blue immersed mesh marks displaced volume, not liquid inside the sealed hull.

Upright metacentric height describes local initial stability. The full curve comes from waterline-clipped polygons and may reverse sign at large heel. A displaced ballast creates a nonzero upright moment; positive initial height does not imply upright equilibrium or finite-angle safety. A hull without displacement capacity shows a fully immersed candidate and blocks acquisition.

Acquire seventeen imposed angles within four degrees. A cubic raw-moment fit estimates the initial slope and includes an offset; mass and gravity are independently known. This is a finite-angle slope estimate, with geometry-dependent truncation error. Perturbations are deterministic, bounded by the selected gain in newton metres; no confidence interval or external calibration is claimed.

Hydrostatic, watertight prismatic hull with imposed heel and heave equilibrium; no roll dynamics, flooding, trim or ship-safety certification.

Start with the essentials

Focus question: Does floating mean stable?
One-sentence intuition: Compare the full physical model, its approximation and the separately acquired record.

Core mathematical model

Vertical force balance

\rho g\nabla=mg

A floating equilibrium requires enough displaced volume. A sealed overloaded hull cannot satisfy this balance.

Displaced centre

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

Clip the rotated cross-section at the waterline; integrate its area and centroid.

Finite-heel restoring moment

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

For a floating hull, positive moment opposes clockwise heel. Negative arm assists it.

Initial metacentric height

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

The waterplane second moment is about its centroid. This local height does not replace a finite-heel curve.

Initial slope and raw inference

\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 radians. Fit a free offset and cubic over four degrees, using raw moments and independently known mass.

Common difficulties

Reference and acquisition

Typical misconceptionA displayed reference is an observed measurement.

Better mental modelRecord sensor channels explicitly. Changing the configuration clears observations; no statistical certification follows from deterministic perturbations.

Run the experiment

01
Move weight down
Compare Low ballast with High ballast, unstable.
What to observe: Both can float; their local restoring slopes differ in sign.
02
Look beyond the initial slope
Drag the gold heel selector through large angles.
What to observe: The geometry curve and local sine approximation diverge.
03
Move the ballast sideways
Select Off-centre ballast and inspect zero heel.
What to observe: Positive initial stability can coexist with a nonzero upright moment.
04
Measure before inferring
Acquire the small-heel sweep, add perturbations and export CSV.
What to observe: The slope estimate uses raw moments; clearing them removes the estimate.